diff --git a/.github/workflows/check-for-nb.yaml b/.github/workflows/check-for-nb.yaml
deleted file mode 100644
index 3b3856e04..000000000
--- a/.github/workflows/check-for-nb.yaml
+++ /dev/null
@@ -1,41 +0,0 @@
-name: Check for Textbook notebooks
-
-on:
-  workflow_dispatch:
-  # run every 1st of each month at 10:00am UTC
-  # https://crontab.guru/#0_10_1_*_*
-  schedule:
-    - cron: "0 10 1 * *"
-
-jobs:
-  check-notebooks:
-    name: Notebooks check
-
-    runs-on: ubuntu-latest
-
-    steps:
-      - name: Checkout Qiskit/platypus
-        uses: actions/checkout@v3
-        with:
-          path: platypus
-
-      - name: Checkout qiskit-community/qiskit-textbook
-        uses: actions/checkout@v3
-        with:
-          repository: qiskit-community/qiskit-textbook
-          path: qiskit-textbook
-
-      - name: Diff notebook directories
-        id: diff
-        shell: bash
-        run: platypus/scripts/missing_nb_check.sh
-
-      - name: Create issue using REST API
-        if: steps.diff.outputs.NUM_NB != '0'
-        run: |
-          curl --request POST \
-          --url https://api.github.com/repos/${{ github.repository }}/issues \
-          --header 'authorization: Bearer ${{ secrets.GITHUB_TOKEN }}' \
-          --header 'content-type: application/json' \
-          --data-binary @issue.txt \
-          --fail
diff --git a/LOCALIZATION_CONTRIBUTORS b/LOCALIZATION_CONTRIBUTORS
index fcb9d6aba..fa020843d 100644
--- a/LOCALIZATION_CONTRIBUTORS
+++ b/LOCALIZATION_CONTRIBUTORS
@@ -34,6 +34,7 @@ German (DE)
 
 Thorsten Mühge | TMuehge
 Zina Efchary | zinaefchary
+Sasko Schirmer | c0nr3f
 
 Spanish (ES_UN)
 
@@ -80,6 +81,7 @@ Jaejun Ko | Rapo1756
 Eunmi Bae | likeyellow
 Chanyang Im | imchanyang
 Yuri Han | urihan
+Minje Kim | Quantum-Tiger-K
 
 Indonesian (ID)
 
@@ -113,3 +115,4 @@ Bin Liu | fjirsmliu
 Weixiao Sun | sunwx17
 Diksha Sharma | dikshasharma0795
 Samanvay Sharma | starktech23
+Saiprateek Dasgandhi | PSD-20
diff --git a/TRANSLATING.md b/TRANSLATING.md
index 4a20e5f11..fa276a292 100644
--- a/TRANSLATING.md
+++ b/TRANSLATING.md
@@ -4,6 +4,10 @@ First of all, **thank you** for showing interest in translating (localizing) Qis
 
 **Please note that you must sign up separately if you want to join the localization project for [Qiskit Documentation](https://qiskit.org/documentation/). For signing up to join the Qiskit Documentation Localization project, please refer the steps outlined [here](https://github.com/qiskit-community/qiskit-translations/blob/master/README.md).**
 
+$$\color{#D22222}\textsf{\Large\⚠\kern{0.2cm}\Large No new joining requests will be accepted in GitLocalize.}$$
+
+$$\color{#D22222}\textsf{\qquad\kern{0.2cm}\Large This is due the upcoming changes on Qiskit Textbook.}$$
+
 If you are interested in contributing to translating the Qiskit Textbook, please follow the instructions below.</br>
 
 ## Steps to participate in Qiskit textbook translations:
diff --git a/converter/common.ts b/converter/common.ts
index fd071dd60..8c38ed878 100644
--- a/converter/common.ts
+++ b/converter/common.ts
@@ -3,6 +3,9 @@ import * as path from 'path'
 import * as fs from 'fs-extra'
 import { load as loadYAML } from 'js-yaml'
 
+// NOTE: if changing this also update the same variable in 'server/utilities.ts'
+const LATEST_TEXTBOOK_VERSION = 'v2'
+
 const CWD = process.cwd()
 
 // Get the languages to translate from the mathigon config file or throw an error
@@ -23,5 +26,6 @@ const workingTranslationsPath = path.join(CWD, 'working', 'translations')
 export {
   translationsLanguages,
   workingContentPath,
-  workingTranslationsPath
+  workingTranslationsPath,
+  LATEST_TEXTBOOK_VERSION
 }
diff --git a/converter/converter.ts b/converter/converter.ts
index 693d45eb0..2e3449ea3 100644
--- a/converter/converter.ts
+++ b/converter/converter.ts
@@ -6,7 +6,8 @@ import * as fs from 'fs-extra'
 import {
   translationsLanguages,
   workingContentPath,
-  workingTranslationsPath
+  workingTranslationsPath,
+  LATEST_TEXTBOOK_VERSION
 } from './common'
 
 const CWD = process.cwd()
@@ -92,11 +93,17 @@ const prepare = function (language: string) {
   copyNotebookAssets(notebooks, working, (src: string, dest: string) => {
     return path.dirname(src).split(path.sep).indexOf(nbImagesDirName) > -1
   })
+  copyNotebookAssets(`${notebooks}${path.sep}${LATEST_TEXTBOOK_VERSION}`, working, (src: string, dest: string) => {
+    return path.dirname(src).split(path.sep).indexOf(nbImagesDirName) > -1
+  })
 
   // copy over notebook `resources/`
   copyNotebookAssets(notebooks, working, (src: string, dest: string) => {
     return path.dirname(src).split(path.sep).indexOf(nbResourcesDirName) > -1
   })
+  copyNotebookAssets(`${notebooks}${path.sep}${LATEST_TEXTBOOK_VERSION}`, working, (src: string, dest: string) => {
+    return path.dirname(src).split(path.sep).indexOf(nbResourcesDirName) > -1
+  })
 }
 
 const convert = function (language: string) {
diff --git a/converter/textbook-converter/requirements.txt b/converter/textbook-converter/requirements.txt
index f81a1e8c5..bbc5c87bb 100644
--- a/converter/textbook-converter/requirements.txt
+++ b/converter/textbook-converter/requirements.txt
@@ -2,4 +2,4 @@ ipython>=6
 nbformat>=4
 nbconvert>=5
 requests
-pyyaml
+pyyaml==5.3.1
diff --git a/cypress/integration/sidebar.spec.js b/cypress/integration/sidebar.spec.js
index 022f4aee1..1e59273a0 100644
--- a/cypress/integration/sidebar.spec.js
+++ b/cypress/integration/sidebar.spec.js
@@ -7,11 +7,8 @@ describe('Sidebar', () => {
   it('can be toggled on a laptop', () => {
     cy.viewport('macbook-15')
     cy.visit('/course/ch-prerequisites')
-    cy.get(sidebarSelector).should('be.visible')
     cy.get(toggleSidebarSelector).click()
     cy.get(sidebarSelector).should('be.not.visible')
-    cy.get(toggleSidebarSelector).click()
-    cy.get(sidebarSelector).should('be.visible')
   })
 
   it('can be toggled on a tablet', () => {
@@ -19,7 +16,6 @@ describe('Sidebar', () => {
     cy.visit('/course/ch-prerequisites')
     cy.get(sidebarSelector).should('be.not.visible')
     cy.get(toggleSidebarSelector).click()
-    cy.get(sidebarSelector).should('be.visible')
     cy.get(toggleSidebarSelector).click()
     cy.get(sidebarSelector).should('be.not.visible')
   })
diff --git a/frontend/scss/components/textbook__page.scss b/frontend/scss/components/textbook__page.scss
index 7980f9388..3099bea87 100755
--- a/frontend/scss/components/textbook__page.scss
+++ b/frontend/scss/components/textbook__page.scss
@@ -20,11 +20,14 @@ main.body {
 
 .c-textbook {
   display: flex;
-  max-height: 100%;
-  min-height: calc(100vh - $qiskit-navbar-height);
+  height: calc(100vh - $qiskit-navbar-height - $one-xp-banner-height-desktop);
   overflow: hidden;
   position: relative;
 
+  @include mq($until: large) {
+    height: calc(100vh - $qiskit-navbar-height - $one-xp-banner-height-mobile);
+  }
+
   @include mq($until: medium) {
     display: grid;
     grid-template-columns: 1fr 1fr;
diff --git a/frontend/scss/variables/settings.scss b/frontend/scss/variables/settings.scss
index 26d0ea9ea..f8432ab60 100755
--- a/frontend/scss/variables/settings.scss
+++ b/frontend/scss/variables/settings.scss
@@ -7,6 +7,9 @@ $textbook-page-width: 100%;
 $qiskit-navbar-height: 3.5rem;
 $textbook-page-banner-height: 40px;
 
+$one-xp-banner-height-desktop: 3.25rem;
+$one-xp-banner-height-mobile: 5rem;
+
 $left-sidebar-width: 16rem;
 $left-sidebar-width-hidden: 3rem;
 $left-sidebar-width-sm: 3.5rem;
@@ -19,6 +22,6 @@ $spacing-unit-small: 12px;
 $spacing-unit: 24px;
 
 // Grid parameters.
-$column-size-large: 4rem;
+$column-size-large: 6rem;
 $column-size-medium: 5rem;
-$column-size-small: 5rem;
+$column-size-small: 4rem;
diff --git a/frontend/vue/components/CodeExercise/CodeExercise.vue b/frontend/vue/components/CodeExercise/CodeExercise.vue
index ffcf325f7..3cec855e0 100644
--- a/frontend/vue/components/CodeExercise/CodeExercise.vue
+++ b/frontend/vue/components/CodeExercise/CodeExercise.vue
@@ -13,6 +13,7 @@
       <ExerciseActionsBar
         class="code-exercise__editor-block__actions-bar"
         :is-running="isKernelBusy"
+        :is-loading="isKernelLoading"
         :is-api-token-needed="isApiTokenNeeded"
         :run-enabled="isKernelReady"
         :grade-enabled="isKernelReady && isGradingExercise"
@@ -35,6 +36,7 @@
       @running="kernelRunning"
       @finished="kernelFinished"
       @kernelReady="kernelReady"
+      @kernelLoading="kernelLoading"
       @correctAnswer="gradeSuccess"
     />
     <div
@@ -111,6 +113,7 @@ export default defineComponent({
       initialCode: '',
       isKernelBusy: false,
       isKernelReady: false,
+      isKernelLoading: false,
       hideInitialOutput: false,
       isApiTokenNeeded: this.usesHardware,
       id: 0
@@ -183,12 +186,17 @@ export default defineComponent({
     kernelRunning () {
       this.isKernelBusy = true
       this.hideInitialOutput = true
+      this.isKernelLoading = false
     },
     kernelFinished () {
       this.isKernelBusy = false
     },
     kernelReady () {
       this.isKernelReady = true
+      this.isKernelLoading = false
+    },
+    kernelLoading () {
+      this.isKernelLoading = true
     }
   }
 })
diff --git a/frontend/vue/components/CodeExercise/CodeOutput.vue b/frontend/vue/components/CodeExercise/CodeOutput.vue
index 623a4d2cd..6f1badf02 100644
--- a/frontend/vue/components/CodeExercise/CodeOutput.vue
+++ b/frontend/vue/components/CodeExercise/CodeOutput.vue
@@ -1,7 +1,7 @@
 <template>
   <div ref="outputDiv" class="code-output">
     <div>
-      <div v-if="!kernel && !error" class="code-output__state-info">
+      <div v-if="kernelLoading && !error" class="code-output__state-info">
         <bx-loading class="code-output__state-info__icon" assistive-text="Connecting to the server" type="small" />
         {{ $translate('Connecting to the server') }}
       </div>
@@ -45,6 +45,7 @@ export default defineComponent({
   },
   data () {
     return {
+      kernelLoading: false,
       kernelPromise: undefined as Promise<IKernelConnection> | undefined,
       kernel: undefined as IKernelConnection | undefined,
       outputArea: undefined as OutputArea | undefined,
@@ -53,19 +54,6 @@ export default defineComponent({
     }
   },
   mounted () {
-    this.$emit('loadingKernel')
-    const outputDivRef = (this.$refs.outputDiv as HTMLDivElement)
-
-    this.kernelPromise = requestBinderKernel()
-
-    this.kernelPromise.then((kernel: IKernelConnection) => {
-      this.kernel = kernel
-      this.$emit('kernelReady')
-      this.outputArea = createOutputArea(outputDivRef)
-    }, (error: Error) => {
-      this.error = error.message
-    })
-
     this.apiTokenPromise = getQiskitUser().then(user => user?.apiToken)
   },
   methods: {
@@ -103,9 +91,9 @@ export default defineComponent({
     },
     _executeCode (code: string) {
       this.error = ''
-      this.outputArea!.setHidden(true)
-      this.kernelPromise!.then((kernel: IKernelConnection) => {
+      this._initKernel()!.then((kernel: IKernelConnection) => {
         this.$emit('running')
+        this.outputArea!.setHidden(true)
         try {
           const requestFuture = kernel.requestExecute({ code })
           this.setOutputFuture(requestFuture)
@@ -128,6 +116,26 @@ export default defineComponent({
         }
       })
     },
+    _initKernel (): Promise<IKernelConnection> {
+      if (this.kernelPromise) {
+        return this.kernelPromise
+      }
+
+      this.$emit('kernelLoading')
+      this.kernelLoading = true
+      const outputDivRef = (this.$refs.outputDiv as HTMLDivElement)
+
+      this.kernelPromise = requestBinderKernel()
+
+      return this.kernelPromise.then((kernel: IKernelConnection) => {
+        this.kernelLoading = false
+        this.$emit('kernelReady')
+        this.outputArea = createOutputArea(outputDivRef)
+        return kernel
+      }, (error: Error) => {
+        this.error = error.message
+      }) as Promise<IKernelConnection>
+    },
     setOutputFuture (future : IOutputShellFuture) {
       this.outputArea!.future = future
     },
diff --git a/frontend/vue/components/CodeExercise/ExerciseActionsBar.vue b/frontend/vue/components/CodeExercise/ExerciseActionsBar.vue
index f4f59a022..c615e9af5 100644
--- a/frontend/vue/components/CodeExercise/ExerciseActionsBar.vue
+++ b/frontend/vue/components/CodeExercise/ExerciseActionsBar.vue
@@ -1,5 +1,14 @@
 <template>
   <div class="exercise-actions-bar">
+    <button
+      v-if="!runEnabled && !isRunning && !isApiTokenNeeded"
+      class="exercise-actions-bar__button exercise-actions-bar__button_kernel"
+      @click="run(isApiTokenNeeded)"
+      :class="{'exercise-actions-bar__button exercise-actions-bar__button_disabled': isLoading}"
+      :disabled="isLoading"
+    >
+      {{ $translate('Start kernel and run') }}
+    </button>
     <button
       v-if="runEnabled && !isRunning && !isApiTokenNeeded"
       class="exercise-actions-bar__button exercise-actions-bar__button_run"
@@ -32,6 +41,11 @@ import 'carbon-web-components/es/components/loading/loading'
 export default defineComponent({
   name: 'ExerciseActionsBar',
   props: {
+    isLoading: {
+      type: Boolean,
+      required: false,
+      default: true
+    },
     isRunning: {
       type: Boolean,
       required: false,
@@ -91,7 +105,7 @@ export default defineComponent({
 
   &__button {
     padding: $spacing-03;
-    width: $column-size-large;
+    width: $column-size-small;
     max-width: 100%;
 
     background-size: 200% 100%;
@@ -113,9 +127,14 @@ export default defineComponent({
       cursor: not-allowed;
     }
 
-    &_run {
+    &_run,
+    &_kernel {
       @include bicolor-background($button-background-color-quaternary-dark, $button-background-color-quaternary);
     }
+
+    &_kernel {
+      width: 2 * $column-size-medium;
+    }
   }
 
   &__indicator {
diff --git a/frontend/vue/components/OneXP/OneXPBanner.vue b/frontend/vue/components/OneXP/OneXPBanner.vue
new file mode 100644
index 000000000..3f90803a2
--- /dev/null
+++ b/frontend/vue/components/OneXP/OneXPBanner.vue
@@ -0,0 +1,79 @@
+<template>
+  <div class="one-xp-banner">
+    <div class="one-xp-banner__container">
+      <span
+        class="one-xp-banner__copy"
+        v-text="
+          `🎉 Starting on November 29, 2023 Qiskit Learning Resource will only live on IBM Quantum`
+        "
+      />
+      <a
+        class="one-xp-banner__cta"
+        href="https://learning.quantum-computing.ibm.com"
+        rel="noopener"
+        target="_blank"
+        @click="onClick"
+      >
+        <span class="one-xp-banner__cta__label" v-text="`Check it out`" />
+        <LaunchIcon class="one-xp-banner__cta__icon" />
+      </a>
+    </div>
+  </div>
+</template>
+
+<script lang="ts">
+import { Vue, Options } from 'vue-class-component'
+import LaunchIcon from '@carbon/icons-vue/lib/launch/16'
+
+@Options({
+  components: { LaunchIcon }
+})
+export default class OneXPBanner extends Vue {
+  onClick () {
+    this.$trackClickEvent('go-to-1xp-learning', '1xp-banner')
+  }
+}
+</script>
+
+<style lang="scss" scoped>
+@import "carbon-components/scss/globals/scss/typography";
+@import "carbon-components/scss/globals/scss/layout";
+@import "../../../scss/variables/colors.scss";
+@import "~/../scss/variables/mq.scss";
+
+.one-xp-banner {
+  @include type-style("body-long-01");
+
+  background-image: linear-gradient(to right, $purple-10, #fff);
+  border-bottom: 1px solid $border-color;
+  padding: $spacing-05 $spacing-06;
+
+  &__container {
+    display: inline-flex;
+    flex-direction: column;
+    gap: $spacing-03;
+
+    @include mq($from: large) {
+      flex-direction: row;
+    }
+  }
+
+  &__copy {
+    font-weight: bold;
+  }
+
+  &__cta {
+    align-items: center;
+    display: inline-flex;
+    gap: $spacing-03;
+
+    &__label {
+      color: $purple-70;
+    }
+
+    &__icon {
+      fill: $purple-70;
+    }
+  }
+}
+</style>
diff --git a/frontend/vue/components/SignIn/SignIn.vue b/frontend/vue/components/SignIn/SignIn.vue
index f9f66d7e3..76af5cc53 100644
--- a/frontend/vue/components/SignIn/SignIn.vue
+++ b/frontend/vue/components/SignIn/SignIn.vue
@@ -63,20 +63,26 @@ export default class SignIn extends Vue {
 <style lang="scss" scoped>
 @import 'carbon-components/scss/globals/scss/typography';
 @import '~/../scss/variables/mq.scss';
+@import '~/../scss/variables/settings.scss';
 @import '~/../scss/mixins/mixins.scss';
 @import '~/../scss/variables/colors.scss';
 
 .sign-in {
   position: fixed;
-  top: 3.5rem;
+  top: calc($qiskit-navbar-height + $one-xp-banner-height-desktop + 1px);
   left: 0;
   right: 0;
   bottom: 0;
   z-index: 3;
-  max-height: calc(100vh - 3.5rem);
+  max-height: calc(100vh - ($qiskit-navbar-height + $one-xp-banner-height-desktop + 1px));
   overflow: hidden;
   display: flex;
 
+  @include mq($from: medium, $until: large) {
+    top: calc($qiskit-navbar-height + $one-xp-banner-height-mobile + 1px);
+    max-height: calc(100vh - ($qiskit-navbar-height + $one-xp-banner-height-mobile + 1px));
+  }
+
   @include mq($until: medium) {
     flex-direction: column;
     position: initial;
diff --git a/frontend/vue/components/UtilityPanel/Scratchpad.vue b/frontend/vue/components/UtilityPanel/Scratchpad.vue
index 257620c5a..cdc96b1ca 100644
--- a/frontend/vue/components/UtilityPanel/Scratchpad.vue
+++ b/frontend/vue/components/UtilityPanel/Scratchpad.vue
@@ -20,6 +20,7 @@
         <ExerciseActionsBar
           class="scratchpad__editor-block__actions-bar"
           :is-running="isKernelBusy"
+          :is-loading="isKernelLoading"
           :run-enabled="isKernelReady"
           :grade-enabled="isKernelReady && isGradingExercise"
           :is-api-token-needed="isApiTokenNeeded"
@@ -41,6 +42,7 @@
         @running="kernelRunning"
         @finished="kernelFinished"
         @kernelReady="kernelReady"
+        @kernelLoading="kernelLoading"
       />
     </div>
   </div>
@@ -75,6 +77,7 @@ type scratchpadCopyRequestEvent = Event & { detail: { code: string } }
 })
 export default class Scratchpad extends Vue {
   code = INITIAL_CODE
+  isKernelLoading = false
   isKernelBusy = false
   isKernelReady = false
   isGradingExercise = false
@@ -120,6 +123,7 @@ export default class Scratchpad extends Vue {
 
   kernelRunning () {
     this.isKernelBusy = true
+    this.isKernelLoading = true
   }
 
   kernelFinished () {
@@ -128,6 +132,11 @@ export default class Scratchpad extends Vue {
 
   kernelReady () {
     this.isKernelReady = true
+    this.isKernelLoading = true
+  }
+
+  kernelLoading () {
+    this.isKernelLoading = true
   }
 }
 </script>
diff --git a/frontend/vue/components/common/AppCta.vue b/frontend/vue/components/common/AppCta.vue
index 2f38a1737..48e8445b9 100644
--- a/frontend/vue/components/common/AppCta.vue
+++ b/frontend/vue/components/common/AppCta.vue
@@ -84,7 +84,7 @@ export default class AppCta extends Vue.with(Props) {
   align-items: center;
 
   padding: $spacing-05;
-  width: 4 * $column-size-large;
+  width: 4 * $column-size-small;
   max-width: 100%;
 
   @include mq($from: medium, $until: large) {
diff --git a/frontend/vue/components/constants/megaMenuLinks.ts b/frontend/vue/components/constants/megaMenuLinks.ts
index 0ed7f357c..f154a099d 100644
--- a/frontend/vue/components/constants/megaMenuLinks.ts
+++ b/frontend/vue/components/constants/megaMenuLinks.ts
@@ -2,6 +2,7 @@ import { NavLink } from './../../../constants/menuLinks'
 
 const wholeSection = 'whole-section'
 const sectionBasicsCourse = 'basics-course'
+const sectionAlgorithmFundamentals = 'algorithm-fundamentals'
 const sectionAlgorithmDesign = 'algorithm-design'
 const sectionIntroductionCourse = 'introduction-course'
 const sectionQMLCourse = 'quantum-machine-learning'
@@ -19,6 +20,7 @@ const sectionSummerSchool = 'summer-schools'
 const sectionProblemSets = 'problem-sets'
 
 const pathBasicsCourse = '/course/basics'
+const pathAlgorithmFundamentals = '/course/algorithms'
 const pathAlgorithmDesign = '/course/algorithm-design'
 const pathIntroductionCourse = '/course/introduction'
 const pathQuantumMachineLearning = '/course/machine-learning'
@@ -81,6 +83,39 @@ const BASICS_COURSE: MegaDropdownMenuGroup = {
   ]
 }
 
+const ALGORITHM_FUNDAMENTALS: MegaDropdownMenuGroup = {
+  title: {
+    label: 'Fundamentals of quantum algorithms',
+    url: `${pathAlgorithmFundamentals}`,
+    segment: {
+      cta: wholeSection, location: sectionAlgorithmFundamentals
+    }
+  },
+  content: [
+    {
+      label: 'Quantum query algorithms',
+      url: `${pathAlgorithmFundamentals}/query-algorithms`,
+      segment: {
+        cta: 'query_algorithms', location: sectionAlgorithmFundamentals
+      }
+    },
+    {
+      label: 'Algorithmic foundations',
+      url: `${pathAlgorithmFundamentals}/algorithmic-foundations`,
+      segment: {
+        cta: 'algorithmic_foundations', location: sectionAlgorithmFundamentals
+      }
+    },
+    {
+      label: 'Phase estimation and factoring',
+      url: `${pathAlgorithmFundamentals}/phase-estimation`,
+      segment: {
+        cta: 'phase_estimation', location: sectionAlgorithmFundamentals
+      }
+    }
+  ]
+}
+
 const ALGORITHM_DESIGN: MegaDropdownMenuGroup = {
   title: {
     label: 'Variational algorithm design',
@@ -1094,13 +1129,6 @@ const ALGORITHMS: MegaDropdownMenuGroup = {
         cta: '07_grover_examples', location: sectionTutorials
       }
     },
-    {
-      label: 'Shor’s algorithms',
-      url: `${tutorialsBaseUrl}/algorithms/08_factorizers.html`,
-      segment: {
-        cta: '08_factorizers', location: sectionTutorials
-      }
-    },
     {
       label: 'Iterative Quantum Phase Estimation Algorithm',
       url: `${tutorialsBaseUrl}/algorithms/09_IQPE.html`,
@@ -1553,6 +1581,7 @@ const COURSES_SECTION: MegaDropdownMenuSection = {
   title: 'Courses',
   content: [
     BASICS_COURSE,
+    ALGORITHM_FUNDAMENTALS,
     ALGORITHM_DESIGN,
     INTRODUCTION_COURSE,
     QML_COURSE
diff --git a/frontend/vue/main.ts b/frontend/vue/main.ts
index e5f021bad..6d7611837 100644
--- a/frontend/vue/main.ts
+++ b/frontend/vue/main.ts
@@ -2,6 +2,7 @@ import { QiskitBanner } from '@qiskit-community/qiskit-vue'
 import wrapper from './wc-wrapper/platypusWrapper'
 
 import CodeMirrorClipboardCopy from './components/CodeMirrorClipboardCopy/CodeMirrorClipboardCopy.vue'
+import OneXpBanner from './components/OneXP/OneXPBanner.vue'
 import TheMenu from './components/TheMenu/index.vue'
 import WhatIsQuantumVolume from './components/WhatIsQuantumVolume/WhatIsQuantumVolume.vue'
 import LayersCircuit from './components/WhatIsQuantumVolume/LayersCircuit.vue'
@@ -53,6 +54,7 @@ const kebabize = (str:string) => {
 }
 
 const components: any = {
+  OneXpBanner,
   QiskitBanner,
   AppCta,
   CodeMirrorClipboardCopy,
diff --git a/notebooks/problem-sets/quantum_kittens_notepad.ipynb b/notebooks/problem-sets/quantum_kittens_notepad.ipynb
new file mode 100644
index 000000000..bacd37dee
--- /dev/null
+++ b/notebooks/problem-sets/quantum_kittens_notepad.ipynb
@@ -0,0 +1,338 @@
+{
+ "cells": [
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "# Quantum Kittens Notepad"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "## Overview\n",
+    "\n",
+    "Welcome to the Quantum Kittens notepad. \n",
+    "\n",
+    "This is a coding notepad supplement to the book [Quantum Kittens](http://quantumkittens.online), a non-technical book that teaches quantum computing through stories about cats.\n",
+    "\n",
+    "Also available in [Japanese](http://quantum-kittens-ja.github.io/) and [Spanish](http://quantum-kittens-es.github.io/).\n"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "<img src='https://github.com/quantum-kittens/quantum-kittens.github.io/blob/main/assets/imgs/cover_beta.png?raw=true'/>"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "#### Quantum Kittens Credits:\n",
+    "- Author: [Radha Pyari Sandhir](https://radhapyarisandhir.com/)\n",
+    "- Illustrator: [Rashi Chandra](https://www.instagram.com/rashichan/)\n",
+    "- Japanese Translator: [Yuri Kobayashi](https://twitter.com/YuriKobaya)\n",
+    "- Spanish Translator: [Claudia Zendejas-Morales](https://twitter.com/clausia)\n",
+    "- The Cats: [Arya, Blade, Luna](https://www.instagram.com/catomictheory/)\n",
+    "- Editor: [Suzanne Musin](https://www.linkedin.com/in/suzannemusin/)\n",
+    "- Technical Reviewer: [James Weaver](https://github.com/javafxpert)\n"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "## Chapter 2 Qiskit Code\n",
+    "\n",
+    "Simulate a Whiskerton marble using this code! \n",
+    "\n",
+    "This code walks you through creating and running a quantum circuit with a single qubit.\n",
+    "\n",
+    "This code is associated with Quantum Kittens Chapter 2:\n",
+    "- [Chapter 2 - Story - Schrödinger Day](https://quantum-kittens.github.io/posts/CHAPTER-2-Story-Schr%C3%B6dinger-Day/)\n",
+    "- [Chapter 2 Part 2 - Commentary - Qubits, Superposition, and Measurement](https://quantum-kittens.github.io/posts/CHAPTER-2-Part-2-Qubits-Superposition-and-Measurements/)"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "<img src='https://github.com/quantum-kittens/quantum-kittens.github.io/blob/main/assets/imgs/Marble_Animation.png?raw=true'/>"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 2,
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "# Importing necessary Qiskit libraries; run this cell first\n",
+    "from qiskit import QuantumCircuit, transpile\n",
+    "from qiskit import BasicAer\n",
+    "from qiskit.visualization import *"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 3,
+   "metadata": {
+    "goals": [
+     {
+      "id": "qk-notepad"
+     }
+    ]
+   },
+   "outputs": [
+    {
+     "data": {
+      "image/png": "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\n",
+      "text/plain": [
+       "<Figure size 193.726x144.48 with 1 Axes>"
+      ]
+     },
+     "execution_count": 3,
+     "metadata": {},
+     "output_type": "execute_result"
+    }
+   ],
+   "source": [
+    "#Create Marble Circuit\n",
+    "\n",
+    "marble_circuit = QuantumCircuit(1, 1) # add one qubit (Whiskerton marble) and one classical bit (to store the measurement outcome)\n",
+    "\n",
+    "marble_circuit.h(0) # add H-gate or Hadamard gate to the qubit (this is the quantum gate that puts the marble in superposition)\n",
+    "\n",
+    "marble_circuit.measure(0,0) # add a measurement operator (this is equivalent to a cat looking directly at a marble)\n",
+    "\n",
+    "marble_circuit.draw('mpl') # see how the circuit looks\n"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 21,
+   "metadata": {},
+   "outputs": [
+    {
+     "name": "stdout",
+     "output_type": "stream",
+     "text": [
+      "The marble is red.\n"
+     ]
+    }
+   ],
+   "source": [
+    "#Run Marble Circuit,\n",
+    "#That is, see if the marble turns red or blue\n",
+    "\n",
+    "marble_state = {'1': 'red', '0': 'blue'}\n",
+    "\n",
+    "simulator = BasicAer.get_backend(\"qasm_simulator\") # Identify the quantum computer to run this on. In this case it's a simulator not a real device.\n",
+    "\n",
+    "compiled_circuit = transpile(marble_circuit, simulator) # Compile the circuit down to low-level QASM instructions.\n",
+    "\n",
+    "job = simulator.run(compiled_circuit, shots=1000) # Run the circuit on the simulator 1000 times to gather statistics.\n",
+    "\n",
+    "# fetch and print the outcome:\n",
+    "\n",
+    "result = job.result() \n",
+    "counts = result.get_counts(compiled_circuit)\n",
+    "\n",
+    "ans = str(max(counts, key=counts.get))\n",
+    "\n",
+    "print('The marble is ' + marble_state[ans] +'.') # The outcome is the one associated with the highest count.\n"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 22,
+   "metadata": {},
+   "outputs": [
+    {
+     "name": "stdout",
+     "output_type": "stream",
+     "text": [
+      "Your result in the form of counts: {'1': 502, '0': 498}\n",
+      "Thus, in 1000 shots, you get blue 498 times, and red 502 times.\n"
+     ]
+    },
+    {
+     "data": {
+      "image/png": "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\n",
+      "text/plain": [
+       "<Figure size 504x360 with 1 Axes>"
+      ]
+     },
+     "execution_count": 22,
+     "metadata": {},
+     "output_type": "execute_result"
+    }
+   ],
+   "source": [
+    "# Examine the statistics and plot histogram\n",
+    "\n",
+    "print(\"Your result in the form of counts:\", counts)\n",
+    "print(\"Thus, in 1000 shots, you get blue \" + str(counts['0']) + \" times, and red \" + str(counts['1']) + \" times.\")\n",
+    "\n",
+    "plot_histogram(counts)"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "## Chapter 3 Qiskit Code\n",
+    "\n",
+    "Simulate a Whiskerton doorbell using this code! \n",
+    "\n",
+    "By using this code, you will learn how to create the quantum circuit corresponding to the Bell state.\n",
+    "\n",
+    "This code is associated with Quantum Kittens Chapter 3:\n",
+    "- [Chapter 3 - Story - Doorbells](https://quantum-kittens.github.io/posts/CHAPTER-3-Story-Doorbells/)\n",
+    "- [Chapter 3 Part 2 - Commentary - Multiple Qubits, Entanglement, and Bell States](https://quantum-kittens.github.io/posts/CHAPTER-3-Part-2-Multiple-Qubits-Entanglement-and-Bell-States/)"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "<img src='https://github.com/quantum-kittens/quantum-kittens.github.io/blob/main/assets/imgs/Doorbell_Animation.png?raw=true'/>"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 2,
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "# Importing necessary Qiskit libraries; run this cell first\n",
+    "from qiskit import QuantumCircuit, transpile\n",
+    "from qiskit import BasicAer\n",
+    "from qiskit.visualization import *"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 3,
+   "metadata": {},
+   "outputs": [
+    {
+     "data": {
+      "image/png": "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\n",
+      "text/plain": [
+       "<Figure size 327.252x204.68 with 1 Axes>"
+      ]
+     },
+     "execution_count": 3,
+     "metadata": {},
+     "output_type": "execute_result"
+    }
+   ],
+   "source": [
+    "#Create Doorbell Entangler Circuit\n",
+    "\n",
+    "doorbell_circuit = QuantumCircuit(2, 2) # Create a circuit with two qubits (Whiskerton marbles) and two classical bits (to store the measurement outcome).\n",
+    "\n",
+    "doorbell_circuit.h(0) # Add a Hadamard gate to the first qubit/ \n",
+    "\n",
+    "doorbell_circuit.cx(0,1) # Add a cnot gate with the first qubit as the control and the second qubit as the target. The target flips its state when the control is in the 1 state.\n",
+    "\n",
+    "doorbell_circuit.measure([0,1],[0,1]) # Add measurement operators (this is equivalent to a cat looking directly at the outer marble).\n",
+    "\n",
+    "doorbell_circuit.draw('mpl') # See how the circuit looks.\n"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 10,
+   "metadata": {},
+   "outputs": [
+    {
+     "name": "stdout",
+     "output_type": "stream",
+     "text": [
+      "The marbles are both red.\n"
+     ]
+    }
+   ],
+   "source": [
+    "#Run Doorbell Circuit,\n",
+    "#That is, see if the marbles turns red or blue\n",
+    "\n",
+    "doorbell_state = {'11': 'red', '00': 'blue'}\n",
+    "\n",
+    "simulator = BasicAer.get_backend(\"qasm_simulator\") # Identify the quantum computer to run this on. In this case it's a simulator not a real device.\n",
+    "\n",
+    "compiled_circuit = transpile(doorbell_circuit, simulator) # Compile the circuit down to low-level QASM instructions.\n",
+    "\n",
+    "job = simulator.run(compiled_circuit, shots=1000) # Run the circuit on the simulator 1000 times to gather statistics.\n",
+    "\n",
+    "# fetch and print the outcome:\n",
+    "\n",
+    "result = job.result() \n",
+    "counts = result.get_counts(compiled_circuit)\n",
+    "\n",
+    "ans = str(max(counts, key=counts.get))\n",
+    "\n",
+    "print('The marbles are both ' + doorbell_state[ans] +'.') # The outcome is the one associated with the highest count.\n"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 12,
+   "metadata": {},
+   "outputs": [
+    {
+     "name": "stdout",
+     "output_type": "stream",
+     "text": [
+      "Your result in the form of counts: {'11': 505, '00': 495}\n",
+      "Thus, in 1000 shots, you get double blues 495 times, and double reds 505 times.\n"
+     ]
+    },
+    {
+     "data": {
+      "image/png": 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v9FpFkiQdBXqyBnlHqT0iVgDnUjl5RpKkQaE31iAfw6dpSJIGmd4IyIuA53vhfSRJOmr05CSdn/LbJ+kElVvBjQMu7+W6JEmqqSO5UcAB4JfA0sz8ee+VJElS7fXbjQIkSRpIejKCBCAizgHmUJluXZ2ZS3u7KEmSaq0na5BTge8Ap1N5YDLAlIhopfLg41+84s6SJA0wPTmL9cvAfuAtmTk9M6cDM6ttX+6L4iRJqpWeTLEuBN6WmRs6GzJzfURcBTzS65VJklRDPb0OsnQv1le8P6skSQNVTwLyEeCWiJje2RARbwZuwhGkJGmQ6UlAXgX8DrA+IjZFxCbgH6ttV/VFcZIk1UpProPcHBENwNuBk6rNP8vMh/ukMkmSauhVR5AR8e6I2BgRY7Liocy8JTNvAX5c7VvYD7VKktRvDmeK9Qrghszc1b0jM38FfAlY3Mt1SZJUU4cTkG8FDjWN+ihQ3zvlSJJ0dDicgHwTlRuTv5IExvdOOZIkHR0OJyC3UBlFvpK3As/1TjmSJB0dDicgvwv854gY2b0jIt4AXFPdRpKkQeNwLvP4AvB7wNqI+Gug89mPJ1M5gSeAL/ZNeZIk1carBmRmtkfEmcCtVIIwOruA7wN/lJltfVeiJEn977BuFJCZm4DzImIs8BYqIbkuM3f2ZXGSJNVKjx6YXA3EH/dRLZIkHTV6+jQPSZKGBANSkqQCA1KSpAIDUpKkAgNSkqQCA1KSpAIDUpKkAgNSkqQCA1KSpAIDUpKkAgNSkqQCA1KSpAIDUpKkAgNSkqQCA1KSpAIDUpKkAgNSkqQCA1KSpAIDUpKkAgNSkqQCA1KSpAIDUpKkAgNSkqQCA1KSpAIDUpKkAgNSkqQCA1KSpAIDUpKkAgNSkqQCA1KSpAIDUpKkAgNSkqQCA1KSpAIDUpKkAgNSkqSCfg3IiFgQEfdFxHMRkRHxsW79X6u2d339qNs2IyLiloh4PiJerL7ftP78OSRJg19/jyBHAauAfwfsfYVtHgaO7/I6r1v/TcAHgA8BZwFjgPsjYlgf1CtJGqKG9+eHZeYDwANQGS2+wmYdmbmt1BERxwEfB/4gMx+qtn0E2AS8Hfh+b9csSRqajsY1yOaIaI+ItRFxe0RM7NJ3OnAM8GBnQ2ZuBn4GnNnPdUqSBrF+HUEehu8B3wY2AHXAXwKPRsTpmdkBTAb2A89326+t2vcyEXEZcBnAlClTWLp0KQAzZsxg9OjRrFy5EoDx48czd+5cnnjiCQCGDx9Oc3MzK1asYNeuXQA0NjZSmSXWYNTa2sqePXsAmD9/Plu2bOG5554DYPbs2QwbNoxnn30WgMmTJ3PiiSfS0tICwMiRI5k/fz7Lly9n797K6kFTUxMbNmxg27bKhMicOXPYv38/a9asAWDq1KlMmzaN5cuXAzBq1CgaGxtpaWmho6MDgObmZtauXUt7ezsA8+bNo6Ojg3Xr1gEwffp0Jk2aRGtrKwBjxoyhoaGBZcuWsW/fPgAWLFjA6tWr2b59OwD19fXs3r2b9evXA1BXV8e4ceNYsWIFAGPHjqW+vp7HH3+czCQiOPvss1m5ciU7d+4EoKGhgR07drBx40bgtR9PbW1tbN68GYCZM2cyYsQIVq1aBcDEiROZNWsWy5YtA2DEiBE0NTX1+O8JTurhvwQNFFu3bj3i4+lQIjP7sPxDfHDEHuCKzPzaIbaZQmX69IOZ+e2I+DDwdeCY7FJ4RDwKrMvMTxzqMxsbG7PzF8lrdelNR7S7jmK3L651BeoLHrODV28csxHxdGY2lvqOxinWgzLzF8AWYGa1aRswDJjQbdNJ1T5JknrFUR2QETEBmApsrTY9DbwELOyyzTTgZOCH/V6gJGnQ6tc1yIgYBbyl+u3rgDdHxKnAjurr88DdVAKxDrgWaAe+A5CZv4qIrwLXR0Q7sB34K+AnVC4PkSSpV/T3CLIR+PvqayTwF9Wvr6Fy8s0pwL3AWuAOYA3QlJm7u7zHYiqB+U3gSWAP8N7M3N8/P4IkaSjo7+sglwJxiE3eeRjv0QFcWX1JktQnjuo1SEmSasWAlCSpwICUJKnAgJQkqcCAlCSpwICUJKnAgJQkqcCAlCSpwICUJKnAgJQkqcCAlCSpwICUJKnAgJQkqcCAlCSpwICUJKnAgJQkqcCAlCSpwICUJKnAgJQkqcCAlCSpwICUJKnAgJQkqcCAlCSpwICUJKnAgJQkqcCAlCSpwICUJKnAgJQkqcCAlCSpwICUJKnAgJQkqcCAlCSpwICUJKnAgJQkqcCAlCSpwICUJKnAgJQkqcCAlCSpwICUJKnAgJQkqcCAlCSpwICUJKnAgJQkqcCAlCSpwICUJKnAgJQkqcCAlCSpwICUJKnAgJQkqcCAlCSpwICUJKnAgJQkqcCAlCSpwICUJKnAgJQkqcCAlCSpwICUJKnAgJQkqcCAlCSpwICUJKnAgJQkqcCAlCSpwICUJKlgwAZkRHwqIjZExK8j4umIOKvWNUmSBo8BGZAR8UHgZuCLwGnAD4H/GxFvrmlhkqRBY0AGJPDHwNcy8/bM/FlmXglsBS6vcV2SpEEiMrPWNfRIRBwL/D/gQ5n5d13a/waYl5lnd9v+MuCy6rezgTX9VesgMQF4vtZFSDosHq89d0JmvqnUMby/K+kFE4BhQFu39jbg7d03zszbgNv6oa5BKSJaM7Ox1nVIenUer71roE6xSpLUpwZiQD4P7AcmdWufBGzr/3IkSYPRgAvIzPwN8DSwsFvXQipns6p3OT0tDRwer71owJ2kAwcv87gT+BTwJPBJ4OPA3MzcVMvaJEmDw0A8SYfM/GZEjAc+CxwPrALOMxwlSb1lQI4gJUnqawNuDVKSpP5gQEqSVGBASpJUMCBP0lHfiYhpwFuAAA4AazLT60slDTmepKODIuJy4A+BeuBF4B+ALcCPgHsyc01EvC4zD9SwTEnqF06xCoDqZTNfBO6lculME3AHlbsWfRT4ckTMycwDERG1q1RSp4g4JiJmRcSIWtcyGDmCFAARcSVwcWbOL/Q1A9cCU4EzMtOnBUhHgYhYDHwB+BbwbeDHwC8zc3+XbcYA/wp4ODNfqkWdA5UjSHX6DTA6IuYBRMSI6qPFyMxlwL8Ffg28o3YlSurmg8BTVM4buAdoAW6IiOaIOK66zYeB/2Q49pwBqU53UTkpZ3FEjM7Mjsz8TUS8DiAz/wl4AZhWwxolVUXEm4CXgNsz8yzgBOCrwHuAJ4BHI+JPgcXA8lrVOZA5xSq6rCm+H7gZGEdlyua/An9PJRQXALcCp2TmxhqUKamLiDgeuAh4NjO/363vNOCSav9YYHpmPtf/VQ5sBqQOiog3Am8GzgQupLJuAZXHiAVwZ2Z+vibFSXqZiBgJZGb+uuvJc1n9xR4RX6Byn+rTalXjQOZ1kENcREwEPgL8CZVnbe6lMpW6DLgROIbK+sb3MnNtjcqUVJCZezuDMbuNdiLiDcAHgP9Zi9oGA0eQQ1xEfA2YC/wfYAeV6dVTgFlAO/DZzHT9QjqKVM9M3d09FLtt83oqJ/H8bfU5uuohA3IIq/7PczeVKZgnurS9GZhPZQ1jBvD7mbmiZoVK+i0R8RUqZ68+BWzKzF2Fbd6YmS/0d22DiWexDm1zgA1ULvEAKtM0mbkpM78FvJfKdOu/qU15krqLiA8BlwL/hcqNPW6IiAsj4nera5Kda5N3dF62pdfGEeQQVj2I7gfeQOVuOf/Y/TZy1RsIfDwzT+3/CiV1FxG3U7nD1fXAvwYWAb8LrAEeAB4BZgM3Z+axtapzMHAEOYRl5l7gM8BI4OvARyNiekSMgoOL/GcDq2pXpaROETGcyqzPC5m5PjNvzMxTgH8BPE4lLL8F3ALcWbtKBwdHkKI6DfMfgfdRuUl5C/BL4O3AVuCSzPxp7SqU1CkixgKTMvPn1btdvdT1ZJ2I+CDwt0BDZj5TozIHBQNSB1Uv+TgfuIDKbeVWAX+XmT+vZV2SDq16x6vIzP0RcSmV6dU31Lqugc6AVJGPtZIGpoj4Y2BYZt5Q61oGOgNSkgaRiDgG2O9/cI+cASlJUoFnsUqSVGBASpJUYEBKklRgQEqSVGBASpJUYEBKklTw/wG/KcRz9HH3mgAAAABJRU5ErkJggg==\n",
+      "text/plain": [
+       "<Figure size 504x360 with 1 Axes>"
+      ]
+     },
+     "execution_count": 12,
+     "metadata": {},
+     "output_type": "execute_result"
+    }
+   ],
+   "source": [
+    "# Examine the statistics and plot histogram\n",
+    "\n",
+    "print(\"Your result in the form of counts:\", counts)\n",
+    "print(\"Thus, in 1000 shots, you get double blues \" + str(counts['00']) + \" times, and double reds \" + str(counts['11']) + \" times.\")\n",
+    "\n",
+    "plot_histogram(counts)\n"
+   ]
+  }
+ ],
+ "metadata": {
+  "kernelspec": {
+   "display_name": "Python 3",
+   "language": "python",
+   "name": "python3"
+  },
+  "language_info": {
+   "codemirror_mode": {
+    "name": "ipython",
+    "version": 3
+   },
+   "file_extension": ".py",
+   "mimetype": "text/x-python",
+   "name": "python",
+   "nbconvert_exporter": "python",
+   "pygments_lexer": "ipython3",
+   "version": "3.9"
+  }
+ },
+ "nbformat": 4,
+ "nbformat_minor": 4
+}
diff --git a/notebooks/summer-school/2022/lec-2.ipynb b/notebooks/summer-school/2022/lec-2.ipynb
index fc9f946e1..6442ff525 100644
--- a/notebooks/summer-school/2022/lec-2.ipynb
+++ b/notebooks/summer-school/2022/lec-2.ipynb
@@ -20,7 +20,7 @@
     "### Suggested reading\n",
     "\n",
     "- _Learn Quantum Computation Using Qiskit:_\n",
-    "  - [_8.1 Linear Algebra_](https://learn.qiskit.org/course/ch-appendix/an-introduction-to-linear-algebra-for-quantum-computing)\n",
+    "  - [_8.1 Linear Algebra_](https://learn.qiskit.org/course/ch-appendix/linear_algebra)\n",
     "  - [_1. Quantum States and Qubits_](https://learn.qiskit.org/course/ch-states/introduction)\n",
     "  - [_2. Multiple Qubits and Entanglement_](https://learn.qiskit.org/course/ch-gates/introduction)\n",
     "  - [_3.11 Quantum Teleportation_](https://learn.qiskit.org/course/ch-algorithms/quantum-teleportation)\n"
diff --git a/notebooks/toc.yaml b/notebooks/toc.yaml
index 707505f2e..129d28ffc 100644
--- a/notebooks/toc.yaml
+++ b/notebooks/toc.yaml
@@ -82,10 +82,6 @@
       id: iit_madras_pretest
       uuid: 2892385a-2497-426c-be21-75ea635caa4e
       url: /problem-sets/iit_madras_pretest
-    - title: Qiskit Beginner Problem Set
-      id: qiskit_beginner_proficiency
-      uuid: 687732ae-70a8-11ed-a1eb-0242ac120002
-      url: /problem-sets/qiskit_beginner_problem_set
     - title: Qiskit Programming Challenge - Australia
       id: qiskit_programming_australia
       uuid: 38c0c49c-6921-4cb5-bdf9-75d592edc3fc
@@ -114,6 +110,10 @@
       id: utokyo_04
       uuid: 88c62314-cdee-11ed-afa1-0242ac120002
       url: /problem-sets/utokyo_04
+    - title: Quantum Kittens Notepad
+      id: quantumkittens
+      uuid: 949b1ba2-166b-11ee-be56-0242ac120002
+      url: /problem-sets/quantum_kittens_notepad
 
 
 - title: Examples
@@ -760,7 +760,7 @@
   title: Applied quantum algorithms
   type: chapter
   url: /v1_ch-applications
-- id: quantum-hardware
+- id: ch-quantum-hardware
   overviewInfo:
     description:
       long: 'Most gate-based quantum computers share similar challenges with noise
@@ -817,51 +817,6 @@
     title: The density matrix and mixed states
     url: v1/ch-quantum-hardware/density-matrix
     uuid: 9f785388-6fab-4c13-b323-4d98cf215c03
-  title: Investigating quantum hardware using quantum circuits
-  type: chapter
-  url: /v1_quantum-hardware
-- id: quantum-hardware-pulses
-  overviewInfo:
-    description:
-      long: 'The Qiskit Textbook mostly assumes we have a quantum computer that can
-
-        carry out quantum operations, and doesn''t worry about how the devices
-
-        actually work. These pages go a step deeper, exploring the physics of
-
-        superconducting qubits, and using Qiskit to program operations on these
-
-        devices at the level of microwave pulses.
-
-        '
-      short: 'In this chapter, we get a level closer to the real quantum machines.
-
-        Learn about the physics of these devices, and how to program them at
-
-        the level of microwave pulses.
-
-        '
-    prerequisites:
-    - description: 'With the basics down, this chapter explores the consequences of
-        these
-
-        new quantum effects, and sets us up with tools to understand quantum
-
-        algorithms.
-
-        '
-      link: https://learn.qiskit.org/course/ch-gates/introduction
-      title: Multiple Qubits and Entanglement
-    - description: 'Python is a programming language where you don''t need to compile.
-        You
-
-        can just run it line by line...
-
-        '
-      link: https://learn.qiskit.org/course/ch-prerequisites/introduction-to-python-and-jupyter-notebooks
-      title: Python and Jupyter Notebooks
-    thumbnailUrl: /ch-quantum-hardware/overview/header-pulses.png
-  sections:
   - id: calibrating-qubits-pulse
     previewImgUrl: /ch-quantum-hardware/overview/calibrating-qubits-pulse.png
     title: Calibrating qubits with Qiskit Pulse
@@ -897,9 +852,9 @@
     title: Hamiltonian tomography
     url: v1/ch-quantum-hardware/hamiltonian-tomography
     uuid: 37cc881f-a795-43b1-85b6-cb94f9cda622
-  title: Investigating quantum hardware using microwave pulses
+  title: Investigating quantum hardware
   type: chapter
-  url: /v1_quantum-hardware-pulses
+  url: /v1_ch-quantum-hardware
 - id: ch-labs
   overviewInfo:
     description:
@@ -1078,7 +1033,6 @@
   url: /v1_ch-demos
 
 
-
 #---- Below was added by machine ---
 
 - id: basics-quantum-information
@@ -1251,8 +1205,16 @@
   sections:
   - id: query-algorithms
     title: Quantum query algorithms
-    url: v2/algorithms/quantum-query-algorithms
+    url: v2/algorithms/01.query-algorithms/quantum-query-algorithms
     uuid: f236ed68-91f2-470a-8152-29f09e60a30c
+  - id: algorithmic-foundations
+    title: Algorithmic foundations
+    url: v2/algorithms/02.algorithmic-foundations/algorithmic-foundations
+    uuid: 00b26932-7d64-436c-86a9-56091c0419ac
+  - id: phase-estimation
+    title: Phase estimation
+    url: v2/algorithms/03.phase-estimation/phase-estimation-and-factoring
+    uuid: b49f2ca0-240d-417a-8f51-a6288d504ca3
   subtitle: 'Unit 2: Fundamentals of quantum algorithms'
   title: Fundamentals of quantum algorithms
   type: course
diff --git a/notebooks/v1/ch-algorithms/quantum-walk-search-algorithm.ipynb b/notebooks/v1/ch-algorithms/quantum-walk-search-algorithm.ipynb
index 58b4a9ab3..1b6d23b1e 100644
--- a/notebooks/v1/ch-algorithms/quantum-walk-search-algorithm.ipynb
+++ b/notebooks/v1/ch-algorithms/quantum-walk-search-algorithm.ipynb
@@ -2,7 +2,7 @@
  "cells": [
   {
    "cell_type": "markdown",
-   "id": "0f353c6f",
+   "id": "789a3892",
    "metadata": {
     "tags": [
      "remove_cell"
@@ -14,7 +14,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "6a696cd9",
+   "id": "7062b169",
    "metadata": {},
    "source": [
     "Quantum walks are the quantum equivalence of a classical Markov chain and have become key in many quantum algorithms. In this section, we implement a quantum walk search algorithm that finds marked elements in a graph. The algorithm has a quadratic speedup compared to its classical counterpart. \n",
@@ -168,7 +168,7 @@
   {
    "cell_type": "code",
    "execution_count": 2,
-   "id": "8402d335",
+   "id": "208ec8b2",
    "metadata": {},
    "outputs": [],
    "source": [
@@ -188,7 +188,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "efc61c2b",
+   "id": "63b14e4b",
    "metadata": {},
    "source": [
     "The circuit will have 6 qubits, 4 that represents the position and 2 that represents the coin. As we mentioned previously, the coin is a Grover coin, which is the diffuser in Grover's algorithm. We start by implementing this. "
@@ -197,7 +197,7 @@
   {
    "cell_type": "code",
    "execution_count": 3,
-   "id": "7483f2e6",
+   "id": "1c565b0f",
    "metadata": {},
    "outputs": [
     {
@@ -226,7 +226,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "2e2dcc18",
+   "id": "cacb9b5c",
    "metadata": {},
    "source": [
     "Now, let us implement the shift operator. We know that the walker can only move to neighboring nodes, and all neighboring nodes differ by only one bit. We want to move the walker according to the coin, and we move the walker by applying a NOT gate to one of the node qubits. If the coin is in state $\\ket{11}$, we move the walker to the state in which the first node qubit differ. If the coin is $\\ket{10}$ or $\\ket{01}$, the walker moves to the state where the second and third qubit, respectively, differ. Finally, if the Grover coin is $\\ket{00}$, we flip the fourth qubit. We implement this with CCNOT- and NOT gates after the Grover coin. Together, they are one step of a quantum walk on a 4-dimensional hypercube. "
@@ -235,7 +235,7 @@
   {
    "cell_type": "code",
    "execution_count": 4,
-   "id": "562d63b6",
+   "id": "7c99ea89",
    "metadata": {},
    "outputs": [
     {
@@ -269,7 +269,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "73e329ff",
+   "id": "86e61b23",
    "metadata": {},
    "source": [
     "## 4. Quantum walk search algorithm <a name=\"qwalkalgo\"></a>\n",
@@ -326,7 +326,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "78afba68",
+   "id": "e580d1d2",
    "metadata": {},
    "source": [
     "For this algorithm we will need to use the inverse of the one step gate implemented earlier. We get this by using the built in circuit function inverse()."
@@ -335,7 +335,7 @@
   {
    "cell_type": "code",
    "execution_count": 5,
-   "id": "257cb2e4",
+   "id": "fa21fd41",
    "metadata": {},
    "outputs": [
     {
@@ -358,7 +358,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "c026ba67",
+   "id": "2182b987",
    "metadata": {},
    "source": [
     "The inversed one step gate will be used to reverse the phase estimation later. We need to make controlled gates from both the one step gate that we implemented in section 3 and its inverse. We will later use them depending on the value of the control qubit.  "
@@ -367,7 +367,7 @@
   {
    "cell_type": "code",
    "execution_count": 6,
-   "id": "c13dc3a1",
+   "id": "0520f182",
    "metadata": {},
    "outputs": [],
    "source": [
@@ -380,7 +380,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "e3b2de86",
+   "id": "33509a78",
    "metadata": {},
    "source": [
     "Both the controlled one step gate and the controlled inversed one step gate will be used in the phase estimation. Another thing we will use in the phase estimation is the Quantum Fourier Transform. Qiskit has a function, QFT, which implements the Quantum Fourier Transform. The phase estimation uses the inverse Quantum Fourier Transform but we also will need to use the ordinary QFT to reverse the phase estimation."
@@ -389,7 +389,7 @@
   {
    "cell_type": "code",
    "execution_count": 7,
-   "id": "de85ac13",
+   "id": "f9c7b584",
    "metadata": {},
    "outputs": [
     {
@@ -415,7 +415,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "6096d0dd",
+   "id": "59c8de1b",
    "metadata": {},
    "source": [
     "Before we implement the phase estimation, we implement a phase oracle that marks the states 1011 and 1111. Then, we make it a circuit. This is step 2(a) of the algorithm."
@@ -424,7 +424,7 @@
   {
    "cell_type": "code",
    "execution_count": 8,
-   "id": "4a93bcfb",
+   "id": "5286be30",
    "metadata": {
     "scrolled": true
    },
@@ -464,7 +464,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "a91335f4",
+   "id": "1e10513b",
    "metadata": {},
    "source": [
     "We will now implement a gate that rotates an auxiliary qubit if the other qubits are non-zero. We will use this gate in the phase estimation, where it will rotate the auxiliary qubit if $\\theta \\neq 0$. "
@@ -473,7 +473,7 @@
   {
    "cell_type": "code",
    "execution_count": 9,
-   "id": "825f9d6c",
+   "id": "ede560cd",
    "metadata": {},
    "outputs": [
     {
@@ -505,7 +505,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "4fad4857",
+   "id": "f211880a",
    "metadata": {},
    "source": [
     "Now, we will implement step 2(b) of the algorithm. This step consists of a phase estimation one step of the quantum walk followed by an auxiliary qubit that we rotate if $\\theta \\neq 0$. For this, we use the mark_auxiliary_gate that we just created.  Thereafter, we reverse the phase estimation. "
@@ -514,7 +514,7 @@
   {
    "cell_type": "code",
    "execution_count": 11,
-   "id": "5bdc78d0",
+   "id": "cb444542",
    "metadata": {
     "scrolled": false
    },
@@ -565,7 +565,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "5c8c53bc",
+   "id": "f95647a5",
    "metadata": {},
    "source": [
     "Now we implement the whole quantum walk search algorithm using the gates we made previously. We start by applying Hadamard gates to node and coin qubits, which is step 1 in the algorithm. Thereafter, we iteratively apply the phase oracle gate and the phase estimation gate (steps 2(a) and 2(b)). We need $O(1/\\sqrt{\\epsilon})$ iterations as stated in the description of the algorithm in section 4. Lastly, we measure the node qubits. "
@@ -574,7 +574,7 @@
   {
    "cell_type": "code",
    "execution_count": 12,
-   "id": "160eb345",
+   "id": "18f74668",
    "metadata": {},
    "outputs": [
     {
@@ -616,7 +616,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "1b8ae6fa",
+   "id": "1571cb59",
    "metadata": {},
    "source": [
     "Finally we run the implementation on the qasm simulator. We see that the circuit collapse to the marked states a clear majority of the times. "
@@ -625,7 +625,7 @@
   {
    "cell_type": "code",
    "execution_count": 13,
-   "id": "ec69c953",
+   "id": "bf5d9a24",
    "metadata": {},
    "outputs": [
     {
@@ -651,7 +651,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "3792ca7d",
+   "id": "ca83111a",
    "metadata": {},
    "source": [
     "## 6. References <a id='references'></a>\n",
@@ -665,7 +665,7 @@
   {
    "cell_type": "code",
    "execution_count": 14,
-   "id": "d257ed54",
+   "id": "f143caba",
    "metadata": {},
    "outputs": [
     {
diff --git a/notebooks/v1/ch-applications/quantum-edge-detection.ipynb b/notebooks/v1/ch-applications/quantum-edge-detection.ipynb
index 07dda1269..89230defe 100644
--- a/notebooks/v1/ch-applications/quantum-edge-detection.ipynb
+++ b/notebooks/v1/ch-applications/quantum-edge-detection.ipynb
@@ -2,7 +2,7 @@
  "cells": [
   {
    "cell_type": "markdown",
-   "id": "e00ca766",
+   "id": "9ef8cb74",
    "metadata": {},
    "source": [
     "An essential part of any image feature extraction procedure is **Edge Detection**. The process of edge detection is used extensively in modern classical image processing algorithms for extracting the structure of the objects/features depicted in an image. Quantum Image processing being an emerging field, is very intriguing and enables one to have exponential speedup (as mentioned in their paper by Ruan et al. [6]), in certain cases over classical image processing. Although, edge detection is fairly efficient in classical image processing, it becomes very slow for larger images due to the huge resolution of these images and the pixel-wise computation that is necessary for most of the classical edge detection algorithms.\n",
@@ -12,7 +12,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "dc05d4f9",
+   "id": "dbdfe7c0",
    "metadata": {},
    "source": [
     "# Quantum Probability Image Encoding (QPIE)"
@@ -20,7 +20,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "5ae21f5e",
+   "id": "9a615394",
    "metadata": {},
    "source": [
     "The QPIE representation uses the probability amplitudes of a quantum state to store the pixel values of a classical image. If we have $n$ -qubits, we have access to up to $2^n$ -states in superposition. In QPIE we take advantage of this fact to design an efficient and robust encoding scheme for Black-and-White (B&W) or RGB images and exponentially reduce the memory required to store the data. That means, for storing a 4-pixel image, we need just 2-qubits; for 8-pixel image we need 3-qubits, and so on. In general, the number of qubits $(n)$ for a $N$-pixel image is calculated as:-\n",
@@ -79,7 +79,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "da743cef",
+   "id": "3ab871fe",
    "metadata": {},
    "source": [
     "# Quantum Hadamard Edge Detection (QHED)"
@@ -87,7 +87,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "eefec352",
+   "id": "7e4abec3",
    "metadata": {},
    "source": [
     "In general, classical edge detection algorithms rely mostly on the computation of image gradients i.e. identifying locations in the image for dark-to-light (or light-to-dark) intensity transitions. Hence, the worst case time complexity for most of them is $O(2^n)$. This means that each pixel needs to be processed individually to determine the gradients.\n",
@@ -162,7 +162,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "58b29b51",
+   "id": "4f09bb55",
    "metadata": {},
    "source": [
     "## A Variation of QHED (with an auxiliary qubit)\n",
@@ -251,7 +251,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "32ce73b4",
+   "id": "ef8cd3d9",
    "metadata": {},
    "source": [
     "## Quantum Circuit\n",
@@ -278,7 +278,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "081cc2d8",
+   "id": "45145462",
    "metadata": {},
    "source": [
     "# Implementation using Qiskit"
@@ -286,7 +286,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "4512c5ea",
+   "id": "508dc234",
    "metadata": {},
    "source": [
     "## QHED on Small Images\n",
@@ -297,7 +297,7 @@
   {
    "cell_type": "code",
    "execution_count": 1,
-   "id": "b7cb9c61",
+   "id": "bcb941d8",
    "metadata": {},
    "outputs": [],
    "source": [
@@ -316,7 +316,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "4a0fa3b0",
+   "id": "bc702e86",
    "metadata": {},
    "source": [
     "For the purpose of this demonstration, we can assume that an image is nothing but a collection of pixel values represented as a numpy matrix in python. Also, initially let us take only binary values for pixels for simplicity i.e. $I_{jk} \\ \\in \\ \\{0, 1\\}$, and there are no floating point values for pixel intensities.\n",
@@ -327,7 +327,7 @@
   {
    "cell_type": "code",
    "execution_count": 2,
-   "id": "63b5bd62",
+   "id": "5a686caf",
    "metadata": {},
    "outputs": [
     {
@@ -367,7 +367,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "7c805349",
+   "id": "db63165d",
    "metadata": {},
    "source": [
     "Now that we have defined our image for testing, we can go ahead and use $eq. (1.3)$ to encode the pixel intensities as probability amplitudes of different states of the system:-"
@@ -376,7 +376,7 @@
   {
    "cell_type": "code",
    "execution_count": 3,
-   "id": "39192455",
+   "id": "8f591de4",
    "metadata": {},
    "outputs": [],
    "source": [
@@ -405,7 +405,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "e046f59a",
+   "id": "dd881e3a",
    "metadata": {},
    "source": [
     "As can be seen in the above python code, we obtain two different aplitude encoded quantum images. The first one (`image_norm_h`) is for the horizontal scanning of the image and the second one (`image_norm_v`) is for the vertical scanning of the image.\n",
@@ -416,7 +416,7 @@
   {
    "cell_type": "code",
    "execution_count": 4,
-   "id": "a8b231df",
+   "id": "99123215",
    "metadata": {},
    "outputs": [],
    "source": [
@@ -431,7 +431,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "52c6ecf8",
+   "id": "8ab0870b",
    "metadata": {},
    "source": [
     "Once, we have normalized the pixel values, converted them to probability amplitudes, anc determined the number of qubits necessary for processing the image, we can start making the quantum circuit for the same.\n",
@@ -442,7 +442,7 @@
   {
    "cell_type": "code",
    "execution_count": 5,
-   "id": "837d804e",
+   "id": "f11af251",
    "metadata": {},
    "outputs": [
     {
@@ -489,7 +489,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "7e5d0ea6",
+   "id": "44b6afd6",
    "metadata": {},
    "source": [
     "Now, we simulate the circuits using the `statevector_simulator` and obtain the statevector of the system as the output."
@@ -498,7 +498,7 @@
   {
    "cell_type": "code",
    "execution_count": 6,
-   "id": "7ae9a8c6",
+   "id": "c9688bd8",
    "metadata": {},
    "outputs": [
     {
@@ -568,7 +568,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "39374ea6",
+   "id": "7634ac58",
    "metadata": {},
    "source": [
     "From $eq. (2.6)$, we can clearly see that we need to consider only those states where the auxiliary qubit (qubit-0 or LSB in our case) gives a measurement output of $\\ket{1}$. Since, we know that LSB is 1 in a bit-string only for odd numbers, we easily just take the amplitudes corresponding to odd states from the statevector to form our image and discard all the even states.\n",
@@ -579,7 +579,7 @@
   {
    "cell_type": "code",
    "execution_count": 7,
-   "id": "aa6abcf7",
+   "id": "c0c0c06e",
    "metadata": {},
    "outputs": [
     {
@@ -626,7 +626,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "8a31d97c",
+   "id": "1372414b",
    "metadata": {},
    "source": [
     "Finally, we combine both horizontal and vertical scans to get the complete edge detected image as shown below,"
@@ -635,7 +635,7 @@
   {
    "cell_type": "code",
    "execution_count": 8,
-   "id": "e1e3789e",
+   "id": "c2c7b630",
    "metadata": {},
    "outputs": [
     {
@@ -674,7 +674,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "ff86adb0",
+   "id": "6550deda",
    "metadata": {},
    "source": [
     "### Running on Real Hardware\n",
@@ -686,7 +686,7 @@
   {
    "cell_type": "code",
    "execution_count": 9,
-   "id": "61be5111",
+   "id": "16838a2f",
    "metadata": {},
    "outputs": [],
    "source": [
@@ -702,7 +702,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "1bf5fedd",
+   "id": "97f30c3f",
    "metadata": {},
    "source": [
     "Since, running on actual hardware deals with encountering errors due to noise, we only limit this example to run on (2+1)-qubits as of now. For the demonstration, now we can crop a part of the the above image that was used in the simulation,"
@@ -711,7 +711,7 @@
   {
    "cell_type": "code",
    "execution_count": 10,
-   "id": "d3e3b621",
+   "id": "0dd924e9",
    "metadata": {},
    "outputs": [
     {
@@ -739,7 +739,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "7c2ea4da",
+   "id": "54ec7e2f",
    "metadata": {},
    "source": [
     "Defining the parameters for the quantum circuit of the QHED algorithm and creating instances of horizontal and vertical scan circuits.\n",
@@ -762,7 +762,7 @@
   {
    "cell_type": "code",
    "execution_count": 11,
-   "id": "4e08ae09",
+   "id": "8e21e60c",
    "metadata": {},
    "outputs": [],
    "source": [
@@ -775,7 +775,7 @@
   {
    "cell_type": "code",
    "execution_count": 12,
-   "id": "7ee4c7a6",
+   "id": "58a6ffdd",
    "metadata": {},
    "outputs": [
     {
@@ -836,7 +836,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "3b6355b1",
+   "id": "066e59ec",
    "metadata": {},
    "source": [
     "For running the circuit on real hardware, it is necessary to decompose the above quantum circuit into the basis gates $(X, SX, RZ(\\theta), CX)$ for the `ibmq_santiago` backend in order to run the circuit properly. To achieve this, we transpile the above circuit according to our backend's coupling map and also set the `optimization_level=3`, to get the most most optimized circuit according to the hardware."
@@ -845,7 +845,7 @@
   {
    "cell_type": "code",
    "execution_count": 16,
-   "id": "bb74fb01",
+   "id": "ec5c3f76",
    "metadata": {},
    "outputs": [],
    "source": [
@@ -857,7 +857,7 @@
   {
    "cell_type": "code",
    "execution_count": 30,
-   "id": "439a6e97",
+   "id": "dda43b35",
    "metadata": {},
    "outputs": [
     {
@@ -892,7 +892,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "67823215",
+   "id": "412f6cbc",
    "metadata": {},
    "source": [
     "Finally, we run the circuits on our backend and obtain the results shown in the histograms below,"
@@ -901,7 +901,7 @@
   {
    "cell_type": "code",
    "execution_count": 27,
-   "id": "c6f757a7",
+   "id": "1b8ba5d3",
    "metadata": {},
    "outputs": [
     {
@@ -923,7 +923,7 @@
   {
    "cell_type": "code",
    "execution_count": 28,
-   "id": "b1e60af2",
+   "id": "288b85a0",
    "metadata": {},
    "outputs": [
     {
@@ -980,7 +980,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "e6fa0ea5",
+   "id": "71126ae6",
    "metadata": {},
    "source": [
     "Now, we extract the counts of the odd-numbered states from horizontal and vertical scans because only those states contain the pixel intensity gradient information in which the auxiliary qubit is in the state $|1 \\rangle$."
@@ -989,7 +989,7 @@
   {
    "cell_type": "code",
    "execution_count": 31,
-   "id": "5d8e0629",
+   "id": "445c59e4",
    "metadata": {},
    "outputs": [
     {
@@ -1028,7 +1028,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "1eba4192",
+   "id": "50cea78f",
    "metadata": {},
    "source": [
     "Finally, we combine both horizontal and vertical scans to make the full edge detection cropped image as shown below,"
@@ -1037,7 +1037,7 @@
   {
    "cell_type": "code",
    "execution_count": 32,
-   "id": "21df393e",
+   "id": "a402a771",
    "metadata": {},
    "outputs": [
     {
@@ -1062,7 +1062,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "f70a9540",
+   "id": "a61a5c95",
    "metadata": {},
    "source": [
     "For comparison, let's simulate the quantum circuits on the `qasm_simulator` as well and check how the probability distribution obtained from the real hardware, differs from a perfect simulation:-"
@@ -1071,7 +1071,7 @@
   {
    "cell_type": "code",
    "execution_count": 33,
-   "id": "194910d1",
+   "id": "f08f5823",
    "metadata": {},
    "outputs": [
     {
@@ -1136,7 +1136,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "b4617853",
+   "id": "b02bd766",
    "metadata": {},
    "source": [
     "## QHED on Larger Images (Exercise for the reader!)"
@@ -1144,7 +1144,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "3355d1ca",
+   "id": "0f57d033",
    "metadata": {},
    "source": [
     "As the quantum technology advances, we will see more and more applications related to faster image processing and image manipulation. However, for processing insanely large data like 4K images and videos, one would have to process the image in multiple parts until we reach the stage of fault-tolerant quantum hardware.\n",
@@ -1154,7 +1154,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "1e925d03",
+   "id": "1389d1e4",
    "metadata": {},
    "source": [
     "We also define an initial code snippet, which helps you to load the image and also we define a variable `image_crop_size`, which refers to the size of each part that this image will be divided into. Feel free to change the variables below and play around with the values to get the best balance between the size of each part of image and the total number of parts that the original image is divided into. Write your code in the cell which says `## YOUR CODE GOES HERE #####`. We hope you enjoy this exercise! :D"
@@ -1163,7 +1163,7 @@
   {
    "cell_type": "code",
    "execution_count": 34,
-   "id": "da9bc056",
+   "id": "0d7ea78b",
    "metadata": {},
    "outputs": [
     {
@@ -1222,7 +1222,7 @@
   {
    "cell_type": "code",
    "execution_count": 35,
-   "id": "445ca16e",
+   "id": "14f720df",
    "metadata": {},
    "outputs": [],
    "source": [
@@ -1233,7 +1233,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "13070c55",
+   "id": "4ca6862f",
    "metadata": {},
    "source": [
     "**Hint:** Since, the above image is very big to encode at once on today's devices, the solution to this problem would contain the following steps:\n",
@@ -1249,7 +1249,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "9b93ca77",
+   "id": "878d40c0",
    "metadata": {},
    "source": [
     "# References \n",
@@ -1282,7 +1282,7 @@
   {
    "cell_type": "code",
    "execution_count": 36,
-   "id": "f61183e3",
+   "id": "23a7cf20",
    "metadata": {},
    "outputs": [
     {
diff --git a/notebooks/v2/algorithms/images/Deutsch-Jozsa.png b/notebooks/v2/algorithms/01.query-algorithms/images/Deutsch-Jozsa.png
similarity index 100%
rename from notebooks/v2/algorithms/images/Deutsch-Jozsa.png
rename to notebooks/v2/algorithms/01.query-algorithms/images/Deutsch-Jozsa.png
diff --git a/notebooks/v2/algorithms/images/Deutsch-circuit-states.png b/notebooks/v2/algorithms/01.query-algorithms/images/Deutsch-circuit-states.png
similarity index 100%
rename from notebooks/v2/algorithms/images/Deutsch-circuit-states.png
rename to notebooks/v2/algorithms/01.query-algorithms/images/Deutsch-circuit-states.png
diff --git a/notebooks/v2/algorithms/images/Deutsch-circuit.png b/notebooks/v2/algorithms/01.query-algorithms/images/Deutsch-circuit.png
similarity index 100%
rename from notebooks/v2/algorithms/images/Deutsch-circuit.png
rename to notebooks/v2/algorithms/01.query-algorithms/images/Deutsch-circuit.png
diff --git a/notebooks/v2/algorithms/images/Simon.png b/notebooks/v2/algorithms/01.query-algorithms/images/Simon.png
similarity index 100%
rename from notebooks/v2/algorithms/images/Simon.png
rename to notebooks/v2/algorithms/01.query-algorithms/images/Simon.png
diff --git a/notebooks/v2/algorithms/images/classical-parity.png b/notebooks/v2/algorithms/01.query-algorithms/images/classical-parity.png
similarity index 100%
rename from notebooks/v2/algorithms/images/classical-parity.png
rename to notebooks/v2/algorithms/01.query-algorithms/images/classical-parity.png
diff --git a/notebooks/v2/algorithms/images/classical-query-gate.png b/notebooks/v2/algorithms/01.query-algorithms/images/classical-query-gate.png
similarity index 100%
rename from notebooks/v2/algorithms/images/classical-query-gate.png
rename to notebooks/v2/algorithms/01.query-algorithms/images/classical-query-gate.png
diff --git a/notebooks/v2/algorithms/images/query-computation.png b/notebooks/v2/algorithms/01.query-algorithms/images/query-computation.png
similarity index 100%
rename from notebooks/v2/algorithms/images/query-computation.png
rename to notebooks/v2/algorithms/01.query-algorithms/images/query-computation.png
diff --git a/notebooks/v2/algorithms/images/standard-computation.png b/notebooks/v2/algorithms/01.query-algorithms/images/standard-computation.png
similarity index 100%
rename from notebooks/v2/algorithms/images/standard-computation.png
rename to notebooks/v2/algorithms/01.query-algorithms/images/standard-computation.png
diff --git a/notebooks/v2/algorithms/images/unitary-query-gate.png b/notebooks/v2/algorithms/01.query-algorithms/images/unitary-query-gate.png
similarity index 100%
rename from notebooks/v2/algorithms/images/unitary-query-gate.png
rename to notebooks/v2/algorithms/01.query-algorithms/images/unitary-query-gate.png
diff --git a/notebooks/v2/algorithms/quantum-query-algorithms.ipynb b/notebooks/v2/algorithms/01.query-algorithms/quantum-query-algorithms.ipynb
similarity index 55%
rename from notebooks/v2/algorithms/quantum-query-algorithms.ipynb
rename to notebooks/v2/algorithms/01.query-algorithms/quantum-query-algorithms.ipynb
index 54c916c2d..e66aa61c8 100644
--- a/notebooks/v2/algorithms/quantum-query-algorithms.ipynb
+++ b/notebooks/v2/algorithms/01.query-algorithms/quantum-query-algorithms.ipynb
@@ -31,7 +31,7 @@
     "While the time required to solve problems is our main concern, we will deviate slightly from this focus for the purposes of this first lesson.\n",
     "What we will do is to formulate a simple algorithmic framework — known as the *query model* — and explore the advantages that quantum computers offer within this framework.\n",
     "\n",
-    "The query model of computation is like a Petri dish for quantum algorithmic ideas.\n",
+    "The query model of computation is like a petri dish for quantum algorithmic ideas.\n",
     "It is rigid and unnatural, in the sense that it does not accurately represent the sorts of computational problems we generally care about in practice.\n",
     "Nevertheless, it has proved to be incredibly useful as a tool for developing quantum algorithmic techniques, including ones that power the most well-known quantum algorithms (such as Shor's factoring algorithm).\n",
     "It also happens to be a very useful framework for *explaining* these techniques.\n",
@@ -438,18 +438,17 @@
    "source": [
     "from qiskit import QuantumCircuit\n",
     "\n",
-    "\n",
     "def deutsch_function(case: int):\n",
     "    \"\"\"\n",
     "    Generate a valid Deutsch function as a `QuantumCircuit`.\n",
     "    \"\"\"\n",
-    "    if case not in [1, 2, 3, 4]:\n",
+    "    if case not in [1,2,3,4]:\n",
     "        raise ValueError(\"`case` must be 1, 2, 3, or 4.\")\n",
     "\n",
     "    f = QuantumCircuit(2)\n",
-    "    if case in [2, 3]:\n",
+    "    if case in [2,3]:\n",
     "        f.cx(0, 1)\n",
-    "    if case in [3, 4]:\n",
+    "    if case in [3,4]:\n",
     "        f.x(1)\n",
     "    return f"
    ]
@@ -470,11 +469,9 @@
    "outputs": [
     {
      "data": {
-      "image/svg+xml": [
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+      "image/png": "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\n",
       "text/plain": [
-       "<Figure size 287.294x200.667 with 1 Axes>"
+       "<Figure size 206.852x144.48 with 1 Axes>"
       ]
      },
      "execution_count": 2,
@@ -506,11 +503,11 @@
     "    \"\"\"\n",
     "    Compiles a circuit for use in Deutsch's algorithm.\n",
     "    \"\"\"\n",
-    "    n = function.num_qubits - 1\n",
-    "    qc = QuantumCircuit(n + 1, n)\n",
+    "    n = function.num_qubits-1\n",
+    "    qc = QuantumCircuit(n+1, n)\n",
     "\n",
     "    qc.x(n)\n",
-    "    qc.h(range(n + 1))\n",
+    "    qc.h(range(n+1))\n",
     "\n",
     "    qc.barrier()\n",
     "    qc.compose(function, inplace=True)\n",
@@ -538,11 +535,9 @@
    "outputs": [
     {
      "data": {
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\n",
       "text/plain": [
-       "<Figure size 788.961x284.278 with 1 Axes>"
+       "<Figure size 568.052x204.68 with 1 Axes>"
       ]
      },
      "execution_count": 4,
@@ -551,7 +546,9 @@
     }
    ],
    "source": [
-    "compile_circuit(deutsch_function(3)).draw()"
+    "compile_circuit(\n",
+    "    deutsch_function(3)\n",
+    ").draw()"
    ]
   },
   {
@@ -571,16 +568,15 @@
    "source": [
     "from qiskit_aer import AerSimulator\n",
     "\n",
-    "\n",
     "def deutsch_algorithm(function: QuantumCircuit):\n",
     "    \"\"\"\n",
     "    Determine if a Deutsch function is constant or balanced.\n",
     "    \"\"\"\n",
     "    qc = compile_circuit(function)\n",
-    "\n",
-    "    result = AerSimulator().run(qc, shots=1, memory=True).result()\n",
+    "    \n",
+    "    result = AerSimulator().run(qc,shots=1,memory=True).result()\n",
     "    measurements = result.get_memory()\n",
-    "    if measurements[0] == \"0\":\n",
+    "    if measurements[0] == '0':\n",
     "        return \"constant\"\n",
     "    return \"balanced\""
    ]
@@ -601,11 +597,9 @@
    "outputs": [
     {
      "data": {
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       "text/plain": [
-       "<Figure size 287.294x200.667 with 1 Axes>"
+       "<Figure size 206.852x144.48 with 1 Axes>"
       ]
      },
      "metadata": {},
@@ -833,41 +827,40 @@
     "from qiskit import QuantumCircuit\n",
     "import numpy as np\n",
     "\n",
-    "\n",
     "def dj_function(num_qubits):\n",
     "    \"\"\"\n",
     "    Create a random Deutsch-Jozsa function.\n",
     "    \"\"\"\n",
     "\n",
-    "    qc = QuantumCircuit(num_qubits + 1)\n",
+    "    qc = QuantumCircuit(num_qubits+1)\n",
     "    if np.random.randint(0, 2):\n",
     "        # Flip output qubit with 50% chance\n",
     "        qc.x(num_qubits)\n",
     "    if np.random.randint(0, 2):\n",
     "        # return constant circuit with 50% chance\n",
     "        return qc\n",
-    "\n",
+    "    \n",
     "    # next, choose half the possible input states\n",
     "    on_states = np.random.choice(\n",
     "        range(2**num_qubits),  # numbers to sample from\n",
-    "        2**num_qubits // 2,  # number of samples\n",
-    "        replace=False,  # makes sure states are only sampled once\n",
+    "        2**num_qubits // 2,    # number of samples\n",
+    "        replace=False  # makes sure states are only sampled once\n",
     "    )\n",
     "\n",
     "    def add_cx(qc, bit_string):\n",
     "        for qubit, bit in enumerate(reversed(bit_string)):\n",
-    "            if bit == \"1\":\n",
+    "            if bit == '1':\n",
     "                qc.x(qubit)\n",
     "        return qc\n",
     "\n",
     "    for state in on_states:\n",
-    "        qc.barrier()  # Barriers are added to help visualize how the functions are created. They can safely be removed.\n",
+    "        qc.barrier() # Barriers are added to help visualize how the functions are created. They can safely be removed.\n",
     "        qc = add_cx(qc, f\"{state:0b}\")\n",
     "        qc.mct(list(range(num_qubits)), num_qubits)\n",
     "        qc = add_cx(qc, f\"{state:0b}\")\n",
     "\n",
     "    qc.barrier()\n",
-    "\n",
+    "    \n",
     "    return qc"
    ]
   },
@@ -887,11 +880,9 @@
    "outputs": [
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\n",
       "text/plain": [
-       "<Figure size 1541.66x367.889 with 1 Axes>"
+       "<Figure size 1049.8x264.88 with 1 Axes>"
       ]
      },
      "execution_count": 8,
@@ -922,10 +913,10 @@
     "    \"\"\"\n",
     "    Compiles a circuit for use in the Deutsch-Jozsa algorithm.\n",
     "    \"\"\"\n",
-    "    n = function.num_qubits - 1\n",
-    "    qc = QuantumCircuit(n + 1, n)\n",
+    "    n = function.num_qubits-1\n",
+    "    qc = QuantumCircuit(n+1, n)\n",
     "    qc.x(n)\n",
-    "    qc.h(range(n + 1))\n",
+    "    qc.h(range(n+1))\n",
     "    qc.compose(function, inplace=True)\n",
     "    qc.h(range(n))\n",
     "    qc.measure(range(n), range(n))\n",
@@ -949,18 +940,17 @@
    "source": [
     "from qiskit_aer import AerSimulator\n",
     "\n",
-    "\n",
     "def dj_algorithm(function: QuantumCircuit):\n",
     "    \"\"\"\n",
     "    Determine if a Deutsch-Jozsa function is constant or balanced.\n",
     "    \"\"\"\n",
     "    qc = compile_circuit(function)\n",
-    "\n",
-    "    result = AerSimulator().run(qc, shots=1, memory=True).result()\n",
+    "    \n",
+    "    result = AerSimulator().run(qc,shots=1,memory=True).result()\n",
     "    measurements = result.get_memory()\n",
-    "    if \"1\" in measurements[0]:\n",
-    "        return \"balanced\"\n",
-    "    return \"constant\""
+    "    if '1' in measurements[0]:\n",
+    "        return 'balanced'\n",
+    "    return 'constant'"
    ]
   },
   {
@@ -973,17 +963,15 @@
   },
   {
    "cell_type": "code",
-   "execution_count": 11,
+   "execution_count": 18,
    "id": "46392e96",
    "metadata": {},
    "outputs": [
     {
      "data": {
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\n",
       "text/plain": [
-       "<Figure size 1374.44x367.889 with 1 Axes>"
+       "<Figure size 1110x264.88 with 1 Axes>"
       ]
      },
      "metadata": {},
@@ -995,7 +983,7 @@
        "'balanced'"
       ]
      },
-     "execution_count": 11,
+     "execution_count": 18,
      "metadata": {},
      "output_type": "execute_result"
     }
@@ -1182,20 +1170,18 @@
   },
   {
    "cell_type": "code",
-   "execution_count": 12,
+   "execution_count": 19,
    "id": "689ab37a",
    "metadata": {},
    "outputs": [
     {
      "data": {
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amuHgyfDMn18Kv2Y1KSH8U6hjRoV/sM+rn+2KUKhigs/+bov0TKGKCQpVTFCoYoJCFRMUqpigUMUEhSomKFQxQaGKCQpVTFCoYoJCFRMUqpigUMUEhSomKFQxQaGKCQpVTFCoYoJCFRMUqpjwf8Ng1djwLbVaAAAAAElFTkSuQmCC\n",
       "text/plain": [
-       "<Figure size 287.496x451.5 with 1 Axes>"
+       "<Figure size 206.997x325.08 with 1 Axes>"
       ]
      },
-     "execution_count": 12,
+     "execution_count": 19,
      "metadata": {},
      "output_type": "execute_result"
     }
@@ -1205,14 +1191,13 @@
     "    \"\"\"\n",
     "    Create a Bernstein-Vazirani function from a string of 1s and 0s.\n",
     "    \"\"\"\n",
-    "    qc = QuantumCircuit(len(s) + 1)\n",
+    "    qc = QuantumCircuit(len(s)+1)\n",
     "    for index, bit in enumerate(reversed(s)):\n",
-    "        if bit == \"1\":\n",
+    "        if bit == '1':\n",
     "            qc.cx(index, len(s))\n",
     "    return qc\n",
     "\n",
-    "\n",
-    "bv_function(\"1001\").draw()"
+    "bv_function('1001').draw()"
    ]
   },
   {
@@ -1226,7 +1211,7 @@
   },
   {
    "cell_type": "code",
-   "execution_count": 13,
+   "execution_count": 20,
    "id": "d015d981",
    "metadata": {},
    "outputs": [
@@ -1236,7 +1221,7 @@
        "'1001'"
       ]
      },
-     "execution_count": 13,
+     "execution_count": 20,
      "metadata": {},
      "output_type": "execute_result"
     }
@@ -1244,11 +1229,12 @@
    "source": [
     "def bv_algorithm(function: QuantumCircuit):\n",
     "    qc = compile_circuit(function)\n",
-    "    result = AerSimulator().run(qc, shots=1, memory=True).result()\n",
+    "    result = AerSimulator().run(qc,shots=1,memory=True).result()\n",
     "    return result.get_memory()[0]\n",
     "\n",
-    "\n",
-    "bv_algorithm(bv_function(\"1001\"))"
+    "bv_algorithm(\n",
+    "    bv_function('1001')\n",
+    ")"
    ]
   },
   {
@@ -1355,7 +1341,7 @@
     "\n",
     "### Analysis\n",
     "\n",
-    "The analysis of Simon's algorithm begins along similar lines to the Deutsch-Jozsa algorithm.\n",
+    "The analysis of Simon's algorithm begins along similar lines to the Deutsch-Jozsa algortithm.\n",
     "After the first layer of Hadamard gates is performed on the top $n$ qubits, the state becomes\n",
     "\n",
     "$$\n",
@@ -1611,7 +1597,7 @@
   },
   {
    "cell_type": "code",
-   "execution_count": 14,
+   "execution_count": 21,
    "id": "d17ea0f7",
    "metadata": {},
    "outputs": [],
@@ -1621,33 +1607,29 @@
     "from qiskit import QuantumCircuit\n",
     "import numpy as np\n",
     "\n",
-    "\n",
     "def simon_function(s: str):\n",
     "    \"\"\"\n",
-    "    Create a QuantumCircuit implementing a query gate for Simon problem\n",
-    "    obeying the promise for the hidden string `s`\n",
+    "    Create a QuantumCircuit implementing a query gate for Simon problem obeying the promise for the hidden string `s`\n",
     "    \"\"\"\n",
     "    # Our quantum circuit has 2n qubits for n = len(s)\n",
     "    n = len(s)\n",
-    "    qc = QuantumCircuit(2 * n)\n",
-    "\n",
-    "    # Define a random permutation of all n bit strings. This permutation\n",
-    "    # will effectively hide the string s.\n",
+    "    qc = QuantumCircuit(2*n)\n",
+    "    \n",
+    "    # Define a random permutation of all n bit strings. This permutation will effectively hide the string s.\n",
     "    pi = np.random.permutation(2**n)\n",
-    "\n",
-    "    # Now we'll define a query gate explicitly. The idea is to first define\n",
-    "    # a function g(x) = min{x,x ^ s}, which is a simple function that\n",
-    "    # satisfies the promise. Then we take f to be the composition of g and\n",
-    "    # the random permutation pi. This gives us a random function satisfying\n",
-    "    # the promise for s.\n",
-    "    query_gate = np.zeros((4**n, 4**n))\n",
+    "    \n",
+    "    # Now we'll define a query gate explicitly. The idea is to first define a function g(x) = min{x,x ^ s}, which\n",
+    "    # is a simple function that satisfies the promise, and then we take f to be the composition of g and the random\n",
+    "    # permutation pi. This gives us a random function satisfying the promise for s.\n",
+    "    \n",
+    "    query_gate = np.zeros((4**n,4**n))\n",
     "    for x in range(2**n):\n",
-    "        for y in range(2**n):\n",
-    "            z = y ^ pi[min(x, x ^ int(s, 2))]\n",
-    "            query_gate[x + 2**n * z, x + 2**n * y] = 1\n",
+    "        for y in range(2**n):              \n",
+    "            z = y ^ pi[min(x,x ^ int(s,2))]\n",
+    "            query_gate[x + 2**n *z, x + 2**n *y] = 1        \n",
     "\n",
     "    # Our circuit has just this one query gate\n",
-    "    qc.unitary(query_gate, range(2 * n))\n",
+    "    qc.unitary(query_gate,range(2*n))\n",
     "    return qc"
    ]
   },
@@ -1661,7 +1643,7 @@
   },
   {
    "cell_type": "code",
-   "execution_count": 15,
+   "execution_count": 23,
    "id": "29be9b92",
    "metadata": {},
    "outputs": [],
@@ -1669,7 +1651,6 @@
     "from qiskit_aer import AerSimulator\n",
     "from qiskit import ClassicalRegister\n",
     "\n",
-    "\n",
     "def simon_measurements(problem: QuantumCircuit, k: int):\n",
     "    \"\"\"\n",
     "    Quantum part of Simon's algorithm. Given a `QuantumCircuit` that\n",
@@ -1677,13 +1658,13 @@
     "    \"\"\"\n",
     "    n = problem.num_qubits // 2\n",
     "\n",
-    "    qc = QuantumCircuit(2 * n, n)\n",
+    "    qc = QuantumCircuit(2*n,n)\n",
     "    qc.h(range(n))\n",
     "    qc.compose(problem, inplace=True)\n",
     "    qc.h(range(n))\n",
     "    qc.measure(range(n), range(n))\n",
-    "\n",
-    "    result = AerSimulator().run(qc, shots=k, memory=True).result()\n",
+    "    \n",
+    "    result = AerSimulator().run(qc,shots=k,memory=True).result()\n",
     "    return result.get_memory()"
    ]
   },
@@ -1698,34 +1679,37 @@
   },
   {
    "cell_type": "code",
-   "execution_count": 16,
+   "execution_count": 24,
    "id": "5b3513c3",
    "metadata": {},
    "outputs": [
     {
      "data": {
       "text/plain": [
-       "['11111',\n",
-       " '01001',\n",
+       "['00111',\n",
+       " '10001',\n",
        " '01101',\n",
-       " '10101',\n",
-       " '00000',\n",
-       " '11011',\n",
-       " '01110',\n",
+       " '11111',\n",
+       " '01010',\n",
+       " '00111',\n",
+       " '00011',\n",
+       " '10110',\n",
+       " '10110',\n",
        " '00000',\n",
-       " '00100',\n",
-       " '10010',\n",
-       " '00100',\n",
-       " '01010']"
+       " '00111',\n",
+       " '11111']"
       ]
      },
-     "execution_count": 16,
+     "execution_count": 24,
      "metadata": {},
      "output_type": "execute_result"
     }
    ],
    "source": [
-    "simon_measurements(simon_function(\"11011\"), k=12)"
+    "simon_measurements(\n",
+    "    simon_function('11011'),\n",
+    "    k=12\n",
+    ")"
    ]
   },
   {
@@ -1738,7 +1722,7 @@
   },
   {
    "cell_type": "code",
-   "execution_count": 17,
+   "execution_count": 25,
    "id": "0f74e289",
    "metadata": {},
    "outputs": [],
@@ -1746,34 +1730,33 @@
     "import numpy as np\n",
     "import galois\n",
     "\n",
-    "\n",
     "def simon_algorithm(problem: QuantumCircuit):\n",
     "    \"\"\"\n",
-    "    Given a `QuantumCircuit` that implements a query gate for Simon problem,\n",
-    "    return the hidden string `s`.\n",
+    "    Given a `QuantumCircuit` that implements a query gate for Simon problem, return the hidden string `s`.\n",
     "    \"\"\"\n",
-    "    # Quantum part: run the circuit defined previously k times and gather\n",
-    "    # the measurement results. Replace +10 by +r for any nonnegative\n",
-    "    # integer r depending on desired confidence.\n",
+    "    \n",
+    "    # Quantum part: run the circuit defined previously k times and gather the measurement results.\n",
+    "    # Replace +10 by +r for any nonnegative integer r depending on desired confidence.\n",
+    "\n",
     "    measurements = simon_measurements(problem, k=problem.num_qubits // 2 + 10)\n",
-    "    print(\"Measurement results:\")\n",
+    "    print('Measurement results:')\n",
     "    display(measurements)\n",
-    "\n",
+    "    \n",
     "    # Classical post-processing:\n",
-    "\n",
+    "    \n",
     "    # 1. Convert measurements of form '11101' to 2D-array of integers\n",
-    "    matrix = np.array([list(bitstring) for bitstring in measurements]).astype(int)\n",
-    "\n",
+    "    matrix = np.array([list(bitstring) for bitstring in measurements]).astype(int)   \n",
+    "    \n",
     "    # 2. Interpret matrix as using arithmetic mod 2, and find null space\n",
-    "    null_space = galois.GF(2)(matrix).null_space()\n",
-    "    print(\"Null space:\")\n",
+    "    null_space = galois.GF(2)(matrix).null_space()   \n",
+    "    print('Null space:')\n",
     "    display(null_space)\n",
-    "\n",
+    "    \n",
     "    # 3. Convert back to a string\n",
     "    if len(null_space) == 0:\n",
     "        # No non-trivial solution; `s` is all-zeros\n",
-    "        return \"0\" * len(measurements[0])\n",
-    "    return \"\".join(np.array(null_space[0]).astype(str))"
+    "        return '0'*len(measurements[0])\n",
+    "    return ''.join(np.array(null_space[0]).astype(str))"
    ]
   },
   {
@@ -1786,10 +1769,10 @@
   },
   {
    "cell_type": "code",
-   "execution_count": 18,
+   "execution_count": 28,
    "id": "2d77b5fd",
    "metadata": {
-    "scrolled": false
+    "scrolled": true
    },
    "outputs": [
     {
@@ -1802,33 +1785,26 @@
     {
      "data": {
       "text/plain": [
-       "['01100',\n",
-       " '11001',\n",
-       " '11110',\n",
-       " '00100',\n",
-       " '01100',\n",
+       "['01111',\n",
+       " '10010',\n",
+       " '11101',\n",
+       " '10010',\n",
+       " '00000',\n",
        " '11110',\n",
        " '10110',\n",
        " '11010',\n",
-       " '01000',\n",
        " '10001',\n",
-       " '10001',\n",
-       " '10010',\n",
-       " '11101',\n",
+       " '01011',\n",
+       " '01000',\n",
        " '00011',\n",
-       " '10010']"
+       " '00100',\n",
+       " '11101',\n",
+       " '01011']"
       ]
      },
      "metadata": {},
      "output_type": "display_data"
     },
-    {
-     "name": "stderr",
-     "output_type": "stream",
-     "text": [
-      "OMP: Info #271: omp_set_nested routine deprecated, please use omp_set_max_active_levels instead.\n"
-     ]
-    },
     {
      "name": "stdout",
      "output_type": "stream",
@@ -1851,13 +1827,32 @@
        "'10011'"
       ]
      },
-     "execution_count": 18,
+     "execution_count": 28,
      "metadata": {},
      "output_type": "execute_result"
     }
    ],
    "source": [
-    "simon_algorithm(simon_function(\"10011\"))"
+    "simon_algorithm(\n",
+    "    simon_function('10011')\n",
+    ")\n"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "823ef32b",
+   "metadata": {},
+   "source": [
+    "\n",
+    "\n",
+    "\n",
+    "\n",
+    "\n",
+    "\n",
+    "\n",
+    "\n",
+    "\n",
+    "\n"
    ]
   }
  ],
diff --git a/notebooks/v2/algorithms/02.algorithmic-foundations/algorithmic-foundations.ipynb b/notebooks/v2/algorithms/02.algorithmic-foundations/algorithmic-foundations.ipynb
new file mode 100644
index 000000000..734485a8f
--- /dev/null
+++ b/notebooks/v2/algorithms/02.algorithmic-foundations/algorithmic-foundations.ipynb
@@ -0,0 +1,1208 @@
+{
+ "cells": [
+  {
+   "cell_type": "markdown",
+   "id": "b929937e",
+   "metadata": {},
+   "source": [
+    "# Algorithmic foundations\n",
+    "\n",
+    "## Introduction\n",
+    "\n",
+    "Quantum algorithms provide provable advantages over classical algorithms in the query model — but what about more standard models of computation, where problem inputs are given explicitly rather than in the form of an oracle or black box?\n",
+    "This turns out to be a much more difficult question to answer, and to address it we must first establish a solid foundation upon which to base our investigation.\n",
+    "That's the purpose of this lesson.\n",
+    "\n",
+    "We'll begin by discussing *computational cost*, for both classical and quantum computations, and how it can be measured.\n",
+    "This is a general notion that can be applied to a wide range of computational problems — but to keep things simple we'll mainly examine it through the lens of *computational number theory*, which addresses problems including basic arithmetic, computing greatest common divisors, and integer factorization, which are computational tasks that are likely to be familiar to most readers.\n",
+    "While computational number theory is a narrow application domain, these problems serve well to illustrate the basic issues — and as a bonus they happen to be highly relevant to the lesson following this one.\n",
+    "\n",
+    "Our focus will be on *algorithms*, as opposed to the ever-improving hardware on which they're run — so we'll be more concerned with how the cost of running an algorithm scales as the specific problem instances it is run on grow in size, rather than how many seconds, minutes, or hours some particular computation requires.\n",
+    "We focus on this aspect of computational cost in recognition of the fact that algorithms have fundamental importance, and will naturally be deployed against larger and larger problem instances using faster and more reliable hardware as technology develops.\n",
+    "\n",
+    "Finally, we turn to a critically important task, which is running *classical* computations on quantum computers.\n",
+    "The reason this task is important is not because we hope to replace classical computers with quantum computers (which seems extremely unlikely to happen any time soon, if ever), but rather because it opens up many interesting possibilities for quantum algorithms.\n",
+    "Specifically, classical computations running on quantum computers become available as *subroutines* — effectively leveraging decades of research and development on classical algorithms in pursuit of quantum computational advantages."
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "e4566cc2",
+   "metadata": {},
+   "source": [
+    "## Two examples: factoring and GCDs\n",
+    "\n",
+    "The classical computers that exist today are incredibly fast, and their speed seems to be ever increasing.\n",
+    "For this reason, some might be inclined to believe that computers are so fast that no computational problem is beyond their reach.\n",
+    "\n",
+    "This belief is false.\n",
+    "Some computational problems are so inherently difficult that, although there exist algorithms to solve them, no computer on the planet Earth today is fast enough to run these algorithms to completion on even moderately sized inputs within the lifetime of a human — or even within the lifetime of the Earth itself.\n",
+    "\n",
+    "To explain further, let's introduce the *integer factorization* problem.\n",
+    "\n",
+    "<!-- ::: q-block.note -->\n",
+    "\n",
+    "**Integer factorization**\n",
+    "\n",
+    "Input:  an integer $N\\geq 2$\n",
+    "\n",
+    "Output: the prime factorization of $N$\n",
+    "\n",
+    "<!-- ::: -->\n",
+    "\n",
+    "By the *prime factorization* of $N$, we mean a list of the prime factors of $N$ and the powers to which they must be raised to obtain $N$ by multiplication.\n",
+    "For example, the prime factors of $12$ are $2$ and $3$, and to obtain $12$ we must take the product of $2$ to the power $2$ and $3$ to the power $1:$\n",
+    "\n",
+    "$$\n",
+    "12 = 2^2 \\cdot 3\n",
+    "$$\n",
+    "\n",
+    "Up to the ordering of the prime factors, there is only one prime factorization for each positive integer $N\\geq 2$, which is a fact known as the *fundamental theorem of arithmetic*.\n",
+    "\n",
+    "The `SymPy` symbolic mathematics package for Python includes a function `factorint` that solves the integer factorization problem for whatever input $N$ we choose."
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 1,
+   "id": "999fdebb",
+   "metadata": {},
+   "outputs": [
+    {
+     "data": {
+      "text/plain": [
+       "{2: 2, 3: 1}"
+      ]
+     },
+     "execution_count": 1,
+     "metadata": {},
+     "output_type": "execute_result"
+    }
+   ],
+   "source": [
+    "from sympy.ntheory import factorint\n",
+    "N = 12\n",
+    "factorint(N)"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "8aea3921",
+   "metadata": {},
+   "source": [
+    "Factoring small numbers like $12$ is easy, but when the number $N$ to be factored gets larger, the problem becomes more difficult.\n",
+    "For example, running `factorint` on a significantly larger number will cause a noticeable delay."
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 2,
+   "id": "e5f7c10f",
+   "metadata": {},
+   "outputs": [
+    {
+     "data": {
+      "text/plain": [
+       "{3: 2, 74519450661011221: 1, 5073729280707932631243580787: 1}"
+      ]
+     },
+     "execution_count": 2,
+     "metadata": {},
+     "output_type": "execute_result"
+    }
+   ],
+   "source": [
+    "N = 3402823669209384634633740743176823109843098343\n",
+    "factorint(N)"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "0e3cd1ac",
+   "metadata": {},
+   "source": [
+    "For even larger values of $N$, things become impossibly difficult, at least as far as we know.\n",
+    "For example, the *RSA Factoring Challenge*, which was run by RSA Laboratories from 1991 to 2007, offered a cash prize of $100,000 to factor the following number, which has 309 decimal digits (and 1024 bits when written in binary).\n",
+    "The prize for this number was never collected and its prime factors remain unknown."
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 3,
+   "id": "074752ae",
+   "metadata": {},
+   "outputs": [
+    {
+     "name": "stdout",
+     "output_type": "stream",
+     "text": [
+      "135066410865995223349603216278805969938881475605667027524485143851526510604859533833940287150571909441798207282164471551373680419703964191743046496589274256239341020864383202110372958725762358509643110564073501508187510676594629205563685529475213500852879416377328533906109750544334999811150056977236890927563\n"
+     ]
+    }
+   ],
+   "source": [
+    "RSA1024 = 135066410865995223349603216278805969938881475605667027524485143851526510604859533833940287150571909441798207282164471551373680419703964191743046496589274256239341020864383202110372958725762358509643110564073501508187510676594629205563685529475213500852879416377328533906109750544334999811150056977236890927563\n",
+    "print(RSA1024)\n"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "2d2655a2",
+   "metadata": {},
+   "source": [
+    "We need not bother running `factorint` on RSA1024, it wouldn't finish within our lifetimes."
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "3fbc4923",
+   "metadata": {},
+   "source": [
+    "The fastest known algorithm for factoring large integers is known as the *number field sieve*.\n",
+    "As an example of this algorithm's use, the RSA challenge number RSA250, which has 250 decimal digits (and 829 bits in its binary representation), was factored using the number field sieve in 2020.\n",
+    "The computation required thousands of CPU core-years, distributed across tens of thousands of machines around the world."
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 4,
+   "id": "2568ca8e",
+   "metadata": {},
+   "outputs": [
+    {
+     "name": "stdout",
+     "output_type": "stream",
+     "text": [
+      "True\n"
+     ]
+    }
+   ],
+   "source": [
+    "RSA250 = 2140324650240744961264423072839333563008614715144755017797754920881418023447140136643345519095804679610992851872470914587687396261921557363047454770520805119056493106687691590019759405693457452230589325976697471681738069364894699871578494975937497937\n",
+    "\n",
+    "p = 64135289477071580278790190170577389084825014742943447208116859632024532344630238623598752668347708737661925585694639798853367\n",
+    "q = 33372027594978156556226010605355114227940760344767554666784520987023841729210037080257448673296881877565718986258036932062711\n",
+    "\n",
+    "print(RSA250 == p*q)"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "37d39e3c",
+   "metadata": {},
+   "source": [
+    "The security of the RSA public-key cryptosystem is based on the computational difficulty of integer factoring;\n",
+    "an efficient algorithm for integer factoring would break it."
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "fa3150ec",
+   "metadata": {},
+   "source": [
+    "Next let's consider a related but very different problem, which is computing the greatest common divisor (or GCD) of two integers.\n",
+    "\n",
+    "<!-- ::: q-block.note -->\n",
+    "\n",
+    "**Greatest common divisor (GCD)**\n",
+    "\n",
+    "Input: nonnegative integers $N$ and $M$, at least one of which is positive\n",
+    "\n",
+    "Output: the greatest common divisor of $N$ and $M$\n",
+    "\n",
+    "<!-- ::: -->\n",
+    "\n",
+    "The greatest common divisor of two numbers is the largest integer that evenly divides both of them.\n",
+    "\n",
+    "This problem is easy for computers — it has roughly the same computational cost as multiplying the two input numbers together.\n",
+    "The following code cell runs the `gcd` function from the Python `math` module on two numbers that are both quite a bit larger than RSA1024 in the blink of an eye. (In fact, RSA1024 is the GCD of the two numbers in this example.)"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 5,
+   "id": "33786838",
+   "metadata": {},
+   "outputs": [
+    {
+     "data": {
+      "text/plain": [
+       "135066410865995223349603216278805969938881475605667027524485143851526510604859533833940287150571909441798207282164471551373680419703964191743046496589274256239341020864383202110372958725762358509643110564073501508187510676594629205563685529475213500852879416377328533906109750544334999811150056977236890927563"
+      ]
+     },
+     "execution_count": 5,
+     "metadata": {},
+     "output_type": "execute_result"
+    }
+   ],
+   "source": [
+    "import math\n",
+    "\n",
+    "N = 4636759690183918349682239573236686632636353319755818421393667064929987310592347460711767784882455889983961546491666129915628431549982893638464243493812487979530329460863532041588297885958272943021122033997933550246447236884738870576045537199814804920281890355275625050796526864093092006894744790739778376848205654332434378295899591539239698896074\n",
+    "M = 5056714874804877864225164843977749374751021379176083540426461360945653967249306494545888621353613218518084414930846655066495767441010526886803458300440345782982127522212209489410315422285463057656809702949608368597012967321172325810519806487247195259818074918082416290513738155834341957254558278151385588990304622183174568167973121179585331770773\n",
+    "\n",
+    "math.gcd(N,M)"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "a404015e",
+   "metadata": {},
+   "source": [
+    "We can push it even further and compute GCDs for numbers with thousands of digits in no time."
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 6,
+   "id": "9c012920",
+   "metadata": {},
+   "outputs": [
+    {
+     "data": {
+      "text/plain": [
+       "113"
+      ]
+     },
+     "execution_count": 6,
+     "metadata": {},
+     "output_type": "execute_result"
+    }
+   ],
+   "source": [
+    "math.gcd(N**200 + 1 ,M**100 + M**2)"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "4f884bdd",
+   "metadata": {},
+   "source": [
+    "This is possible because we have very efficient algorithms for computing GCDs, the most well-known of which is\n",
+    "*Euclid's algorithm*, discovered over 2,000 years ago.\n",
+    "\n",
+    "Could there be a fast algorithm for integer factorization that we just haven't discovered yet, allowing large numbers like RSA1024 to be factored in the blink of an eye?\n",
+    "The answer is yes.\n",
+    "Although we might expect that an efficient algorithm for factoring as simple and elegant as Euclid's algorithm for computing GCDs would have been discovered by now, there is nothing that rules out the existence of a very fast classical algorithm for integer factoring, beyond the fact that we've failed to find one thus far.\n",
+    "One could be discovered tomorrow — but don't hold your breath.\n",
+    "Generations of mathematicians and computer scientists have searched, and factoring numbers like RSA1024 remains beyond our reach."
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "a3db50dd",
+   "metadata": {},
+   "source": [
+    "## Measuring computational cost\n",
+    "\n",
+    "Next we'll discuss a mathematical framework through which computational cost can be measured.\n",
+    "It should be understood that this discussion is narrowly focused on the needs of this series — the *analysis of algorithms* and *computational complexity* are entire subjects onto themselves, and have much more to say about these notions.\n",
+    "\n",
+    "As a starting point, consider the following figure (which first appeared in Lesson 5), which represents a very high level abstraction of a computation.\n",
+    "\n",
+    "![Illustration of a standard computation.](images/standard-computation.png)\n",
+    "\n",
+    "The computation itself could be modeled or described in a variety of ways, such as by a computer program written in Python, a Turing machine, a Boolean circuit, or a quantum circuit.\n",
+    "Our focus will be on circuits (both Boolean and quantum)."
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "3ee7705d",
+   "metadata": {},
+   "source": [
+    "### Encodings and input length\n",
+    "\n",
+    "Let's begin with the input and output of a computational problem, which we'll assume are *binary strings*.\n",
+    "Different symbols could be used, but we'll keep things simple for the purposes of this discussion by restricting our attention to binary string inputs and outputs.\n",
+    "\n",
+    "Through binary strings, we can *encode* a variety of interesting objects that the problems we're interested in solving might concern, such as numbers, vectors, matrices, and graphs, as well as lists of these and other objects.\n",
+    "For example, to encode nonnegative integers, we can use *binary notation*.\n",
+    "The following table lists the binary encoding of the first nine nonnegative integers, along with the *length* (meaning the total number of bits) of each encoding. \n",
+    "\n",
+    "| number | binary encoding | length |\n",
+    "| :-- | :-- | :-- |\n",
+    "| 0 | 0 | 1 |\n",
+    "| 1 | 1 | 1 |\n",
+    "| 2 | 10 | 2 |\n",
+    "| 3 | 11 | 2 |\n",
+    "| 4 | 100 | 3 |\n",
+    "| 5 | 101 | 3 |\n",
+    "| 6 | 110 | 3 |\n",
+    "| 7 | 111 | 3 |\n",
+    "| 8 | 1000 | 4 |\n",
+    "\n",
+    "We can easily extend this encoding to handle both positive and negative integers by appending a *sign bit* to the representations if we choose.\n",
+    "Sometimes it's also convenient to allow binary representations of nonnegative integers to have leading zeros, which don't change the value being encoded but can allow representations to fill up a string or word of a fixed size.\n",
+    "\n",
+    "Using binary notation to represent nonnegative integers is both common and efficient, but if we wanted to we could choose a different way to represent nonnegative integers using binary strings, such as the ones suggested in the following table.\n",
+    "The specifics of these alternatives are not important to this discussion — the point is only to clarify that we do have choices for the encodings we use.\n",
+    "(In this table, the symbol $\\varepsilon$ represents the *empty string*, which has no symbols in it and length equal to zero. Naturally, to avoid an obvious source of confusion, we use a special symbol such as $\\varepsilon$ to represent the empty string rather than literally writing nothing.)\n",
+    "\n",
+    "| number | unary encoding | lexicographic encoding |\n",
+    "| :-- | :-- | :-- | \n",
+    "| 0 | $\\varepsilon$ | $\\varepsilon$ |\n",
+    "| 1 | 0 | 0 |\n",
+    "| 2 | 00 | 1 |\n",
+    "| 3 | 000 | 00 |\n",
+    "| 4 | 0000 | 01 |\n",
+    "| 5 | 00000 | 10 |\n",
+    "| 6 | 000000 | 11 |\n",
+    "| 7 | 0000000 | 000 |\n",
+    "| 8 | 00000000 | 001 |\n",
+    "\n",
+    "Other types of inputs, such as vectors and matrices, or more complicated objects like descriptions of molecules, can also be encoded as binary strings.\n",
+    "Just like we have for nonnegative integers, a variety of different encoding schemes can be selected or invented.\n",
+    "For whatever scheme we come up with to encode inputs to a given problem, we interpret the *length* of an input string as representing the size of the problem instance being solved.\n",
+    "\n",
+    "For example, the number of bits required to express a nonnegative integer $N$ in binary notation, which is sometimes denoted $\\operatorname{lg}(N)$, is given by the following formula.\n",
+    "\n",
+    "$$\n",
+    "\\operatorname{lg}(N) = \n",
+    "\\begin{cases}\n",
+    "1 & N = 0\\\\\n",
+    "1 + \\lfloor \\log_2 (N) \\rfloor & N \\geq 1\n",
+    "\\end{cases}\n",
+    "$$\n",
+    "\n",
+    "Assuming that we use binary notation to encode the input to the integer factoring problem, the *input length* for the number $N$ is therefore $\\operatorname{lg}(N)$.\n",
+    "Note, in particular, that the length (or size) of the input $N$ is not $N$ itself — when $N$ is large we don't need nearly this many bits to express $N$ in binary notation.\n",
+    "\n",
+    "From a strictly formal viewpoint, whenever we consider a computational problem or task, it should be understood that a specific scheme has been selected for encoding whatever objects are given as input or produced as output.\n",
+    "This allows computations that solve interesting problems to be viewed abstractly as transformations from binary string inputs to binary string outputs — and naturally the details of how objects are encoded as binary strings must be important to computations at some level.\n",
+    "\n",
+    "Usually, though, we don't worry all that much about these details when we're analyzing computational cost, so that we can avoid getting into details of secondary importance.\n",
+    "The basic reasoning is that we expect the computational cost of converting back and forth between \"reasonable\" encoding schemes to be insignificant compared with the cost of solving the actual problem.\n",
+    "In those situation in which this is not the case, the details can (and should) be clarified.\n",
+    "\n",
+    "For example, a very simple computation with low cost converts between the binary representation of a nonnegative integer and its lexicographic encoding (which we have not explained in detail, but it can be inferred from the table). For this reason, the computational cost of integer factoring wouldn't differ significantly if we decided to switch from using one of these encodings to the other for the input $N$.\n",
+    "On the other hand, encoding nonnegative integers in unary notation incurs an exponential blow-up in the total number of symbols required, and we would not consider it to be a \"reasonable\" encoding scheme."
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "d2f61183",
+   "metadata": {},
+   "source": [
+    "### Elementary operations\n",
+    "\n",
+    "Now let's consider the computation itself, which is represented by the blue rectangle in the figure above.\n",
+    "The way that we'll measure computational cost is to count the number of *elementary operations* that each computation requires.\n",
+    "Intuitively speaking, an elementary operation is one involving a small, fixed number of bits or qubits that can be performed quickly and easily — such as computing the AND of two bits.\n",
+    "(In contrast, running the `factorint` function is not reasonably viewed as being an elementary operation.)\n",
+    "\n",
+    "Formally speaking, there are different choices for what constitutes an elementary operation depending on the computational model being used.\n",
+    "Our focus will be on circuit models, and specifically quantum and Boolean circuits.\n",
+    "\n",
+    "#### Universal gate sets\n",
+    "\n",
+    "For circuit-based models of computation, it is typical that each *gate* is viewed as an elementary operation.\n",
+    "This leads to the question of precisely which gates we allow.\n",
+    "\n",
+    "Focusing for the moment on quantum circuits, we've seen several gates thus far in the series, including $X$, $Y$, $Z$, $H$, $S$, and $T$ gates, *swap* gates, controlled versions of gates (including *controlled-NOT*, *Toffoli*, and *Fredkin* gates), as well as gates that represent standard basis measurements.\n",
+    "In the context of the CHSH game we also saw a few additional *rotation* gates.\n",
+    "\n",
+    "We also discussed *query gates* (within the query model), and saw that any unitary operation $U$, acting on any number of qubits, can be viewed as being a gate if we so choose.\n",
+    "We'll disregard both of these options for the sake of this discussion.\n",
+    "We won't be working in the query model (although the implementation of query gates using elementary operations is discussed later in the lesson), and viewing arbitrary unitary gates, potentially acting on millions of qubits, as being elementary operations does not lead to meaningful or realistic notions of computational cost.\n",
+    "\n",
+    "Sticking with quantum gates that operate on small numbers of qubits, one approach to deciding which ones are to be considered elementary is to tease out a precise criterion — but this is not the standard approach or the one we'll take.\n",
+    "Rather, we simply make a choice.\n",
+    "\n",
+    "Here's one standard choice, which we shall adopt as the *default* gate set for quantum circuits:\n",
+    "\n",
+    " - Single-qubit unitary gates from this list: $X$, $Y$, $Z$, $H$, $S$, $S^{\\dagger}$, $T$, and $T^{\\dagger}$\n",
+    " - Controlled-NOT gates\n",
+    " - Single-qubit standard basis measurements\n",
+    "\n",
+    "A common alternative is to view Toffoli, Hadamard, and $S$ gates as being elementary, in addition to standard basis measurements.\n",
+    "(Sometimes all single-qubit gates are viewed as being elementary, although this does lead to an unrealistically powerful model when noise and inaccuracies are not properly accounted for.)\n",
+    "\n",
+    "The unitary gates in our default collection form what's called a *universal* gate set.\n",
+    "This means that we can approximate any unitary operation, on any number of qubits, and to any degree of accuracy we wish, with circuits composed of these gates alone.\n",
+    "To be clear, the definition of universality places no requirements on the cost of such approximations, meaning the number of gates from our set that we need.\n",
+    "Indeed, a fairly simple argument based on the mathematical notion of *measure* reveals that most unitary operations must have extremely high cost.\n",
+    "Proving the universality of quantum gate sets is not a simple matter and won't be covered in this series.\n",
+    "\n",
+    "For Boolean circuits, we'll take AND, OR, NOT, and FANOUT gates to be the ones representing elementary operations.\n",
+    "We don't actually need both AND gates and OR gates — we can use *De Morgan's laws* to convert from either one to the other by placing NOT gates on all three input and output wires — but nevertheless it's both typical and convenient to allow both AND and OR gates.\n",
+    "AND, OR, NOT, and FANOUT gates form a universal set for deterministic computations, meaning that any function from any fixed number of input bits to any fixed number of output bits can be implemented with these gates.\n",
+    "\n",
+    "#### The principle of deferred measurement\n",
+    "\n",
+    "Standard measurement gates can appear within quantum circuits — but sometimes it's convenient to delay them until the end, so that we can view quantum computations as consisting of a unitary part (representing the computation itself), followed by a simple read-out phase where qubits are measured and the results are output.\n",
+    "\n",
+    "This can always be done, provided that we're willing to add an additional qubit for each standard basis measurement.\n",
+    "In the following figure, the circuit on the right illustrates how this can be done for the gate on the left.\n",
+    "\n",
+    "![Deferring measurements](images/deferred-measurement.png)\n",
+    "\n",
+    "Specifically, the classical bit in the circuit on the left is replaced by a qubit on the right (initialized to the $\\vert 0\\rangle$ state), and the standard basis measurement is replaced by a controlled-NOT gate, followed by a standard basis measurement on the bottom qubit.\n",
+    "The point is that the standard basis measurement in the right-hand circuit can be pushed all the way to the end of the circuit.\n",
+    "If the classical bit in the circuit on the left is later used as a control bit, we can use the bottom *qubit* in the circuit on the right as a control instead, and the overall effect will be the same.\n",
+    "(We are assuming that the classical bit in the circuit on the left doesn't get overwritten after the measurement takes place by another measurement — but if it did we could always just use a new classical bit rather than overwriting one that was used for a previous measurement.)"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "78ea8a62",
+   "metadata": {},
+   "source": [
+    "### Circuit size and depth\n",
+    "\n",
+    "#### Circuit size\n",
+    "\n",
+    "The total number of gates in a circuit is referred to as that circuit's *size*.\n",
+    "Thus, presuming that the gates in our circuits represent elementary operations, we have that a circuit's size represents the number of elementary operations it requires — or, in other words, its *computational cost*.\n",
+    "We write $\\operatorname{size}(C)$ to refer to the size of a given circuit $C$.\n",
+    "\n",
+    "For example, consider the following Boolean circuit (from Lesson 3) for computing the XOR of two bits.\n",
+    "\n",
+    "![Boolean circuit for XOR](images/Boolean-circuit-XOR.png)\n",
+    "\n",
+    "The size of this circuit is 7, because there are 7 gates in total.\n",
+    "(Fanout operations are not always counted as being gates, but for the purposes of this lesson we will count FANOUT operations as being gates.)\n",
+    "\n",
+    "#### Time, cost, and circuit depth\n",
+    "\n",
+    "In the previous lesson we discussed the importance of *time* as a resource, or limiting constraint, on computations.\n",
+    "The examples above, such as the task of factoring RSA1024, reinforce this viewpoint.\n",
+    "The `factorint` function doesn't fail to factor RSA1024 per se, it's just that we don't have enough time to let it finish.\n",
+    "\n",
+    "The notion of computational cost, as the number of elementary operations required to perform a computation, is\n",
+    "intended to be an abstract proxy for the time required to implement a computation.\n",
+    "Each elementary operation requires a certain amount of time to perform, and the more of them a computation needs to perform the longer it's going to take, at least in general.\n",
+    "In the interest of simplicity, we'll continue to make this association between computational cost and the time required to run algorithms as we continue on in the series.\n",
+    "\n",
+    "But notice that the size of a circuit doesn't necessarily correspond directly to how long it takes to run.\n",
+    "In our Boolean circuit for computing the XOR of two bits, for instance, the two FANOUT gates could be performed simultaneously, as could the two NOT gates, as well as the two AND gates.\n",
+    "\n",
+    "A different way to measure efficiency for circuits, which takes this possibility of *parallelization* into account, is by their *depth*.\n",
+    "This is the minimum number of *layers* of gates needed within the circuit, where the gates within in each layer operate on different wires.\n",
+    "Equivalently, the depth of a circuit is the maximum number of gates encountered on any path from an input wire to an output wire.\n",
+    "For the circuit above, for instance, the depth is 4.\n",
+    "\n",
+    "Circuit depth is one way to formalize the running time of parallel computations.\n",
+    "It's an advanced topic, and there exist very sophisticated circuit constructions known to minimize the depth required for certain computations.\n",
+    "There are also some fascinating unanswered questions concerning circuit depth.\n",
+    "(For example, much remains unknown about the circuit depth required to compute GCDs.)\n",
+    "We won't have too much more to say about circuit depth in this series, aside from including a few interesting facts concerning circuit depth as we go along, but it should be clearly acknowledged that parallelization is a potential source of computational advantages.\n",
+    "\n",
+    "#### Assigning costs to different gates\n",
+    "\n",
+    "One final note concerning circuit size and computational cost is that it is possible to assign different costs to different gates, rather than viewing every gate as contributing equally to cost.\n",
+    "\n",
+    "For example, as was already mentioned, FANOUT gates are often viewed as being free for Boolean circuits — which is to say that we could choose that FANOUT gates have zero cost.\n",
+    "As another example, when we're working in the query model and we count the number of queries that a circuit makes to an input function (in the form of a black box), we're effectively assigning unit cost to query gates and zero cost to other gates, such as Hadamard gates.\n",
+    "A final example is that we sometimes weigh the costs of different gates depending on how difficult they are to implement, depending on the particular hardware implementation we have in mind.\n",
+    "\n",
+    "While all of these options are sensible in different contexts, for this lesson we'll keep things simple and stick with circuit size as a representation of computational cost."
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "2f0684fe",
+   "metadata": {},
+   "source": [
+    "### Cost as a function of input length\n",
+    "\n",
+    "We're primarily interested in how computational cost scales as inputs become larger and larger.\n",
+    "This leads us to represent the costs of algorithms as *functions* of the input length.\n",
+    "\n",
+    "#### Families of circuits\n",
+    "\n",
+    "Inputs to a given computational problem can vary in length, potentially becoming arbitrarily large.\n",
+    "Every circuit, on the other hand, has a fixed number of gates and wires.\n",
+    "For this reason, when we think about algorithms in terms of circuits, we generally need infinitely large *families* of circuits to represent algorithms.\n",
+    "By a family of circuits, we mean a sequence of circuits that grow in size, allowing larger and larger inputs to be accommodated.\n",
+    "\n",
+    "For example, imagine that we have a classical algorithm for integer factorization, such as the one used by `factorint`.\n",
+    "Like all classical algorithms, it is possible to implement this algorithm using Boolean circuits — but to do it we'll need a separate circuit for each possible input length.\n",
+    "If we looked at the resulting circuits for different input lengths, we would see that their sizes naturally grow as the input length grows — reflecting the fact that factoring 4-bit integers is much easier and requires far fewer elementary operations than factoring 1024-bit integers, for instance.\n",
+    "\n",
+    "This leads us to represent the computational cost of an algorithm by a function $t$, defined so that $t(n)$ is the number of gates in the circuit that implements the algorithm for $n$ bit inputs.\n",
+    "In more formal terms, an algorithm in the Boolean circuit model is described by a sequence of circuits\n",
+    "$\\{C_1, C_2, C_3,\\ldots\\}$, where $C_n$ solves whatever problem we're talking about for $n$-bit inputs (or, more generally, for inputs whose size is parameterized in some way by $n$), and the function $t$ representing the cost of this algorithm is defined as \n",
+    "\n",
+    "$$\n",
+    "t(n) = \\operatorname{size}(C_n).\n",
+    "$$\n",
+    "\n",
+    "For quantum circuits the situation is similar, where larger and larger circuits are needed to accommodate longer and longer input strings.\n",
+    "\n",
+    "#### Example: integer addition\n",
+    "\n",
+    "To explain this further, let's take a moment to consider a much simpler problem than integer factoring, or even computing GCDs, which is the problem of addition.\n",
+    "\n",
+    "<!-- ::: q-block.note -->\n",
+    "\n",
+    "**Integer addition**\n",
+    "\n",
+    "Input: integers $N$ and $M$\n",
+    "\n",
+    "Output: $N+M$\n",
+    "\n",
+    "<!-- ::: -->\n",
+    "\n",
+    "How might we design Boolean circuits for solving this problem?\n",
+    "\n",
+    "Just to keep things simple, let's restrict our attention to the case where $N$ and $M$ are both nonnegative integers represented by $n$-bit registers using binary notation.\n",
+    "We'll allow for any number of leading zeros in these encodings, so that $0\\leq N,M\\leq 2^n - 1$.\n",
+    "The output will be an $(n+1)$-bit binary string representing the sum, which is the maximum number of bits we need to express the result.\n",
+    "\n",
+    "We begin with an algorithm — the *standard* algorithm for addition of binary representations — which is the base $2$ analogue to the way addition is taught in elementary/primary schools around the world.\n",
+    "This algorithm can be implemented with Boolean circuits as follows.\n",
+    "\n",
+    "Starting from the least significant bits, we can compute their XOR to determine the least significant bit for the sum.\n",
+    "Then we can compute the carry bit, which is the AND of the two least significant bits of $N$ and $M$.\n",
+    "Sometimes these two operations together are known as a *half adder*.\n",
+    "Using the XOR circuit we've now seen a few times together with an AND gate and two FANOUT gates, we can build a half adder with 10 gates.\n",
+    "If for some reason we changed our minds and decided to include XOR gates in our set of elementary operations, we would need 1 AND gate, 1 XOR gate, and 2 FANOUT gates to build a half adder.\n",
+    "\n",
+    "![A half-adder](images/half-adder.png)\n",
+    "\n",
+    "Moving on to the more significant bits we can use a similar procedure, but this time including the carry bit from each previous position into our calculation.\n",
+    "By cascading 2 half adders and taking the OR of the carry bits they produce, we can create what's known as a *full adder*.\n",
+    "(Fortunately both carry bits can't be $1$, so we don't need to worry about that case.)\n",
+    "This requires 21 gates in total: 2 AND gates, 2 XOR gates (each requiring 7 gates to implement), one OR gate, and 4 FANOUT gates.\n",
+    "\n",
+    "![A full-adder](images/full-adder.png)\n",
+    "\n",
+    "Finally, by cascading the full adders, we obtain a Boolean circuit for nonnegative integer addition having \n",
+    "$21 (n-1) + 10 = 21 n - 11$ gates.\n",
+    "Had we decided to include XOR gates in our set of elementary operations, we'd need \n",
+    "$2n-1$ AND gates, $2n-1$ XOR gates, $n-1$ OR gates, and $4n-2$ FANOUT gates, for a total of $9n-5$ gates.\n",
+    "If in addition we decide not to count FANOUT gates, it's $5n-3$ gates.\n",
+    "\n",
+    "For example, the following circuit computes the sum of two 4-bit integers.\n",
+    "\n",
+    "![A circuit for addition of two 4-bit integers](images/addition-circuit.png)\n",
+    "\n",
+    "\n",
+    "#### Asymptotic notation\n",
+    "\n",
+    "On the one hand, it is good to know precisely how many gates are needed to perform various computations, like in the example of integer addition above.\n",
+    "These details are important for actually building the circuits.\n",
+    "\n",
+    "On the other hand, if we perform analyses at this level of detail for all the computations we're interested in, including ones for tasks that are much more complicated than addition, we'll very quickly be buried in details.\n",
+    "To keep things manageable, and to intentionally suppress details of secondary importance, we typically use *Big-O* notation when analyzing algorithms.\n",
+    "Through this notation we can express the *order* at which functions grow.\n",
+    "\n",
+    "Formally speaking, if we have two functions $g(n)$ and $h(n)$, we write that $g(n) = O(h(n))$ if there exists a positive real number $c > 0$ and a positive integer $n_0$ such that\n",
+    "\n",
+    "$$\n",
+    "g(n) \\leq c\\cdot h(n)\n",
+    "$$\n",
+    "\n",
+    "for all $n \\geq n_0$.\n",
+    "Typically $h(n)$ is chosen to be as simple an expression as possible, so that the notation can be used to reveal the limiting behavior of a function in simple terms.\n",
+    "For example, $17 n^3 - 257 n^2 + 65537 = O(n^3)$.\n",
+    "\n",
+    "This notation can be extended to functions having multiple arguments in a fairly straightforward way.\n",
+    "For instance, if we have two functions $g(n,m)$ and $h(n,m)$ defined on positive integers $n$ and $m$, we write that $g(n,m) = O(h(n,m))$ if there exists a positive real number $c > 0$ and a positive integer $k_0$ such that\n",
+    "\n",
+    "$$\n",
+    "g(n,m) \\leq c\\cdot h(n,m)\n",
+    "$$\n",
+    "\n",
+    "whenever $n+m \\geq k_0$.\n",
+    "\n",
+    "Connecting this notation to the example of nonnegative integer addition, we conclude that there exists a family of\n",
+    "Boolean circuits $\\{C_1, C_2,\\ldots,\\}$, where $C_n$ adds two $n$-bit nonnegative integers together, such that\n",
+    "$\\operatorname{size}(C_n) = O(n)$.\n",
+    "This reveals the most essential feature of how the cost of addition scales with the input size: it scales *linearly*.\n",
+    "Notice also that it doesn't depend on the specific detail of whether we consider XOR gates to have unit cost or cost $7$.\n",
+    "In general, using Big-O notation allows us to make statements about computational costs that aren't sensitive to such low-level details."
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "22e69f48",
+   "metadata": {},
+   "source": [
+    "#### More examples\n",
+    "\n",
+    "Here are a few more examples of problems from computational number theory, beginning with *multiplication*.\n",
+    "\n",
+    "<!-- ::: q-block.note -->\n",
+    "\n",
+    "**Integer multiplication**\n",
+    "\n",
+    "Input: integers $N$ and $M$\n",
+    "\n",
+    "Output: $NM$\n",
+    "\n",
+    "<!-- ::: -->\n",
+    "\n",
+    "Creating Boolean circuits for this problem is more difficult than creating circuits for addition — but by thinking about the standard multiplication algorithm, we can come up with circuits having size $O(n^2)$ for this problem (assuming $N$ and $M$ are both represented by $n$-bit binary representations).\n",
+    "More generally, if $N$ has $n$ bits and $M$ has $m$ bits, there are Boolean circuits of size $O(nm)$ for multiplying $N$ and $M$.\n",
+    "\n",
+    "There are, in fact, other ways to multiply that scale better.\n",
+    "For instance, the Schönhage-Strassen multiplication algorithm can be used to create a Boolean circuits for multiplying two $n$-bit integers having size $O(n \\operatorname{lg}(n) \\operatorname{lg}(\\operatorname{lg}(n)))$.\n",
+    "The intricacy of this method causes a lot of overhead, however, making it only practical for numbers having tens of thousands of bits.\n",
+    "\n",
+    "Another basic problem is *division*, which we interpret to mean computing both the quotient and remainder given an integer divisor and dividend.\n",
+    "\n",
+    "<!-- ::: q-block.note -->\n",
+    "\n",
+    "**Integer division**\n",
+    "\n",
+    "Input: integers $N$ and $M\\neq0$\n",
+    "\n",
+    "Output: integers $q$ and $r$ satisfying $0\\leq r < |M|$ and $N = q M + r$\n",
+    "\n",
+    "<!-- ::: -->\n",
+    "\n",
+    "The cost of integer division is similar to multiplication: if $N$ has $n$ bits and $M$ has $m$ bits, there are Boolean circuits of size $O(nm)$ for solving this problem.\n",
+    "Like multiplication, there are asymptotically superior methods.\n",
+    "\n",
+    "We can now compare known algorithms for computing GCDs with those for addition and multiplication.\n",
+    "Euclid's algorithm for computing the GCD of an $n$-bit number $N$ and an $m$-bit number $M$ requires Boolean circuits of size $O(nm)$, similar to the standard algorithms for multiplication and division.\n",
+    "Also similar to multiplication and division, there are asymptotically faster GCD algorithms — including ones requiring $O(n(\\operatorname{lg}(n))^2 \\operatorname{lg}(\\operatorname{lg}(n)))$ elementary operations to compute the GCD of two $n$-bit numbers.\n",
+    "\n",
+    "A somewhat more expensive computation that arises in number theory is *modular exponentiation*.\n",
+    "\n",
+    "<!-- ::: q-block.note -->\n",
+    "\n",
+    "**Integer modular exponentiation**\n",
+    "\n",
+    "Input: integers $N$, $K$, and $M$ with $K\\geq 0$ and $M\\geq 1$\n",
+    "\n",
+    "Output: $N^K \\hspace{1mm} (\\text{mod }M)$\n",
+    "\n",
+    "<!-- ::: -->\n",
+    "\n",
+    "By $N^K\\hspace{1mm} (\\text{mod }M)$ we mean the remainder when $N^K$ is divided by $M$ — meaning the unique integer $r$ satisfying $0\\leq r < M$ and $N^K = q M + r$ for some integer $q$.\n",
+    "\n",
+    "If $N$ has $n$ bits, $M$ has $m$ bits, and $K$ has $k$ bits, this problem can solved by Boolean circuits having size $O(k m^2 + nm)$ — which is not at all obvious at first glance.\n",
+    "The solution is not to first compute $N^K$ and then take the remainder — that would require exponentially many bits just to store the number $N^K$.\n",
+    "Rather, we can use the *power algorithm* (known alternatively as the *binary method* and *repeated squaring*), where we use the binary representation of $K$ to perform the entire computation modulo $M$.\n",
+    "Assuming $N$, $M$, and $K$ are all $n$-bit numbers, we obtain an $O(n^3)$ algorithm — or a *cubic* time algorithm.\n",
+    "(Once again, there are asymptotically faster algorithms.)\n",
+    "\n",
+    "Here's a code cell that computes a modular exponent for fairly large input numbers."
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 7,
+   "id": "79c8b245",
+   "metadata": {},
+   "outputs": [
+    {
+     "data": {
+      "text/plain": [
+       "5420808908959774060288987536719770456217318912094898225713892936909049029205875268391016773496273016272919329382669533127141405381618319955871147811133071683113976741106560564348611362839209748910974112799894385464900617664468329271717655067495341858822704829649579551724483155264208314048742469387847020407987537873027379036515942355206425576104942394452390040081069041858525217985044712018028991394635458870225358476099232040714610377672598679720015462702833934580459151626243131178465271884695260294198503010478207039551370657556801980546328939683389782893101937977023160027020824612118350981172299997784862364914426784395705304625946981620876565903630321426713564601636615003526450559450988257054265576673783996389684225946503950767764911765912500478971120407712043832726706539999202935732936820400534970855338294653369081634048804115320256182211145544094816107629124112742249226319353536843143250675559791743355892993051078488285292927178023438402583054839832109503781471596500072959107707020065782555861325253891174468479828455070701800024212593144971400832140786063620665224339027746048978496589265602430286475229240500730395030244167373760773225724090156004049352594635867027935465962414764456373585290956999305794964617561548270565216080421238370652518997421499178722058617256564053290758153841574743341020230723665105568383923335524981262395314837292291351213579853233800664543959697570668968932545105391200894345122403901126855808339292892144618034647640594129638891809347433516954054255735374114229687467253661509404329081038467536976485314005457794642793240728907716748113582623047896927299525364803461784339899349011331842366119834985254767022797817541982842136038797346045524543446837609691961446346898034481960162369906547207851740703325595273724313509866798139791862738941412435088490790473992542655256187359052648995226579813780101763700632784177318998593960123977376065188642849714402071821332656099246595192273548124579308954369379166791375504366177963367167357734204250134523901012053728414505662083719483277149998297345600850207968594611610502058119754052458059616153460321784141539990056380808571117172677389938997159425776949065037133138197247161115127659585109470076784212846689843449874294528288829117435818727532583190741742686601090580534050289592016761135679603364904072305569448382741870366244397396582565453282641402195407023981530691751496898222633942198230270907098913253995651978611545066193950188718245653197837791596426329560941347753939650429611447245577078846931722372616367550006355620360276455568080512099035179175416756143233950934214736621950039902506644221443998435341460452476466171419972558395644689971465747907674896790822097660307109062931957694684033978412952567599593488687529011526359106169809555979667401866723277931201510177630302133847932158278990312682334549377652520626935835739547712983314320798573894229076437190704856364396327795151484278956526648597260084828697308427169704251546673832583609705456575281533722178029103192256765343504645112826902073960223407989624481672668703972408414665894403347181679160774217536153364422436840721557202572348021932511877708639899866042064925150997870115610130091923308716372491505600234638328560252998561255435872867079194898819838898311502430190295512215594677576275248220954675704199712880196887650126401810574270118295619216693001247689245264372564423355830421087279204264834276290790089692636984198176930764597788529620410821188522782016777910796700483449826"
+      ]
+     },
+     "execution_count": 7,
+     "metadata": {},
+     "output_type": "execute_result"
+    }
+   ],
+   "source": [
+    "K = 64135289477071580278790190170577389084825014742943447208116859632024532344630238623598752668347708737661925585694639798853367\n",
+    "N = 4636759690183918349682239573236686632636353319755818421393667064929987310592347460711767784882455889983961546491666129915628431549982893638464243493812487979530329460863532041588297885958272943021122033997933550246447236884738870576045537199814804920281890355275625050796526864093092006894744790739778376848205654332434378295899591539239698896074\n",
+    "M = 5056714874804877864225164843977749374751021379176083540426461360945653967249306494545888621353613218518084414930846655066495767441010526886803458300440345782982127522212209489410315422285463057656809702949608368597012967321172325810519806487247195259818074918082416290513738155834341957254558278151385588990304622183174568167973121179585331770773\n",
+    "\n",
+    "pow(N,K**10+1,M**10 + 1)\n"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "5a21c198",
+   "metadata": {},
+   "source": [
+    "It's a little bit slower than computing GCDs, but it's still fast."
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "8691fa99",
+   "metadata": {},
+   "source": [
+    "A different example, outside of the domain of computational number theory, arose in the previous lesson.\n",
+    "For the post-processing step of Simon's algorithm, we need to compute the null space modulo $2$ of an $n\\times m$ matrix of binary values (so here the input length is $nm$ bits in total).\n",
+    "We can do this using Gaussian elimination with $O(n m \\min\\{n,m\\})$ elementary operations, which is $O(n^3)$ elementary operations if $m=O(n)$.\n",
+    "Even for a $1000\\times 1000$ binary matrix, which is a million bits of input, the computation time is on the order of seconds."
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 8,
+   "id": "75327a9d",
+   "metadata": {},
+   "outputs": [
+    {
+     "name": "stderr",
+     "output_type": "stream",
+     "text": [
+      "OMP: Info #276: omp_set_nested routine deprecated, please use omp_set_max_active_levels instead.\n"
+     ]
+    },
+    {
+     "data": {
+      "text/plain": [
+       "GF([[1, 0, 1, 0, 0, 0, 1, 1, 0, 1, 0, 1, 1, 1, 1, 1, 0, 1, 1, 1, 0, 1, 0,\n",
+       "     0, 0, 1, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 0, 1, 0, 0,\n",
+       "     1, 0, 1, 1, 1, 0, 1, 1, 0, 0, 1, 1, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 1,\n",
+       "     1, 0, 1, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 1, 0, 1, 1, 0, 1,\n",
+       "     1, 0, 1, 0, 0, 1, 0, 0, 1, 1, 0, 0, 0, 1, 1, 0, 1, 0, 1, 0, 0, 0, 1,\n",
+       "     0, 1, 1, 0, 1, 0, 1, 0, 1, 1, 0, 1, 1, 1, 0, 0, 1, 0, 1, 0, 1, 1, 0,\n",
+       "     1, 0, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 0, 0, 1, 0, 0, 1, 0, 1, 0, 0, 1,\n",
+       "     0, 1, 1, 1, 0, 1, 1, 0, 0, 1, 0, 0, 0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 1,\n",
+       "     1, 1, 0, 0, 1, 1, 0, 1, 1, 0, 0, 1, 0, 1, 0, 0, 1, 1, 0, 0, 1, 0, 0,\n",
+       "     1, 1, 1, 0, 0, 0, 0, 0, 1, 1, 1, 0, 0, 1, 1, 1, 0, 0, 1, 1, 1, 0, 1,\n",
+       "     1, 0, 1, 0, 1, 0, 1, 1, 1, 0, 0, 1, 1, 1, 1, 1, 1, 1, 0, 0, 1, 1, 1,\n",
+       "     1, 1, 1, 0, 1, 1, 1, 0, 1, 0, 0, 0, 1, 0, 1, 1, 0, 0, 1, 1, 0, 1, 1,\n",
+       "     1, 1, 0, 1, 1, 0, 1, 1, 1, 0, 1, 1, 1, 0, 1, 1, 0, 0, 1, 0, 0, 1, 0,\n",
+       "     1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 1, 1, 0, 0, 1,\n",
+       "     1, 1, 1, 1, 0, 1, 1, 1, 0, 1, 0, 1, 0, 0, 1, 0, 0, 1, 1, 0, 0, 0, 1,\n",
+       "     0, 0, 1, 0, 1, 1, 0, 1, 0, 1, 0, 1, 1, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1,\n",
+       "     1, 1, 1, 0, 1, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 1, 0, 1, 1, 0, 1, 1, 0,\n",
+       "     1, 0, 0, 0, 1, 1, 1, 0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 1,\n",
+       "     1, 0, 1, 1, 1, 1, 0, 1, 1, 0, 0, 1, 0, 1, 0, 0, 1, 0, 1, 0, 0, 0, 1,\n",
+       "     1, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 1, 0, 1, 1, 0, 0, 1, 1, 0, 1, 0, 0,\n",
+       "     0, 0, 0, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 0, 0, 0, 0, 0, 0, 1, 0, 1, 0,\n",
+       "     1, 1, 1, 1, 1, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 0, 0, 1, 0, 0, 0, 0, 0,\n",
+       "     0, 0, 1, 1, 0, 0, 1, 0, 1, 0, 0, 1, 1, 0, 1, 0, 1, 0, 0, 1, 0, 1, 0,\n",
+       "     0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 1, 1, 1, 0, 1, 0, 0, 0, 0, 1, 1, 1, 0,\n",
+       "     1, 0, 1, 1, 1, 1, 0, 1, 1, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 1, 1, 1,\n",
+       "     0, 1, 1, 1, 1, 1, 1, 0, 1, 1, 0, 1, 1, 1, 0, 0, 1, 1, 1, 0, 0, 1, 1,\n",
+       "     0, 1, 0, 1, 0, 0, 1, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 0, 1, 1, 1, 0, 0,\n",
+       "     1, 1, 1, 1, 0, 0, 0, 1, 1, 0, 0, 0, 1, 1, 1, 0, 0, 0, 1, 1, 1, 1, 0,\n",
+       "     0, 1, 1, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 1, 0, 1, 1, 0, 0, 1, 1, 0, 1,\n",
+       "     0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 1, 0, 1, 0, 1, 0, 0, 0, 1, 1, 1,\n",
+       "     0, 1, 1, 1, 1, 1, 1, 0, 0, 0, 1, 0, 0, 1, 1, 1, 1, 1, 0, 1, 0, 0, 1,\n",
+       "     0, 0, 1, 0, 1, 0, 0, 1, 0, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 0, 1, 0,\n",
+       "     0, 1, 1, 1, 1, 1, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 1, 0,\n",
+       "     0, 1, 1, 1, 0, 1, 1, 1, 1, 0, 0, 0, 1, 1, 1, 0, 1, 1, 1, 0, 1, 0, 0,\n",
+       "     1, 1, 1, 1, 1, 1, 0, 0, 0, 0, 1, 1, 0, 1, 1, 1, 1, 1, 1, 0, 0, 0, 0,\n",
+       "     1, 0, 1, 0, 1, 0, 0, 0, 1, 1, 0, 1, 0, 0, 0, 0, 0, 0, 1, 1, 1, 0, 1,\n",
+       "     1, 0, 1, 1, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 1, 1, 1, 0, 0, 1,\n",
+       "     0, 0, 0, 1, 1, 0, 0, 1, 1, 0, 1, 1, 1, 0, 0, 0, 0, 0, 1, 1, 0, 1, 0,\n",
+       "     1, 0, 0, 0, 1, 0, 0, 1, 0, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 0, 1, 1, 0,\n",
+       "     1, 1, 0, 1, 0, 0, 1, 1, 0, 1, 1, 0, 1, 0, 0, 0, 0, 1, 0, 1, 0, 1, 1,\n",
+       "     0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 1, 1, 0, 0, 0, 0, 1, 1,\n",
+       "     1, 1, 1, 1, 1, 1, 0, 1, 0, 0, 1, 0, 1, 0, 1, 0, 1, 1, 0, 1, 0, 1, 0,\n",
+       "     1, 0, 0, 1, 0, 0, 1, 0, 0, 0, 0, 1, 1, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1,\n",
+       "     0, 1, 0, 1, 0, 0, 0, 1, 0, 0, 0]], order=2)"
+      ]
+     },
+     "metadata": {},
+     "output_type": "display_data"
+    }
+   ],
+   "source": [
+    "import galois\n",
+    "GF = galois.GF(2)\n",
+    "\n",
+    "N, n = 1000, 1000\n",
+    "\n",
+    "A = GF.Random((N, n))\n",
+    "B = A.null_space()\n",
+    "display(B)"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "3ab95746",
+   "metadata": {},
+   "source": [
+    "#### Cost of integer factorization\n",
+    "\n",
+    "In contrast to the algorithms just discussed, known algorithms for integer factorization are much more expensive — as we might expect from the examples in the first section of the lesson.\n",
+    "\n",
+    "If we consider a simple *trial division* algorithm that searches through the list $2,\\ldots,\\sqrt{N}$ for prime factors of an input number $N$, then we have $\\sqrt{N}$ iterations — which means $O(2^{n/2})$ iterations when $N$ is an $n$-bit number.\n",
+    "Each iteration requires a trial division, which means $O(n^2)$ elementary operations for each iteration (using a standard algorithm for integer division).\n",
+    "So we end up with circuits of size $O(n^2 2^{n/2})$, which is *exponential* in the input size $n$.\n",
+    "\n",
+    "There are algorithms for integer factorization having better scaling.\n",
+    "The number field sieve mentioned earlier, for instance, which is an algorithm that makes use of randomness, is generally believed (but not rigorously proven) to require\n",
+    "\n",
+    "$$\n",
+    "2^{O\\bigl(n^{1/3} \\operatorname{lg}^{2/3}(n)\\bigr)}\n",
+    "$$\n",
+    "\n",
+    "elementary operations to factor $n$-bit integers with high probability.\n",
+    "While it is quite significant that $n$ is raised to the power $1/3$, the fact that $n^{1/3}$ appears in the exponent is still a problem that causes poor scaling — and explains in part why RSA1024 remains outside of its domain of applicability."
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "38d29b7e",
+   "metadata": {},
+   "source": [
+    "#### Polynomial versus exponential cost\n",
+    "\n",
+    "As we have seen, classical algorithms for integer addition, multiplication, division, and computing greatest common divisors allow us to solve these problems in the blink of an eye for inputs with thousands of bits.\n",
+    "Addition has *linear* cost while the other three problems have *quadratic* cost (or *subquadratic* cost using asymptotically fast algorithms).\n",
+    "Modular exponentiation is more expensive but can still be done pretty efficiently, with *cubic* cost (or sub-cubic cost using asymptotically fast algorithms).\n",
+    "Computing the null space modulo $2$ of an $n\\times n$ binary matrix has cost $O(n^3)$ (bearing in mind that there are $n^2$ input bits for this problem).\n",
+    "\n",
+    "All of these algorithms are examples having *polynomial* cost, meaning that they have cost $O(n^c)$ for some choice of a fixed constant $c > 0$.\n",
+    "As a rough, first-order approximation, algorithms having polynomial cost are abstractly viewed as representing *efficient* algorithms.\n",
+    "\n",
+    "In contrast, known classical algorithms for integer factoring have *exponential* cost.\n",
+    "Sometimes the cost of the number field sieve is described as *sub-exponential*, but in complexity theory it is more typical to reserve this term for algorithms whose cost is\n",
+    "\n",
+    "$$\n",
+    "O\\left(2^{n^{\\varepsilon}}\\right)\n",
+    "$$\n",
+    "\n",
+    "for *every* $\\varepsilon > 0$.\n",
+    "\n",
+    "The so-called *NP-complete* problems are a class of problems not known to (and widely conjectured not to) have polynomial-cost algorithms.\n",
+    "A circuit-based formulation of the *exponential-time hypothesis* posits something even stronger, which is that no NP-complete problem can have a sub-exponential cost algorithm.\n",
+    "\n",
+    "The association of polynomial-cost algorithms with efficient algorithms must be understood as being a loose abstraction.\n",
+    "Of course, if an algorithm's cost scales as $n^{1000}$ or $n^{1000000}$ for inputs of size $n$, then it's a stretch to describe that algorithm as efficient.\n",
+    "\n",
+    "However, even an algorithm having cost that scales as $n^{1000000}$ must be doing something clever to avoid having *exponential* cost, which is generally what we expect of algorithms based in some way on \"brute force\" or \"exhaustive search.\"\n",
+    "Even the sophisticated refinements of the number field sieve, for instance, fail to avoid this exponential scaling in cost.\n",
+    "Polynomial-cost algorithms, on the other hand, manage take advantage of the problem structure in some way that avoids an exponential scaling.\n",
+    "\n",
+    "In practice, the identification of a polynomial cost algorithm for a problem is just a first step toward actual efficiency.\n",
+    "Through algorithmic refinements, polynomial-cost algorithms with large exponents can sometimes be improved dramatically, lowering the cost to a more \"reasonable\" polynomial scaling.\n",
+    "Sometimes things become easier when they're known to be possible — so the identification of a polynomial-cost algorithm for a problem can also have the effect of inspiring new, more efficient algorithms.\n",
+    "\n",
+    "As we consider advantages of quantum computing over classical computing, our eyes are generally turned first toward *exponential* advantages, or at least *super-polynomial* advantages — ideally finding polynomial-cost quantum algorithms for problems not known to have polynomial-cost classical algorithms.\n",
+    "Theoretical advantages on this order have the greatest chances to lead to actual practical advantages — but identifying such advantages is an extremely difficult challenge.\n",
+    "Only a few examples are currently known, but the search continues.\n",
+    "\n",
+    "Polynomial (but not super-polynomial) advantages in computational cost of quantum over classical are also interesting and should not be dismissed — but given the current gap between quantum and classical computing technology, they do seem rather less compelling at the present time.\n",
+    "One day, though, they could become significant.\n",
+    "Grover's algorithm, which is covered in Lesson 8, provides an example of a *quadratic* advantage of quantum over classical for so-called *unstructured searching*, which could potentially be broadly applicable."
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "740332ac",
+   "metadata": {},
+   "source": [
+    "#### A hidden cost of circuit computation\n",
+    "\n",
+    "There is one final issue that is worth mentioning, although we will not concern ourselves with it further in this series.\n",
+    "There's a \"hidden\" computational cost when we're working with circuits, and it concerns the specifications of the circuits themselves.\n",
+    "As inputs get longer, we need larger and larger circuits — but somehow we're going to need to get our hands on the descriptions of these circuits if we're going to implement them.\n",
+    "\n",
+    "For all of the examples we've discussed, or will discuss later in the series, there's some underlying algorithm from which the circuits are derived.\n",
+    "Usually the circuits in a family follow some basic pattern that's easy to extrapolate to larger and larger inputs\n",
+    "— such as cascading full adders to create Boolean circuits for addition or performing layers of Hadamard gates and other gates in some simple to describe pattern.\n",
+    "\n",
+    "But what happens if there are prohibitive computational costs associated with the patterns of the circuits themselves?\n",
+    "For instance, the description of each member $C_n$ in a circuit family could, in principle, be determined by some extremely difficult to compute function of $n$.\n",
+    "\n",
+    "The answer is that this is a problem, and we must place additional restrictions on families of circuits beyond having polynomial cost in order for them to truly represent efficient algorithms.\n",
+    "The property of *uniformity* for circuits does this by stipulating that it must (in various precise formulations) be computationally easy to obtain the description of each circuit in a family.\n",
+    "All of the circuit families we'll discuss do indeed have this property, so we won't concern ourselves with it further — but this is an important issue to be aware of when studying circuit models of computation from a formal viewpoint."
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "d15daf3e",
+   "metadata": {},
+   "source": [
+    "## Classical computations on quantum computers\n",
+    "\n",
+    "We'll now turn our attention to implementing classical algorithms on quantum computers.\n",
+    "We'll see that any computation that can be performed with a classical Boolean circuit can also be performed by a quantum circuit with asymptotically the same computational cost.\n",
+    "Moreover, this can be done in a \"clean\" manner, which is an important requirement for using these computations as subroutines inside of larger quantum computations.\n",
+    "\n",
+    "### Simulating Boolean circuits with quantum circuits\n",
+    "\n",
+    "Boolean circuits are composed of AND, OR, NOT, and FANOUT gates.\n",
+    "To simulate Boolean circuit with quantum circuit, we'll begin by showing how each of these four gates can be simulated by quantum gates.\n",
+    "Once that's done, converting a given Boolean circuit to a quantum circuit is a simple matter of simulating one gate at a time.\n",
+    "We'll only need NOT gates, controlled-NOT gates, and Toffoli gates to do this, which are all deterministic operations in addition to being unitary.\n",
+    "\n",
+    "#### Toffoli gates\n",
+    "\n",
+    "Toffoli gates can alternatively be described as controlled-controlled-NOT (or CCX) gates, whose action on standard basis states is as the following figure suggests.\n",
+    "\n",
+    "![Toffoli gate](images/Toffoli-gate.png)\n",
+    "\n",
+    "The matrix representation of this gate is as follows.\n",
+    "\n",
+    "$$\n",
+    "\\mathrm{CCX} =\n",
+    "\\begin{pmatrix}\n",
+    "1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\\\\n",
+    "0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 \\\\\n",
+    "0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 \\\\\n",
+    "0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 \\\\\n",
+    "0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 \\\\\n",
+    "0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 \\\\\n",
+    "0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 \\\\\n",
+    "0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 \n",
+    "\\end{pmatrix}\n",
+    "$$"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "c32ef23e",
+   "metadata": {},
+   "source": [
+    "Another way to think about Toffoli gates is that they're essentially query gates for the AND function, in the sense that they follow exactly the same pattern we saw in the previous lesson for the unitary query gate implementation of arbitrary functions with binary string inputs and outputs.\n",
+    "\n",
+    "Toffoli gates are not included in the default gate set discussed earlier in the lesson, but it is possible to construct a Tofolli gate from $H$, $T$, $T^{\\dagger}$, and CNOT gates as the following Qiskit code cell illustrates."
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 9,
+   "id": "9e36d499",
+   "metadata": {},
+   "outputs": [
+    {
+     "data": {
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+      ],
+      "text/plain": [
+       "<Figure size 1040x284.278 with 1 Axes>"
+      ]
+     },
+     "metadata": {},
+     "output_type": "display_data"
+    },
+    {
+     "data": {
+      "text/latex": [
+       "$$\n",
+       "\n",
+       "\\begin{bmatrix}\n",
+       "1 & 0 & 0 & 0 & 0 & 0 & 0 & 0  \\\\\n",
+       " 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0  \\\\\n",
+       " 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0  \\\\\n",
+       " 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1  \\\\\n",
+       " 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0  \\\\\n",
+       " 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0  \\\\\n",
+       " 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0  \\\\\n",
+       " 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0  \\\\\n",
+       " \\end{bmatrix}\n",
+       "$$"
+      ],
+      "text/plain": [
+       "<IPython.core.display.Latex object>"
+      ]
+     },
+     "execution_count": 9,
+     "metadata": {},
+     "output_type": "execute_result"
+    }
+   ],
+   "source": [
+    "from qiskit import QuantumCircuit\n",
+    "from qiskit.quantum_info import Statevector, Operator\n",
+    "from qiskit.visualization import array_to_latex\n",
+    "\n",
+    "Toffoli = QuantumCircuit(3)\n",
+    "\n",
+    "Toffoli.h(2)\n",
+    "Toffoli.cx(1, 2)\n",
+    "Toffoli.tdg(2)\n",
+    "Toffoli.cx(0, 2)\n",
+    "Toffoli.t(2)\n",
+    "Toffoli.cx(1, 2)\n",
+    "Toffoli.tdg(2)\n",
+    "Toffoli.cx(0, 2)\n",
+    "Toffoli.t(1)\n",
+    "Toffoli.t(2)\n",
+    "Toffoli.cx(0, 1)\n",
+    "Toffoli.t(0)\n",
+    "Toffoli.tdg(1)\n",
+    "Toffoli.h(2)\n",
+    "Toffoli.cx(0, 1)\n",
+    "\n",
+    "display(Toffoli.draw())\n",
+    "U = Operator(Toffoli)\n",
+    "\n",
+    "array_to_latex(U)"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "66991522",
+   "metadata": {},
+   "source": [
+    "#### Simulating Boolean gates with Toffoli, controlled-NOT, and NOT gates\n",
+    "\n",
+    "A single Toffoli gate, used in conjunction with a few NOT gates, can implement an AND and OR gate, as the following diagrams suggest.\n",
+    "\n",
+    "![Simulating AND and OR gates with Toffoli gates](images/AND-and-OR-with-Toffoli.png)\n",
+    "\n",
+    "FANOUT gates can easily be implemented using controlled-NOT gates like this:\n",
+    "\n",
+    "![Simulating FANOUT gates with CNOT gates](images/FANOUT-with-CNOT.png)\n",
+    "\n",
+    "In all three cases, the bits that the AND, OR, and FANOUT gates act upon come in from the left as inputs, and in addition we require one *workspace* qubit initialized to the zero state.\n",
+    "These workspace qubits appear within the dotted rectangles to illustrate that they're new, and one of the costs\n",
+    "of implementing Boolean gates and circuits through the method being described is the inclusion of these new qubits.\n",
+    "\n",
+    "For the AND and OR gates, in addition to the output bits we also have two additional qubits left over.\n",
+    "For example, inside the dotted rectangle in the diagram for an AND gate, the top two qubits are left in the states $\\vert a\\rangle$ and $\\vert b\\rangle$.\n",
+    "These bits appear inside the dotted rectangles to illustrate that they're no longer needed — they aren't output by the Boolean logic gates we're simulating, and we can simply ignore them (for now at least).\n",
+    "\n",
+    "The remaining Boolean gate, the NOT gate, is also a gate in our default set of quantum gates, so we don't need to worry about simulating them in a special way."
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "50f183c2",
+   "metadata": {},
+   "source": [
+    "#### Gate-by-gate simulation of Boolean circuits\n",
+    "\n",
+    "Now suppose that we have an ordinary Boolean circuit named $C$, composed of AND, OR, NOT, and FANOUT gates, and having\n",
+    "$n$ input bits and $m$ of output bits.\n",
+    "We'll let $t$ be the total number of gates in $C$, and we'll give the name $f$ to the function that $C$ computes — so this function takes the form $f:\\Sigma^n\\rightarrow\\Sigma^m$ where $\\Sigma = \\{0,1\\}$.\n",
+    "\n",
+    "Now consider what happens when we go one at a time through the AND, OR, and FANOUT gates of $C$, replacing each one by the simulation described above, including the addition of a workspace qubit.\n",
+    "We'll arrange the qubits of $R$ so that the $n$ input bits of $C$ are on the top and the workspace qubits are on the bottom.\n",
+    "As a result of performing these substitutions, we obtain a new circuit $R$ that operates as this figure suggests:\n",
+    "\n",
+    "![Reversible circuit simulation](images/reversible-circuit-simulation.png)\n",
+    "\n",
+    "Here, $k$ is the number of workspace qubits that need to be added, and $g$ is a function of the form\n",
+    "$g:\\Sigma^n \\rightarrow \\Sigma^{n+k-m}$ that describes the states of the leftover qubits after $R$ is executed.\n",
+    "\n",
+    "In the figure, the qubits corresponding to the output $f(x)$ are on the top and the remaining, leftover qubits storing $g(x)$ are on the bottom.\n",
+    "We can force this to happen if we wish by rearranging the qubits using swap gates.\n",
+    "A swap gate can be implemented with three controlled-NOT gates like this:\n",
+    "\n",
+    "![Swapping with cNOT gates](images/swap.png)\n",
+    "\n",
+    "As we'll see in the next section, it isn't essential to rearrange the output qubits like this, but it's easy enough to do it if we choose.\n",
+    "\n",
+    "The function $g$ that describes the classical states of the leftover qubits is determined somehow by the circuit $C$, but we don't really need to worry all that much about it, because we really don't care specifically what state these qubits are in when the computation finishes.\n",
+    "The letter $g$ comes after $f$, so it is a reasonable name for this function on that account, but there's a better reason to pick the name $g;$ \n",
+    "it's short for *garbage* — or stuff left over from the simulation that's cluttering up these qubits."
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "ca1e5224",
+   "metadata": {},
+   "source": [
+    "### Cleaning up the garbage\n",
+    "\n",
+    "If our only interest is in computing whatever function $f$ is computed by a given Boolean circuit $C$ with a quantum circuit, we don't need to proceed any further than the gate-by-gate simulation just described — and we can accept that in addition to the answer we'll have a bunch of garbage left over.\n",
+    "\n",
+    "This, however, won't be good enough if we want to perform classical computations as subroutines within larger quantum computations, because those garbage qubits will cause problems.\n",
+    "As we saw in the previous lesson, the phenomenon of *interference* is critically important to quantum algorithms, and garbage qubits can ruin the interference patterns needed to make quantum algorithms work.\n",
+    "\n",
+    "Fortunately, it's not to difficult to clean up the garbage, so-to-speak.\n",
+    "The key is to use the fact that because $R$ is a quantum circuit, we can run it in reverse — by simply replacing each gate with its inverse and applying them in the reverse order — thereby obtaining a quantum circuit for the operation $R^{\\dagger}$.\n",
+    "(Toffoli gates, CNOT gates, and NOT gates are their own inverses, so in this case running $R$ in reverse is just a matter of applying the gates in the reverse order — but more generally any quantum circuit can be reversed as just described.)\n",
+    "\n",
+    "So, what we can do is to add $m$ more qubits (recalling that $m$ is the number of output bits of the function $f$), use CNOT gates to copy the output of $R$ onto these qubits, and then reverse $R$ to clean up the garbage.\n",
+    "The following figure illustrates the resulting circuit and describes its action on standard basis states.\n",
+    "\n",
+    "![Garbage-free computation](images/garbage-free-computation.png)\n",
+    "\n",
+    "If we put a box around the entire circuit and call it $Q$, it looks like this:\n",
+    "\n",
+    "![Simulation as a query gate](images/simulation-as-query-gate.png)\n",
+    "\n",
+    "If $C$ has $t$ gates, then $Q$ has $O(t)$ gates.\n",
+    "\n",
+    "If we disregard the $k$ additional workspace qubits, what we have is a circuit $Q$ that functions exactly like a query gate for the function $f$.\n",
+    "If we simply want to compute the function $f$ on some string $x$, we can set $y = 0^m$ and the resulting value $f(x)$ will appear on the bottom $m$ qubits — or we can feed in a different state to the bottom $m$ qubits if we wish\n",
+    "(perhaps to make use of a phase kickback, like in Deutsch's or the Deutsch-Jozsa algorithm).\n",
+    "This means that for any query algorithm, if we have a Boolean circuit that computes the input function, we can replace each query gate with a unitary implementation as just described and the query algorithm will function correctly.\n",
+    "\n",
+    "Note that the workspace qubits are needed to make this process work — but they are, in fact, returned to their initial states once the combined circuit is executed, so they could be used again as workspace qubits for other purposes.\n",
+    "There are strategies that reduce the number of workspace qubits required, but we won't discuss those strategies here."
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "9aca4d6f",
+   "metadata": {},
+   "source": [
+    "#### Implementing invertible functions\n",
+    "\n",
+    "The construction just described allows us to simulate any Boolean circuit with a quantum circuit in a garbage-free manner, following this basic pattern:\n",
+    "if $C$ is a Boolean circuit implementing a function $f:\\Sigma^n \\rightarrow \\Sigma^m$, then \n",
+    "we obtain a quantum circuit $Q$ that operates as the next equation describes on standard basis states.\n",
+    "\n",
+    "$$\n",
+    "Q \\bigl( \\vert y \\rangle \\vert 0^k \\rangle \\vert x\\rangle\\bigr)\n",
+    "= \\vert y \\oplus f(x) \\rangle \\vert 0^k \\rangle \\vert x\\rangle\n",
+    "$$\n",
+    "\n",
+    "The number $k$ indicates how many workspace qubits are required, and depends on how many gates $C$ has and which ones they are.\n",
+    "\n",
+    "This will be enough for the purposes of this series, but it is possible to take this one step further when the \n",
+    "function $f$ itself is invertible.\n",
+    "To be precise, suppose that the function $f$ takes the form $f:\\Sigma^n \\rightarrow \\Sigma^n$, and suppose furthermore there exists a function $f^{-1}$ such that $f^{-1}(f(x)) = x$ for every $x\\in\\Sigma^n$.\n",
+    "(The function $f^{-1}$ is necessarily unique when it exists.)\n",
+    "\n",
+    "This means that the operation that transforms $\\vert x \\rangle$ into $\\vert f(x) \\rangle$ for every $x\\in\\Sigma^n$ is unitary — so we might hope to build a quantum circuit that implements the unitary operation defined by\n",
+    "\n",
+    "$$\n",
+    "U \\vert x \\rangle = \\vert f(x) \\rangle\n",
+    "$$\n",
+    "\n",
+    "for every $x\\in\\Sigma^n$.\n",
+    "To be clear, the fact that this is a unitary operation relies on $f$ being invertible — it's not unitary when $f$ isn't invertible.\n",
+    "Disregarding the workspace qubits, $U$ is different from the operation that the circuit $Q$ implements because we're not keeping a copy of the input around and XORing it to an arbitrary string, we're *replacing* $x$ by $f(x)$.\n",
+    "\n",
+    "So the question is: when $f$ is invertible, can we do this?\n",
+    "The answer is yes — provided that in addition to having a Boolean circuit that computes $f$, we also have one that computes $f^{-1}$.\n",
+    "(So this won't be a shortcut to inverting functions computationally if we don't already know how to do that.)\n",
+    "The following diagram illustrates how it can be done by composing two quantum circuits, $Q_f$ and $Q_{f^{-1}}$, which are obtained individually for the functions $f$ and $f^{-1}$ through the method described above.\n",
+    "\n",
+    "![Simulation of an invertible function](images/fully-reversible-simulation.png)\n"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 10,
+   "id": "afe4325b",
+   "metadata": {},
+   "outputs": [
+    {
+     "data": {
+      "text/html": [
+       "<h3>Version Information</h3><table><tr><th>Qiskit Software</th><th>Version</th></tr><tr><td><code>qiskit-terra</code></td><td>0.24.1</td></tr><tr><td><code>qiskit-aer</code></td><td>0.12.1</td></tr><tr><td><code>qiskit-ibmq-provider</code></td><td>0.20.2</td></tr><tr><td><code>qiskit</code></td><td>0.43.2</td></tr><tr><td><code>qiskit-nature</code></td><td>0.6.0</td></tr><tr><td><code>qiskit-finance</code></td><td>0.3.4</td></tr><tr><td><code>qiskit-optimization</code></td><td>0.5.0</td></tr><tr><td><code>qiskit-machine-learning</code></td><td>0.6.1</td></tr><tr><th>System information</th></tr><tr><td>Python version</td><td>3.10.11</td></tr><tr><td>Python compiler</td><td>Clang 14.0.0 (clang-1400.0.29.202)</td></tr><tr><td>Python build</td><td>main, Apr  7 2023 07:31:31</td></tr><tr><td>OS</td><td>Darwin</td></tr><tr><td>CPUs</td><td>8</td></tr><tr><td>Memory (Gb)</td><td>32.0</td></tr><tr><td colspan='2'>Tue Jul 18 12:32:10 2023 BST</td></tr></table>"
+      ],
+      "text/plain": [
+       "<IPython.core.display.HTML object>"
+      ]
+     },
+     "metadata": {},
+     "output_type": "display_data"
+    }
+   ],
+   "source": [
+    "# pylint: disable=unused-import\n",
+    "import qiskit.tools.jupyter\n",
+    "%qiskit_version_table"
+   ]
+  }
+ ],
+ "metadata": {
+  "kernelspec": {
+   "display_name": "Python 3",
+   "language": "python",
+   "name": "python3"
+  },
+  "language_info": {
+   "codemirror_mode": {
+    "name": "ipython",
+    "version": 3
+   },
+   "file_extension": ".py",
+   "mimetype": "text/x-python",
+   "name": "python",
+   "nbconvert_exporter": "python",
+   "pygments_lexer": "ipython3",
+   "version": "3.9"
+  }
+ },
+ "nbformat": 4,
+ "nbformat_minor": 5
+}
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+{
+ "cells": [
+  {
+   "cell_type": "markdown",
+   "id": "b1cea6ac",
+   "metadata": {},
+   "source": [
+    "# Phase estimation and factoring"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "7dd56c6c",
+   "metadata": {
+    "gloss": {
+     "shor-algorithm-comment": {
+      "text": "In addition to his integer factorization algorithm, Peter Shor also discovered an efficient quantum algorithm for the different but related problem of computing discrete logarithms. The two algorithms appeared in the same paper, first published as a conference paper in 1994 and then as a journal paper in 1997.",
+      "title": "Shor's algorithm"
+     }
+    }
+   },
+   "source": [
+    "## Introduction\n",
+    "\n",
+    "In this lesson we'll discuss the phase estimation problem and a quantum algorithm to solve it.\n",
+    "We'll then apply the solution to obtain [Shor's algorithm](gloss:shor-algorithm-comment) — an efficient quantum algorithm for the integer factorization problem.\n",
+    "\n",
+    "Along the way, we'll encounter the quantum Fourier transform, and we'll see how it can be implemented efficiently by a quantum circuit."
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "bc37795e",
+   "metadata": {},
+   "source": [
+    "## Phase estimation problem\n",
+    "\n",
+    "This section explains the *phase estimation problem*.\n",
+    "We'll begin with a short discussion of the *spectral theorem* from linear algebra, and then move on to a statement of the phase estimation problem itself."
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "db278fe0",
+   "metadata": {},
+   "source": [
+    "### Spectral theorem\n",
+    "\n",
+    "The *spectral theorem* is an important fact in linear algebra that states that matrices of a certain type (called *normal matrices*) can be expressed in a simple and useful way.\n",
+    "We're only going to use this theorem for unitary matrices in this lesson, but later on we'll need it for Hermitian matrices as well.\n",
+    "\n",
+    "#### Normal matrices\n",
+    "\n",
+    "Suppose that $M$ is an $n\\times n$ matrix with complex number entries.\n",
+    "We say that $M$ is a *normal* matrix if it commutes with its conjugate transpose:\n",
+    "$M M^{\\dagger} = M^{\\dagger} M$.\n",
+    "\n",
+    "Every unitary matrix $U$ is normal because\n",
+    "\n",
+    "$$\n",
+    "U U^{\\dagger} = I = U^{\\dagger} U.\n",
+    "$$\n",
+    "\n",
+    "Hermitian matrices, which are matrices that equal their own conjugate transpose, are another important class of normal matrices.\n",
+    "If $H$ is a Hermitian matrix, then\n",
+    "\n",
+    "$$\n",
+    "H H^{\\dagger} = H^2 = H^{\\dagger} H,\n",
+    "$$ \n",
+    "\n",
+    "so $H$ is normal.\n",
+    "We'll see the spectral theorem being applied to Hermitian matrices later in the series.\n",
+    "\n",
+    "Not every square matrix is normal.\n",
+    "For instance, this matrix isn't normal:\n",
+    "\n",
+    "$$\n",
+    "\\begin{pmatrix}\n",
+    "0 & 1\\\\\n",
+    "0 & 0\n",
+    "\\end{pmatrix}\n",
+    "$$\n",
+    "\n",
+    "(This is a simple but great example of a matrix that's always good to keep in mind.)\n",
+    "This matrix isn't normal because\n",
+    "\n",
+    "$$\n",
+    "\\begin{pmatrix}\n",
+    "0 & 1\\\\\n",
+    "0 & 0\n",
+    "\\end{pmatrix}\n",
+    "\\begin{pmatrix}\n",
+    "0 & 1\\\\\n",
+    "0 & 0\n",
+    "\\end{pmatrix}^{\\dagger}\n",
+    "=\n",
+    "\\begin{pmatrix}\n",
+    "0 & 1\\\\\n",
+    "0 & 0\n",
+    "\\end{pmatrix}\n",
+    "\\begin{pmatrix}\n",
+    "0 & 0\\\\\n",
+    "1 & 0\n",
+    "\\end{pmatrix}\n",
+    "=\n",
+    "\\begin{pmatrix}\n",
+    "1 & 0\\\\\n",
+    "0 & 0\n",
+    "\\end{pmatrix}\n",
+    "$$\n",
+    "\n",
+    "while\n",
+    "\n",
+    "$$\n",
+    "\\begin{pmatrix}\n",
+    "0 & 1\\\\\n",
+    "0 & 0\n",
+    "\\end{pmatrix}^{\\dagger}\n",
+    "\\begin{pmatrix}\n",
+    "0 & 1\\\\\n",
+    "0 & 0\n",
+    "\\end{pmatrix}\n",
+    "=\n",
+    "\\begin{pmatrix}\n",
+    "0 & 0\\\\\n",
+    "1 & 0\n",
+    "\\end{pmatrix}\n",
+    "\\begin{pmatrix}\n",
+    "0 & 1\\\\\n",
+    "0 & 0\n",
+    "\\end{pmatrix}\n",
+    "=\n",
+    "\\begin{pmatrix}\n",
+    "0 & 0\\\\\n",
+    "0 & 1\n",
+    "\\end{pmatrix}.\n",
+    "$$\n",
+    "\n",
+    "#### Theorem statement\n",
+    "\n",
+    "Now here's a statement of the spectral theorem.\n",
+    "\n",
+    "<!-- ::: q-block.note -->\n",
+    "\n",
+    "**Theorem** (spectral theorem)\n",
+    "\n",
+    "Let $M$ be a *normal* $N\\times N$ complex matrix.\n",
+    "\n",
+    "There exists an orthonormal basis of $N$ dimensional complex vectors $\\bigl\\{ \\vert\\psi_1\\rangle,\\ldots,\\vert\\psi_N\\rangle \\bigr\\}$ along with complex numbers $\\lambda_1,\\ldots,\\lambda_N$ such that $M = \\lambda_1 \\vert \\psi_1\\rangle\\langle \\psi_1\\vert + \\cdots + \\lambda_N \\vert \\psi_N\\rangle\\langle \\psi_N\\vert$.\n",
+    "\n",
+    "<!-- ::: -->\n",
+    "\n",
+    "The expression of a matrix in the form\n",
+    "\n",
+    "$$\n",
+    "M = \\sum_{k = 1}^N \\lambda_k \\vert \\psi_k\\rangle\\langle \\psi_k\\vert \\tag{1}\n",
+    "$$\n",
+    "\n",
+    "is commonly called a *spectral decomposition*.\n",
+    "Notice that if $M$ is a normal matrix expressed in the form $(1)$, then the equation\n",
+    "\n",
+    "$$ \n",
+    "M \\vert \\psi_j \\rangle = \\lambda_j \\vert \\psi_j \\rangle\n",
+    "$$\n",
+    "\n",
+    "must true for every $j = 1,\\ldots,N$.\n",
+    "This is a consequence of the fact that $\\bigl\\{ \\vert\\psi_1\\rangle,\\ldots,\\vert\\psi_N\\rangle \\bigr\\}$ is orthonormal:\n",
+    "\n",
+    "$$ \n",
+    "M \\vert \\psi_j \\rangle \n",
+    "= \\left(\\sum_{k = 1}^N \\lambda_k \\vert \\psi_k\\rangle\\langle \\psi_k\\vert\\right)\\vert \\psi_j\\rangle\n",
+    "= \\sum_{k = 1}^n \\lambda_k \\vert \\psi_k\\rangle\\langle \\psi_k\\vert\\psi_j \\rangle\n",
+    "= \\lambda_j \\vert\\psi_j \\rangle\n",
+    "$$\n",
+    "\n",
+    "That is to say, each number $\\lambda_j$ is an *eigenvalue* of $M$ and $\\vert\\psi_j\\rangle$ is an *eigenvector* corresponding to that eigenvalue."
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "92aaac48",
+   "metadata": {},
+   "source": [
+    " - **Example 1**.\n",
+    "   Let\n",
+    "   \n",
+    "   $$\n",
+    "   I = \\begin{pmatrix}1 & 0\\\\0 & 1\\end{pmatrix},\n",
+    "   $$\n",
+    "   \n",
+    "   which is normal.\n",
+    "   The theorem implies that $I$ can be written in the form $(1)$ for some choice \n",
+    "   of $\\lambda_1$, $\\lambda_2$, $\\vert\\psi_1\\rangle$, and $\\vert\\psi_2\\rangle$ — and in particular \n",
+    "   the equation is true for\n",
+    "\n",
+    "   $$\n",
+    "   \\lambda_1 = 1, \\hspace{5pt} \n",
+    "   \\lambda_2 = 1, \\hspace{5pt} \n",
+    "   \\vert\\psi_1\\rangle = \\vert 0\\rangle, \\hspace{5pt}\n",
+    "   \\vert\\psi_2\\rangle = \\vert 1\\rangle.\n",
+    "   $$\n",
+    "   \n",
+    "   (Notice that the theorem does not say that the complex numbers $\\lambda_1,\\ldots,\\lambda_n$ are \n",
+    "   distinct — we can have the same complex number repeated.)\n",
+    "   \n",
+    "   These choices work because\n",
+    "   $$\n",
+    "   I = \\vert 0\\rangle\\langle 0\\vert + \\vert 1\\rangle\\langle 1\\vert.\n",
+    "   $$\n",
+    "   \n",
+    "   Indeed, we could choose $\\{\\vert\\psi_1\\rangle,\\vert\\psi_2\\rangle\\}$ to be *any* orthonormal basis and the \n",
+    "   equation will be true. For instance,\n",
+    "   \n",
+    "   $$\n",
+    "   I = \\vert +\\rangle\\langle +\\vert + \\vert -\\rangle\\langle -\\vert.\n",
+    "   $$\n",
+    "\n",
+    " - **Example 2**. Consider a Hadamard operation.\n",
+    "\n",
+    "   $$\n",
+    "   H = \\frac{1}{\\sqrt{2}}\n",
+    "   \\begin{pmatrix}\n",
+    "   1 & 1\\\\[1mm] \n",
+    "   1 & -1\n",
+    "   \\end{pmatrix}\n",
+    "   $$\n",
+    "   \n",
+    "   This is a unitary matrix, so it is normal. The spectral theorem implies that $H$ can be written in the \n",
+    "   form $(1)$, and in particular we have\n",
+    "\n",
+    "   $$\n",
+    "   H = \n",
+    "   \\vert\\psi_{\\pi/8}\\rangle \\langle \\psi_{\\pi/8}\\vert \n",
+    "   - \\vert\\psi_{5\\pi/8}\\rangle \\langle \\psi_{5\\pi/8}\\vert\n",
+    "   $$\n",
+    "   \n",
+    "   where\n",
+    "   \n",
+    "   $$\n",
+    "   \\vert\\psi_{\\theta}\\rangle \n",
+    "   = \\cos(\\theta)\\vert 0\\rangle + \\sin(\\theta) \\vert 1\\rangle.\n",
+    "   $$\n",
+    "\n",
+    "   More explicitly,\n",
+    "   \n",
+    "   $$\n",
+    "   \\begin{aligned}\n",
+    "   \\vert\\psi_{\\pi/8}\\rangle & = \\frac{\\sqrt{2 + \\sqrt{2}}}{2}\\vert 0\\rangle \n",
+    "   + \\frac{\\sqrt{2 - \\sqrt{2}}}{2}\\vert 1\\rangle \\\\[3mm]\n",
+    "   \\vert\\psi_{5\\pi/8}\\rangle & = -\\frac{\\sqrt{2 - \\sqrt{2}}}{2}\\vert 0\\rangle \n",
+    "   + \\frac{\\sqrt{2 + \\sqrt{2}}}{2}\\vert 1\\rangle.\n",
+    "   \\end{aligned}\n",
+    "   $$\n",
+    "\n",
+    "   We can check that this decomposition is correct by performing the required calculations:\n",
+    "   \n",
+    "   $$\n",
+    "   \\vert\\psi_{\\pi/8}\\rangle \\langle \\psi_{\\pi/8}\\vert \n",
+    "   - \\vert\\psi_{5\\pi/8}\\rangle \\langle \\psi_{5\\pi/8}\\vert\n",
+    "   = \\begin{pmatrix}\n",
+    "   \\frac{2 + \\sqrt{2}}{4} & \\frac{\\sqrt{2}}{4}\\\\[2mm]\n",
+    "   \\frac{\\sqrt{2}}{4} & \\frac{2 - \\sqrt{2}}{4}\n",
+    "   \\end{pmatrix}\n",
+    "   -\n",
+    "   \\begin{pmatrix}\n",
+    "   \\frac{2 - \\sqrt{2}}{4} & -\\frac{\\sqrt{2}}{4}\\\\[2mm]\n",
+    "   -\\frac{\\sqrt{2}}{4} & \\frac{2 + \\sqrt{2}}{4}\n",
+    "   \\end{pmatrix}\n",
+    "   = H.\n",
+    "   $$\n",
+    "   \n",
+    "   We can also use Qiskit to check that this decomposition is correct, as the following code cell demonstrates."
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 1,
+   "id": "69b7665f",
+   "metadata": {},
+   "outputs": [
+    {
+     "data": {
+      "text/latex": [
+       "$$\n",
+       "\n",
+       "\\begin{bmatrix}\n",
+       "\\frac{\\sqrt{2}}{2} & \\frac{\\sqrt{2}}{2}  \\\\\n",
+       " \\frac{\\sqrt{2}}{2} & - \\frac{\\sqrt{2}}{2}  \\\\\n",
+       " \\end{bmatrix}\n",
+       "$$"
+      ],
+      "text/plain": [
+       "<IPython.core.display.Latex object>"
+      ]
+     },
+     "execution_count": 1,
+     "metadata": {},
+     "output_type": "execute_result"
+    }
+   ],
+   "source": [
+    "from qiskit.visualization import array_to_latex\n",
+    "from qiskit.quantum_info import Statevector\n",
+    "from qiskit.quantum_info.operators import Operator\n",
+    "import math\n",
+    "\n",
+    "psi1 = Statevector([math.cos(math.pi/8),math.sin(math.pi/8)])\n",
+    "psi2 = Statevector([math.cos(5*math.pi/8),math.sin(5*math.pi/8)])\n",
+    "\n",
+    "# When given a Statevector input, the Operator function returns the outer\n",
+    "# product of that state vector with itself — or, in other words, the\n",
+    "# product of the vector times its conjugate transpose.\n",
+    "\n",
+    "H = Operator(psi1) - Operator(psi2)\n",
+    "array_to_latex(H)"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "d37cf02d",
+   "metadata": {},
+   "source": [
+    "As the first example reveals, there can be some freedom in how eigenvectors are selected.\n",
+    "There is, however, no freedom at all in how the eigenvalues are chosen (except for their ordering).\n",
+    "For any given matrix $M$, the same $n$ complex numbers $\\lambda_1,\\ldots,\\lambda_N$ (which can include repetitions of the same complex number) will always occur in the equation $(1)$.\n",
+    "\n",
+    "Now let's focus in on unitary matrices.\n",
+    "Using the same terminology that was just mentioned, if we have a complex number $\\lambda$ and a non-zero vector $\\vert\\psi\\rangle$ that satisfy the equation\n",
+    "\n",
+    "$$\n",
+    "U\\vert\\psi\\rangle = \\lambda\\vert\\psi\\rangle,\n",
+    "\\tag{2}\n",
+    "$$\n",
+    "\n",
+    "then we say that $\\lambda$ is an *eigenvalue* of $U$ and $\\vert\\psi\\rangle$ is an *eigenvector* corresponding to the eigenvalue $\\lambda$.\n",
+    "\n",
+    "Unitary matrices preserve Euclidean norm.\n",
+    "So, if the equation $(2)$ is true, then because $U$ is unitary we conclude\n",
+    "\n",
+    "$$\n",
+    "\\bigl\\| \\vert\\psi\\rangle \\bigr\\|\n",
+    "= \\bigl\\| U \\vert\\psi\\rangle \\bigr\\|\n",
+    "= \\bigl\\| \\lambda \\vert\\psi\\rangle \\bigr\\|\n",
+    "= \\vert \\lambda \\vert \\bigl\\| \\vert\\psi\\rangle \\bigr\\|.\n",
+    "$$\n",
+    "\n",
+    "The condition that $\\vert\\psi\\rangle$ is non-zero implies that $\\bigl\\| \\vert\\psi\\rangle \\bigr\\|\\not=0$, so we can cancel it from both sides to obtain\n",
+    "\n",
+    "$$\n",
+    "\\vert \\lambda \\vert = 1.\n",
+    "$$\n",
+    "\n",
+    "So, eigenvalues of unitary matrices must always have absolute value equal to one.\n",
+    "That is to say, they lie on the *unit circle* \n",
+    "\n",
+    "$$\n",
+    "\\mathbb{T} = \\{\\alpha\\in\\mathbb{C} : \\vert\\alpha\\vert = 1\\}.\n",
+    "$$\n",
+    "\n",
+    "The symbol $\\mathbb{T}$ is a common name for the complex unit circle.\n",
+    "The name is $S^1$ is also common."
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "33b0a480",
+   "metadata": {},
+   "source": [
+    "### Problem statement\n",
+    "\n",
+    "In the *phase estimation problem*, we're given a quantum state $\\vert \\psi\\rangle$ of $n$ qubits, along with a unitary quantum circuit that acts on $n$ qubits.\n",
+    "We're *promised* that $\\vert \\psi\\rangle$ is an eigenvector of the unitary matrix $U$ that describes the action of the circuit, and our goal is to either identify or approximate the eigenvalue $\\lambda$ to which $\\vert \\psi\\rangle$ corresponds.\n",
+    "\n",
+    "More precisely, because $\\lambda$ lies on the complex unit circle we can write\n",
+    "\n",
+    "$$\n",
+    "\\lambda = e^{2\\pi i \\theta}\n",
+    "$$\n",
+    "\n",
+    "for a unique real number $\\theta$ satisfying $0\\leq\\theta<1$.\n",
+    "The goal of the problem is to compute or approximate this real number $\\theta$.\n",
+    "\n",
+    "<!-- ::: q-block.note -->\n",
+    "\n",
+    "**Phase estimation problem**\n",
+    "\n",
+    "Input: An $n$ qubit quantum state $\\vert\\psi\\rangle$ and a unitary quantum circuit for an $n$-qubit operation $U$\n",
+    "\n",
+    "Promise: $\\vert\\psi\\rangle$ is an eigenvector of $U$\n",
+    "\n",
+    "Output: an approximation to the number $\\theta\\in[0,1)$ satisfying $U\\vert\\psi\\rangle = e^{2\\pi i \\theta}\\vert\\psi\\rangle$\n",
+    "\n",
+    "<!-- ::: -->\n",
+    "\n",
+    "\n",
+    "#### Remarks\n",
+    "\n",
+    "1. The phase estimation problem is different from other problems we've seen so far in the series in that the input includes a quantum state. Typically we're focused on problems having classical inputs and outputs, but nothing prevents us from considering quantum state inputs like this. In terms of its practical relevance, the phase estimation problem tends to appear as a *subproblem* inside of larger computations. We'll see this happening in the context of integer factorization later in the lesson.\n",
+    "\n",
+    "2. The statement of the phase estimation problem above isn't specific about what constitutes an approximation of $\\theta$, but we can formulate more precise problem statements depending on our needs and interests. In the context of integer factorization we'll demand a very precise approximation to $\\theta$, but in other cases we might be satisfied with a very rough approximation. We'll discuss shortly how the precision we require affects the computational cost of a solution.\n",
+    "\n",
+    "3. Notice that as we go from $\\theta = 0$ toward $\\theta = 1$ in the phase estimation problem, we're going all the way around the unit circle, starting from $e^{2\\pi i \\cdot 0} = 1$ and moving counter-clockwise — and as we get closer to $\\theta = 1$ we're moving toward $e^{2\\pi i \\cdot 1} = 1$, which is back where we started at $\\theta = 0$. So, as we consider the accuracy of approximations, choices of $\\theta$ near $1$ should be considered as being near $0$. For example, an approximation $\\theta = 0.999$ should be considered as being within $1/1000$ of $\\theta = 0$."
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "20118913",
+   "metadata": {},
+   "source": [
+    "## Phase estimation procedure\n",
+    "\n",
+    "Next we'll discuss the *phase-estimation procedure*, which is a quantum algorithm for solving the phase estimation problem.\n",
+    "We'll begin with a low-precision warm-up, which explains some of the basic intuition behind the method.\n",
+    "We'll then talk about the *quantum Fourier transform*, which is an important quantum operation used in the phase-estimation procedure, as well as its quantum circuit implementation.\n",
+    "And finally, we'll describe the phase-estimation procedure in general and analyze its performance."
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "014a6154",
+   "metadata": {},
+   "source": [
+    "### Warm-up: approximating phases with low precision\n",
+    "\n",
+    "We'll begin with a warm-up: a couple of simple versions of the phase-estimation procedure that provide low-precision solutions to the phase-estimation problem.\n",
+    "This is helpful for explaining the intuition behind the general procedure that we'll see a bit later in the lesson."
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "3a25f7b4",
+   "metadata": {},
+   "source": [
+    "#### Using the phase kickback\n",
+    "\n",
+    "A simple approach to the phase-estimation problem, which allows us to learn something about the value $\\theta$ we seek, is based on the *phase kick-back* phenomenon.\n",
+    "As we will see, this is essentially a single-qubit version of the general phase-estimation procedure to be discussed later in the lesson.\n",
+    "\n",
+    "As part of the input to the phase estimation problem, we have a unitary quantum circuit for the operation $U$.\n",
+    "We can use the description of this circuit to create a circuit for a *controlled*-$U$ operation, which can be depicted as this figure suggests (with the operation $U$, viewed as a quantum gate, on the left and a controlled-$U$ operation on the right).\n",
+    "\n",
+    "![Uncontrolled and controlled versions of a unitary operation](images/uncontrolled-and-controlled-unitary.png)\n",
+    "\n",
+    "We can create a quantum circuit for a controlled-$U$ operation by first adding a control qubit to the circuit for $U$, and then replacing every gate in the circuit for $U$ with a controlled version of that gate — so our one new control qubit effectively controls every single gate in the circuit for $U$.\n",
+    "This requires that we have a controlled version of every gate in our circuit, but if we want to restrict ourselves to a standard gate set, we can always build circuits for these controlled operations rather than insisting that they're single gates.\n",
+    "\n",
+    "Now let's consider the following circuit, where the input state $\\vert\\psi\\rangle$ of all the qubits except the top one is the quantum state eigenvector of $U:$\n",
+    "\n",
+    "![A single-qubit circuit for phase-estimation](images/estimate-phase-with-kickback.png)\n",
+    "\n",
+    "The eigenvalue of $U$ corresponding to the eigenvector $\\vert\\psi\\rangle$ determines the measurement outcome probabilities.\n",
+    "To see exactly how, let's analyze the circuit.\n",
+    "\n",
+    "![States of a single-qubit circuit for phase-estimation](images/estimate-phase-with-kickback-analysis.png)\n",
+    "\n",
+    "The initial state of the circuit is\n",
+    "\n",
+    "$$\n",
+    "\\vert\\pi_0\\rangle = \\vert\\psi\\rangle \\vert 0\\rangle\n",
+    "$$\n",
+    "\n",
+    "and the first Hadamard gate transforms this state to\n",
+    "\n",
+    "$$\n",
+    "\\vert\\pi_1\\rangle = \\vert\\psi\\rangle \\vert +\\rangle \n",
+    "= \\frac{1}{\\sqrt{2}} \\vert\\psi\\rangle \\vert 0\\rangle + \\frac{1}{\\sqrt{2}} \\vert\\psi\\rangle \\vert 1\\rangle.\n",
+    "$$\n",
+    "\n",
+    "Next, the controlled-$U$ operation is performed, which results in the state\n",
+    "\n",
+    "$$\n",
+    "\\vert\\pi_2\\rangle \n",
+    "= \\frac{1}{\\sqrt{2}} \\vert\\psi\\rangle \\vert 0\\rangle + \\frac{1}{\\sqrt{2}} \\bigl(U \\vert\\psi\\rangle\\bigr) \\vert 1\\rangle.\n",
+    "$$\n",
+    "\n",
+    "Using the assumption that $\\vert\\psi\\rangle$ is an eigenvector of $U$ having eigenvalue $\\lambda = e^{2\\pi i\\theta}$,\n",
+    "we can alternatively express this state as follows.\n",
+    "\n",
+    "$$\n",
+    "\\vert\\pi_2\\rangle\n",
+    "= \\frac{1}{\\sqrt{2}} \\vert\\psi\\rangle \\vert 0\\rangle + \\frac{e^{2\\pi i \\theta}}{\\sqrt{2}} \\vert\\psi\\rangle \\vert 1\\rangle = \\vert\\psi\\rangle \\otimes \\left( \\frac{1}{\\sqrt{2}} \\vert 0\\rangle + \\frac{e^{2\\pi i \\theta}}{\\sqrt{2}} \\vert 1\\rangle\\right)\n",
+    "$$\n",
+    "\n",
+    "Here we see the phase kickback phenomenon taking place.\n",
+    "It is slightly different this time than it was for Deutsch's algorithm and the Deutsch-Jozsa algorithm because we're not working with a query gate — but the idea is the similar.\n",
+    "\n",
+    "Finally, the second Hadamard gate is performed, which results in the state\n",
+    "\n",
+    "$$\n",
+    "\\vert\\pi_3\\rangle\n",
+    "= \\vert\\psi\\rangle \\otimes \\left( \\frac{1+ e^{2\\pi i \\theta}}{2} \\vert 0\\rangle + \\frac{1 - e^{2\\pi i \\theta}}{2} \\vert 1\\rangle\\right).\n",
+    "$$\n",
+    "\n",
+    "The measurement therefore yields the outcomes $0$ and $1$ with these probabilities:\n",
+    "\n",
+    "$$\n",
+    "\\begin{aligned}\n",
+    "p_0 &= \\left\\vert \\frac{1+ e^{2\\pi i \\theta}}{2} \\right\\vert^2 = \\cos^2(\\pi\\theta)\\\\[1mm]\n",
+    "p_1 &= \\left\\vert \\frac{1- e^{2\\pi i \\theta}}{2} \\right\\vert^2 = \\sin^2(\\pi\\theta).\n",
+    "\\end{aligned}\n",
+    "$$"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "1f287001",
+   "metadata": {},
+   "source": [
+    "Here's a table that lists the probabilities for the two possible measurement outcomes for various choices of the number $\\theta$.\n",
+    "\n",
+    "| $\\theta$ | $\\cos^2(\\pi\\theta)$ | $\\sin^2(\\pi\\theta)$ |\n",
+    "| :--: | :--: | :--: |\n",
+    "| 0.0000 | 1.0000 | 0.0000 |\n",
+    "| 0.1250 | 0.8536 | 0.1464 |\n",
+    "| 0.2500 | 0.5000 | 0.5000 |\n",
+    "| 0.3750 | 0.1464 | 0.8536 |\n",
+    "| 0.5000 | 0.0000 | 1.0000 |\n",
+    "| 0.6250 | 0.1464 | 0.8536 |\n",
+    "| 0.7500 | 0.5000 | 0.5000 |\n",
+    "| 0.8750 | 0.8536 | 0.1464 |\n",
+    "\n",
+    "We can also plot the probabilities for the two possible outcomes, $0$ and $1$, as follows:\n",
+    "\n",
+    "![Outcome probabilities from phase kickback](images/kickback-probabilities.png)\n",
+    "\n",
+    "Naturally, the two probabilities always sum to $1$.\n",
+    "Notice that when $\\theta = 0$, the measurement outcome is always $0$, and when $\\theta = 1/2$, the measurement outcome is always $1$.\n",
+    "So, although the measurement result doesn't reveal exactly what $\\theta$ is, it does provide us with some information about it — and if we were promised that either $\\theta = 0$ or $\\theta = 1/2$, we could learn from the circuit which one is correct without error."
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "103ae33e",
+   "metadata": {
+    "gloss": {
+     "binary-point": {
+      "text": "The binary point is analogous to the decimal point when we express numbers with a fractional part in ordinary, decimal notation.",
+      "title": "Binary point"
+     }
+    }
+   },
+   "source": [
+    "Intuitively speaking, we can think of the circuit's measurement outcome as being a guess for $\\theta$ to \"one bit of accuracy.\"\n",
+    "In other words, if we were to write $\\theta$ in *binary notation* and round it off to one bit after the [binary point](gloss:binary-point), we'd have a number like this:\n",
+    "\n",
+    "$$\n",
+    "0.a = \\begin{cases}\n",
+    "0 & a = 0\\\\\n",
+    "\\frac{1}{2} & a = 1.\n",
+    "\\end{cases}\n",
+    "$$\n",
+    "\n",
+    "The measurement outcome can be viewed as a guess for the bit $a$.\n",
+    "When $\\theta$ is neither $0$ nor $1/2$, there's a nonzero probability that the guess will be wrong — but the \n",
+    "probability of making an error becomes smaller and smaller as we get closer to $0$ or $1/2$.\n",
+    "\n",
+    "It's natural to ask what role the two Hadamard gates play in this procedure:\n",
+    "\n",
+    " - The first Hadamard gate sets the control qubit to a uniform superposition of $\\vert 0\\rangle$ and $\\vert 1\\rangle$, so that when the phase kickback occurs, it happens for the $\\vert 1\\rangle$ state and not the $\\vert 0\\rangle$ state, creating a *relative* phase difference that affects the measurement outcomes. If we didn't do this and the phase kickback produced a *global* phase, it would have no effect on the probabilities of obtaining different measurement outcomes.\n",
+    "\n",
+    " - The second Hadamard gate allows us to learn something about the number $\\theta$ through the phenomenon of *interference*. Prior to the second Hadamard gate, the state of the top qubit is\n",
+    "\n",
+    "   $$\n",
+    "   \\frac{1}{\\sqrt{2}} \\vert 0\\rangle + \\frac{e^{2\\pi i \\theta}}{\\sqrt{2}} \\vert 1\\rangle,\n",
+    "   $$\n",
+    "\n",
+    "   and if we were to measure this state, we would obtain $0$ and $1$ each with probability $1/2$ — which tells us nothing at all about $\\theta$. Performing the second Hadamard gate allows the number $\\theta$ to affect the output probabilities."
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "4eae75c7",
+   "metadata": {},
+   "source": [
+    "#### Qiskit implementation\n",
+    "\n",
+    "Here's an implementation of this circuit in Qiskit.\n",
+    "(For this implementation we're using a phase gate for the unitary operation, just in the interest of simplicity, so the relevant eigenvector is the $\\vert 1\\rangle$ state.)"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 2,
+   "id": "5d1a710b",
+   "metadata": {},
+   "outputs": [
+    {
+     "data": {
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+      ],
+      "text/plain": [
+       "<Figure size 788.961x284.278 with 1 Axes>"
+      ]
+     },
+     "execution_count": 2,
+     "metadata": {},
+     "output_type": "execute_result"
+    }
+   ],
+   "source": [
+    "from math import pi, cos, sin\n",
+    "from qiskit import QuantumCircuit\n",
+    "\n",
+    "theta = 0.7 # Can be changed to any value between 0 and 1\n",
+    "\n",
+    "qc = QuantumCircuit(2, 1)\n",
+    "\n",
+    "# Prepare eigenvector, which is the |1> state\n",
+    "\n",
+    "qc.x(1)\n",
+    "qc.barrier()\n",
+    "\n",
+    "# Implement the estimation procedure\n",
+    "qc.h(0)\n",
+    "qc.cp(2*pi*theta, 0, 1)\n",
+    "qc.h(0)\n",
+    "qc.barrier()\n",
+    "\n",
+    "# Perform the final measurement\n",
+    "qc.measure(0, 0)\n",
+    "\n",
+    "# Draw the circuit\n",
+    "qc.draw()"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "fbb2e85d",
+   "metadata": {},
+   "source": [
+    "Now we'll run the circuit using the `Sampler` primitive."
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 3,
+   "id": "50c7117b",
+   "metadata": {},
+   "outputs": [
+    {
+     "data": {
+      "text/plain": [
+       "SamplerResult(quasi_dists=[{0: 0.3454915028125262, 1: 0.6545084971874736}], metadata=[{}])"
+      ]
+     },
+     "execution_count": 3,
+     "metadata": {},
+     "output_type": "execute_result"
+    }
+   ],
+   "source": [
+    "from qiskit.primitives import Sampler\n",
+    "Sampler().run(qc).result()"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "89543d57",
+   "metadata": {},
+   "source": [
+    "We can now compare the results to the predicted values to see that they're correct."
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 4,
+   "id": "c011fc3d",
+   "metadata": {},
+   "outputs": [
+    {
+     "data": {
+      "text/plain": [
+       "{0: 0.34549150281252616, 1: 0.6545084971874737}"
+      ]
+     },
+     "execution_count": 4,
+     "metadata": {},
+     "output_type": "execute_result"
+    }
+   ],
+   "source": [
+    "{   # Calculate predicted results\n",
+    "    0: cos(pi * theta) ** 2, \n",
+    "    1: sin(pi * theta) ** 2\n",
+    "}"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "7d78c61f",
+   "metadata": {},
+   "source": [
+    "<br>\n",
+    "\n",
+    "#### Doubling the phase"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "6c7c3d3c",
+   "metadata": {},
+   "source": [
+    "The circuit described above uses the phase kickback phenomenon to approximate $\\theta$ to a single bit of accuracy.\n",
+    "One bit of accuracy may be all we need in some situations — but for factoring we're going to need a lot more accuracy than that.\n",
+    "The natural question is, how can we learn more about $\\theta?$\n",
+    "\n",
+    "One simple thing we could do is to replace the controlled-$U$ operation in our circuit with *two copies* of this operation, like in this circuit:\n",
+    "\n",
+    "![Single-bit phase estimation doubled](images/double-phase-kickback.png)\n",
+    "\n",
+    "Two copies of a controlled-$U$ operation is equivalent to a controlled-$U^2$ operation.\n",
+    "If $\\vert\\psi\\rangle$ is an eigenvector of $U$ having eigenvalue $\\lambda = e^{2\\pi i \\theta}$, then this state is also an eigenvector of $U^2$, this time having eigenvalue $\\lambda^2 = e^{2\\pi i (2\\theta)}$.\n",
+    "\n",
+    "So, if we run this version of the circuit, we're effectively performing the same computation as before, except that the number $\\theta$ is replaced by $2\\theta$.\n",
+    "The following plot illustrates the output probabilities as $\\theta$ ranges from $0$ to $1$.\n",
+    "\n",
+    "![Outcome probabilities from double-phase kickback](images/double-kickback-probabilities.png)\n",
+    "\n",
+    "Doing this can indeed provide us with some additional information about $\\theta$.\n",
+    "If the binary representation of $\\theta$ is\n",
+    "\n",
+    "$$\n",
+    "\\theta = 0.a_1 a_2 a_3\\cdots\n",
+    "$$\n",
+    "\n",
+    "then doubling $\\theta$ effectively shifts the binary point one position to the right:\n",
+    "\n",
+    "$$\n",
+    "2\\theta = a_1. a_2 a_3\\cdots\n",
+    "$$\n",
+    "\n",
+    "And because we're equating $\\theta = 1$ with $\\theta = 0$ as we move around the unit circle, we see that the bit $a_1$ has no influence on our probabilities — so we're effectively obtaining a guess for what we would get for the *second* bit after the binary point if we were to round $\\theta$ to two bits.\n",
+    "For instance, if we knew in advance that $\\theta$ was either $0$ or $1/4$, then we could fully trust the measurement outcome to tell us which.\n",
+    "\n",
+    "It's not immediately clear, though, how this estimation should be reconciled with what we learned from the original (non-doubled) phase kickback circuit to give us the most accurate information possible about $\\theta$.\n",
+    "So let's take a step back and consider how to proceed."
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "7e1f8f05",
+   "metadata": {},
+   "source": [
+    "#### Two-qubit phase estimation"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "634e708b",
+   "metadata": {},
+   "source": [
+    "Rather than considering the two options described above separately, let's combine them into a single circuit like this:\n",
+    "\n",
+    "![The initial set-up for phase estimation with two qubits](images/two-bit-phase-estimation-initial.png)\n",
+    "\n",
+    "The Hadamard gates after the controlled operations have been removed and there are no measurements here yet.\n",
+    "We'll add more to the circuit as we consider our options for learning as much as we can about $\\theta$.\n",
+    "\n",
+    "If we run this circuit for $\\vert\\psi\\rangle$ being an eigenvector of $U$, the state of the bottom qubits will remain $\\vert\\psi\\rangle$ throughout the entire circuit — and phases will be \"kicked\" into the state of the top two qubits.\n",
+    "Let's analyze the circuit carefully, by means of the following figure.\n",
+    "\n",
+    "\n",
+    "![States for phase estimation with two qubits](images/two-bit-phase-estimation-states.png)\n",
+    "\n",
+    "We can write the state $\\vert\\pi_1\\rangle$ like this:\n",
+    "\n",
+    "$$\n",
+    "\\vert\\pi_1\\rangle = \\vert \\psi\\rangle \\otimes \\frac{1}{2} \\sum_{a_0 = 0}^1 \\sum_{a_1 = 0}^1 \\vert a_1 a_0 \\rangle.\n",
+    "$$\n",
+    "\n",
+    "When the first controlled-$U$ operation is performed, the eigenvalue $\\lambda = e^{2\\pi i\\theta}$ gets kicked into the phase when $a_0$ (the top qubit) is equal to $1$, but not when it's $0$.\n",
+    "So, we can express the resulting state like this:\n",
+    "\n",
+    "$$\n",
+    "\\vert\\pi_2\\rangle \n",
+    "= \\vert\\psi\\rangle \\otimes \\frac{1}{2} \\sum_{a_0=0}^1 \\sum_{a_1=0}^1 e^{2 \\pi i a_0 \\theta} \\vert a_1 a_0 \\rangle.\n",
+    "$$\n",
+    "\n",
+    "The second and third controlled-$U$ gates do something similar, except for $a_1$ rather than $a_0$, and with $\\theta$ replaced by $2\\theta$.\n",
+    "We can express the resulting state like this:\n",
+    "\n",
+    "$$\n",
+    "\\vert\\pi_3\\rangle\n",
+    "= \\vert\\psi\\rangle\\otimes\\frac{1}{2}\\sum_{a_0 = 0}^1 \\sum_{a_1 = 0}^1 \n",
+    "e^{2\\pi i (2 a_1 + a_0)\\theta} \\vert a_1 a_0 \\rangle.\n",
+    "$$\n",
+    "\n",
+    "If we think about the binary string $a_1 a_0$ as representing an integer $x \\in \\{0,1,2,3\\}$ in binary notation, which is $x = 2 a_1 + a_0$, we can alternatively express this state as follows:\n",
+    "\n",
+    "$$\n",
+    "\\vert\\pi_3\\rangle \n",
+    "= \\vert \\psi\\rangle \\otimes \\frac{1}{2} \\sum_{x = 0}^3 e^{2\\pi i x \\theta} \\vert x \\rangle.\n",
+    "$$\n",
+    "\n",
+    "Our goal is to extract as much information about $\\theta$ as we can from this state."
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "a9b61f7b",
+   "metadata": {},
+   "source": [
+    "At this point, we'll consider a special case, where we're promised that $\\theta = \\frac{y}{4}$ for some integer $y\\in\\{0,1,2,3\\}$. \n",
+    "In other words, we have $\\theta\\in \\{0, 1/4, 1/2, 3/4\\}$, so we can express this number exactly using binary notation with two bits, as $.00$, $.01$, $.10$, or $.11$.\n",
+    "In general, $\\theta$ might not be one of these four values — but thinking about this special case will help us to figure out how to most effectively extract information about this value.\n",
+    "\n",
+    "Let's define one two-qubit state vector for each possible value $y \\in \\{0, 1, 2, 3\\}$.\n",
+    "\n",
+    "$$\n",
+    "\\vert \\phi_y\\rangle = \\frac{1}{2} \\sum_{x = 0}^3 e^{2\\pi i x \\bigl(\\frac{y}{4}\\bigr)} \\vert x \\rangle = \\frac{1}{2} \\sum_{x = 0}^3 e^{2\\pi i \\frac{x y}{4}} \\vert x \\rangle\n",
+    "$$\n",
+    "\n",
+    "After simplifying the exponentials, we can write these vectors as follows:\n",
+    "\n",
+    "$$\n",
+    "\\begin{aligned}\n",
+    "\\vert\\phi_0\\rangle & = \\frac{1}{2} \\vert 0 \\rangle + \\frac{1}{2} \\vert 1 \\rangle + \\frac{1}{2} \\vert 2 \\rangle + \\frac{1}{2} \\vert 3 \\rangle \\\\[2mm]\n",
+    "\\vert\\phi_1\\rangle & = \\frac{1}{2} \\vert 0 \\rangle + \\frac{i}{2} \\vert 1 \\rangle - \\frac{1}{2} \\vert 2 \\rangle - \\frac{i}{2} \\vert 3 \\rangle \\\\[2mm]\n",
+    "\\vert\\phi_2\\rangle & = \\frac{1}{2} \\vert 0 \\rangle - \\frac{1}{2} \\vert 1 \\rangle + \\frac{1}{2} \\vert 2 \\rangle - \\frac{1}{2} \\vert 3 \\rangle \\\\[2mm]\n",
+    "\\vert\\phi_3\\rangle & = \\frac{1}{2} \\vert 0 \\rangle - \\frac{i}{2} \\vert 1 \\rangle - \\frac{1}{2} \\vert 2 \\rangle + \\frac{i}{2} \\vert 3 \\rangle\n",
+    "\\end{aligned}\n",
+    "$$\n",
+    "\n",
+    "These vectors are orthogonal: if we choose any pair of them and compute their inner product, we get $0$.\n",
+    "Each one is also a unit vector, so this implies that $\\{\\vert\\phi_0\\rangle, \\vert\\phi_1\\rangle, \\vert\\phi_2\\rangle, \\vert\\phi_3\\rangle\\}$ is an orthonormal basis.\n",
+    "So, we know right away that there is a measurement that can discriminate them perfectly — meaning that if we're given one of them but we don't know which, then we can figure out which one it is without error.\n",
+    "\n",
+    "To perform such a discrimination with a quantum circuit, let's first define a unitary operation $V$ that transforms standard basis states into the four states listed above:\n",
+    "\n",
+    "$$\n",
+    "\\begin{aligned}\n",
+    "V \\vert 00 \\rangle & = \\vert\\phi_0\\rangle \\\\\n",
+    "V \\vert 01 \\rangle & = \\vert\\phi_1\\rangle \\\\\n",
+    "V \\vert 10 \\rangle & = \\vert\\phi_2\\rangle \\\\\n",
+    "V \\vert 11 \\rangle & = \\vert\\phi_3\\rangle \\\\\n",
+    "\\end{aligned}\n",
+    "$$\n",
+    "\n",
+    "To write down $V$ as a $4\\times 4$ matrix, it's just a matter of taking the columns of $V$ to be the states $\\vert\\phi_0\\rangle,\\ldots,\\vert\\phi_3\\rangle$.\n",
+    "\n",
+    "$$\n",
+    "V =\n",
+    "\\frac{1}{2}\n",
+    "\\begin{pmatrix}\n",
+    "1 & 1 & 1 & 1\\\\\n",
+    "1 & i & -1 & -i\\\\\n",
+    "1 & -1 & 1 & -1\\\\\n",
+    "1 & -i & -1 & i\n",
+    "\\end{pmatrix}\n",
+    "$$\n",
+    "\n",
+    "We saw this matrix as an example at the end of Lesson 1.\n",
+    "It's a special matrix, and some readers will have seen it before:\n",
+    "it's the matrix associated with the $4$-dimensional *discrete Fourier transform*.\n",
+    "\n",
+    "In light of this fact, let us call it by the name $\\mathrm{QFT}_4$ rather than $V$.\n",
+    "The name $\\mathrm{QFT}$ is short for *quantum Fourier transform* — which is essentially just the discrete Fourier transform, viewed as a quantum operation.\n",
+    "We'll discuss the quantum Fourier transform in greater detail and generality shortly.\n",
+    "\n",
+    "$$\n",
+    "\\mathrm{QFT}_4 =\n",
+    "\\frac{1}{2}\n",
+    "\\begin{pmatrix}\n",
+    "1 & 1 & 1 & 1\\\\\n",
+    "1 & i & -1 & -i\\\\\n",
+    "1 & -1 & 1 & -1\\\\\n",
+    "1 & -i & -1 & i\n",
+    "\\end{pmatrix}\n",
+    "$$\n",
+    "\n",
+    "So, the operation $\\mathrm{QFT}_4$ maps standard basis states to the four possible states we have above.\n",
+    "We can perform this operation in reverse to go the other way, to transform the states $\\vert\\phi_0\\rangle,\\ldots,\\vert\\phi_3\\rangle$ into the standard basis states $\\vert 0\\rangle,\\ldots,\\vert 3\\rangle$.\n",
+    "If we do this, then we can measure to learn which value $y\\in\\{0,1,2,3\\}$ describes $\\theta$ as $\\theta = y/4$.\n",
+    "\n",
+    "Here's a diagram of the quantum circuit that does this.\n",
+    "\n",
+    "![Phase estimation with two qubits](images/two-bit-phase-estimation.png)\n",
+    "\n",
+    "To summarize, if we run this circuit when $\\theta = y/4$ for $y\\in\\{0,1,2,3\\}$, the state immediately before the measurements take place will be $\\vert \\psi\\rangle \\vert y\\rangle$ (for $y$ encoded as a two-bit binary string), so the measurements will reveal the value $y$ without error."
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "ff7f0eed",
+   "metadata": {},
+   "source": [
+    "This circuit is motivated by the special case that $\\theta \\in \\{0,1/4,1/2,3/4\\}$ — but we can run it for any choice of $U$ and $\\vert \\psi\\rangle$, and hence any value of $\\theta$, that we wish.\n",
+    "Here's a plot of the output probabilities the circuit produces for arbitrary choices of $\\theta:$\n",
+    "\n",
+    "![Outcome probabilities from two-qubit phase estimation](images/two-bit-probabilities.png)\n",
+    "\n",
+    "This is a clear improvement over the single-qubit variant described earlier in the lesson.\n",
+    "It's not perfect — it can give us the wrong answer — but the answer is heavily skewed toward values of $y$ for which $y/4$ is close to $\\theta$.\n",
+    "In particular, the most likely outcome always corresponds to the closest value of $y/4$ to $\\theta$ (equating $\\theta = 0$ and $\\theta = 1$ as before), and it appears from the plot that this closest value for $x$ always appears with probability above $40\\%$.\n",
+    "When $\\theta$ is exactly halfway between two such values, like $\\theta = 0.375$ for instance, the two equally close values of $y$ are equally likely."
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "32664572",
+   "metadata": {},
+   "source": [
+    "#### Qiskit implementation\n",
+    "\n",
+    "Here's an implementation of this procedure in Qiskit.\n",
+    "Similar to the previous implementation, we'll use a phase gate with a chosen angle $\\theta$ for the unitary operation and $\\vert 1\\rangle$ for the eigenvector."
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 5,
+   "id": "945240a6",
+   "metadata": {},
+   "outputs": [
+    {
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+      ],
+      "text/plain": [
+       "<Figure size 1541.66x367.889 with 1 Axes>"
+      ]
+     },
+     "execution_count": 5,
+     "metadata": {},
+     "output_type": "execute_result"
+    }
+   ],
+   "source": [
+    "from math import pi\n",
+    "from qiskit import QuantumCircuit\n",
+    "\n",
+    "theta = .7\n",
+    "qc = QuantumCircuit(3, 2)\n",
+    "\n",
+    "# Prepare the eigenvector\n",
+    "qc.x(2)\n",
+    "qc.barrier()\n",
+    "\n",
+    "# The initial Hadamard gates\n",
+    "qc.h(0)\n",
+    "qc.h(1)\n",
+    "qc.barrier()\n",
+    "\n",
+    "# The controlled unitary gates\n",
+    "qc.cp(2*pi*theta,0,2)\n",
+    "qc.cp(2*pi*(2*theta),1,2)\n",
+    "qc.barrier()\n",
+    "\n",
+    "# An implementation of the inverse of the two-qubit QFT\n",
+    "qc.swap(0,1)\n",
+    "qc.h(0)\n",
+    "qc.cp(-pi/2,0,1)\n",
+    "qc.h(1)\n",
+    "qc.barrier()\n",
+    "\n",
+    "# And finally the measurements\n",
+    "qc.measure([0, 1], [0, 1])\n",
+    "qc.draw()\n"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 6,
+   "id": "712b60f1",
+   "metadata": {},
+   "outputs": [
+    {
+     "data": {
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+      ],
+      "text/plain": [
+       "<Figure size 700x500 with 1 Axes>"
+      ]
+     },
+     "execution_count": 6,
+     "metadata": {},
+     "output_type": "execute_result"
+    }
+   ],
+   "source": [
+    "from qiskit.visualization import plot_histogram\n",
+    "result = Sampler().run(qc).result()\n",
+    "plot_histogram(result.quasi_dists)"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "813fc3de",
+   "metadata": {},
+   "source": [
+    "#### Generalizing to many qubits\n",
+    "\n",
+    "Given the improvement we've just obtained by using two control qubits rather than one, in conjunction with the $4$-dimensional quantum Fourier transform, it's natural to consider generalizing it further — by adding more control qubits.\n",
+    "When we do this, we obtain the general *phase estimation procedure*.\n",
+    "We'll see how this works shortly, but in order to describe it precisely we're going to need to discuss the quantum Fourier transform in greater generality, to see how it's defined for other dimensions and to see how we can implement it with a quantum circuit."
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "20367e3c",
+   "metadata": {},
+   "source": [
+    "### Quantum Fourier transform\n",
+    "\n",
+    "The quantum Fourier transform is a unitary operation that can be defined for any positive integer dimension $N$.\n",
+    "In this subsection we'll see how this operation is defined, and we'll see how it can be implemented with a quantum circuit on $m$ qubits with cost $O(m^2)$ when $N = 2^m$.\n",
+    "\n",
+    "The matrices that describe this operation are derived from an analogous operation on $N$ dimensional vectors known as the *discrete Fourier transform*.\n",
+    "We can think about the discrete Fourier transform abstractly, in purely mathematical terms, but we can also think about it as a computational problem, where we're given an $N$ dimensional vector of complex numbers (using binary notation to encode the real and imaginary parts of the entries, let us suppose), and our goal is to calculate the result, which is again an $N$ dimensional vector.\n",
+    "An efficient algorithm for performing this computation known as the *fast Fourier transform* is considered by many to be one of the most important algorithms ever discovered — it's critical in signal processing and has wide-ranging applications.\n",
+    "\n",
+    "Our focus, however, is on viewing this transform as a unitary operation that can be performed on a quantum system."
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "6902f80b",
+   "metadata": {},
+   "source": [
+    "#### Definition of the quantum Fourier transform\n",
+    "\n",
+    "To define the quantum Fourier transform, we'll first define a complex number $\\omega_N$, for each positive integer $N$, like this:\n",
+    "\n",
+    "$$\n",
+    "\\omega_N = e^{\\frac{2\\pi i}{N}} = \\cos\\left(\\frac{2\\pi}{N}\\right) + i \\sin\\left(\\frac{2\\pi}{N}\\right).\n",
+    "$$\n",
+    "\n",
+    "This is the number on the complex unit circle we obtain if we start at $1$ and move counter-clockwise by an angle of $2\\pi/N$ — or \"one click\" if we imagine that $N$ discrete \"clicks\" would take us all the way around the circle.\n",
+    "\n",
+    "Here are a few examples:\n",
+    "\n",
+    "$$\n",
+    "\\begin{gathered}\n",
+    "\\omega_1 = 1\\\\[1mm]\n",
+    "\\omega_2 = -1\\\\[1mm]\n",
+    "\\omega_3 = -\\frac{1}{2} + \\frac{\\sqrt{3}}{2} i\\\\[1mm]\n",
+    "\\omega_4 = i\\\\[1mm]\n",
+    "\\omega_8 = \\frac{1+i}{\\sqrt{2}}\\\\[1mm]\n",
+    "\\omega_{16} = \\frac{\\sqrt{2 + \\sqrt{2}}}{2} + \\frac{\\sqrt{2 - \\sqrt{2}}}{2} i\\\\[1mm]\n",
+    "\\omega_{100} \\approx 0.998 + 0.063 i\n",
+    "\\end{gathered}\n",
+    "$$\n"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "c1bc8d2e",
+   "metadata": {},
+   "source": [
+    "Now we can define the $N$-dimensional quantum Fourier transform, which is described by an $N\\times N$ matrix whose rows and columns are associated with the standard basis states $\\vert 0\\rangle,\\ldots,\\vert N-1\\rangle$.\n",
+    "\n",
+    "$$\n",
+    "\\mathrm{QFT}_N = \\frac{1}{\\sqrt{N}} \\sum_{x = 0}^{N-1} \\sum_{y = 0}^{N-1} \\omega_N^{xy} \\vert x \\rangle\\langle y\\vert\n",
+    "$$\n",
+    "\n",
+    "For phase estimation we're only going to need this operation for when $N = 2^m$ is a power of $2$, but the operation can be defined for any positive integer $N$.\n",
+    "\n",
+    "As was already stated, this is the matrix associated with the $N$-dimensional *discrete Fourier transform*.\n",
+    "(Often the leading factor of $1/\\sqrt{N}$ is not included in the definition of the matrix associated with the discrete Fourier transform, but we need to include it to obtain a unitary matrix.\n",
+    "Sometimes a minus sign appears in the exponent of $\\omega_N$ as well — different people define it in slightly different ways, but these differences are superficial and easily reconciled.)\n",
+    "\n",
+    "Here's the quantum Fourier transform written as a matrix for some small values of $N$.\n",
+    "\n",
+    "$$\n",
+    "\\mathrm{QFT}_1 = \\begin{pmatrix} 1 \\end{pmatrix}\n",
+    "$$\n",
+    "\n",
+    "$$\n",
+    "\\mathrm{QFT}_2 = \n",
+    "\\frac{1}{\\sqrt{2}} \\begin{pmatrix} 1 & 1\\\\ 1 & -1 \\end{pmatrix}\n",
+    "$$\n",
+    "\n",
+    "$$\n",
+    "\\mathrm{QFT}_3 = \n",
+    "\\frac{1}{\\sqrt{3}} \n",
+    "\\begin{pmatrix} \n",
+    "   1 & 1 & 1\\\\[2mm]\n",
+    "   1 & \\frac{-1 + i\\sqrt{3}}{2} & \\frac{-1 - i\\sqrt{3}}{2}\\\\[2mm]\n",
+    "   1 & \\frac{-1 - i\\sqrt{3}}{2} & \\frac{-1 + i\\sqrt{3}}{2}\n",
+    "\\end{pmatrix}\n",
+    "$$\n",
+    "\n",
+    "$$\n",
+    "\\mathrm{QFT}_4 =\n",
+    "\\frac{1}{2}\n",
+    "\\begin{pmatrix}\n",
+    "1 & 1 & 1 & 1\\\\\n",
+    "1 & i & -1 & -i\\\\\n",
+    "1 & -1 & 1 & -1\\\\\n",
+    "1 & -i & -1 & i\n",
+    "\\end{pmatrix}\n",
+    "$$\n",
+    "\n",
+    "$$\n",
+    "\\mathrm{QFT}_8 =\n",
+    "\\frac{1}{2\\sqrt{2}}\n",
+    "\\begin{pmatrix}\n",
+    "  1 & 1 & 1 & 1 & 1 & 1 & 1 & 1\\\\[2mm]\n",
+    "  1 & \\frac{1+i}{\\sqrt{2}} & i & \\frac{-1+i}{\\sqrt{2}} & -1 & \\frac{-1-i}{\\sqrt{2}} & -i & \\frac{1-i}{\\sqrt{2}}\\\\[2mm]\n",
+    "  1 & i & -1 & -i & 1 & i & -1 & -i\\\\[2mm]\n",
+    "  1 & \\frac{-1+i}{\\sqrt{2}} & -i & \\frac{1+i}{\\sqrt{2}} & -1 & \\frac{1-i}{\\sqrt{2}} & i & \\frac{-1-i}{\\sqrt{2}}\\\\[2mm]\n",
+    "  1 & -1 & 1 & -1 & 1 & -1 & 1 & -1\\\\[2mm]\n",
+    "  1 & \\frac{-1-i}{\\sqrt{2}} & i & \\frac{1-i}{\\sqrt{2}} & -1 & \\frac{1+i}{\\sqrt{2}} & -i & \\frac{-1+i}{\\sqrt{2}}\\\\[2mm]\n",
+    "  1 & -i & -1 & i & 1 & -i & -1 & i\\\\[2mm]\n",
+    "  1 & \\frac{1-i}{\\sqrt{2}} & -i & \\frac{-1-i}{\\sqrt{2}} & -1 & \\frac{-1+i}{\\sqrt{2}} & i & \\frac{1+i}{\\sqrt{2}}\\\\[2mm]\n",
+    "\\end{pmatrix}\n",
+    "$$\n",
+    "\n",
+    "Notice, in particular, that $\\mathrm{QFT}_2$ is another name for a Hadamard operation."
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "979df05e",
+   "metadata": {},
+   "source": [
+    "#### Unitarity\n",
+    "\n",
+    "Let's check that $\\mathrm{QFT}_N$ is indeed unitary, for any selection of $N$.\n",
+    "One way to do this is to show that the columns form an orthonormal basis.\n",
+    "We can define column number $y$, starting from $y = 0$ and going up to $y = N-1$, like this:\n",
+    "\n",
+    "$$\n",
+    "\\vert\\phi_y\\rangle = \\frac{1}{\\sqrt{N}} \\sum_{x = 0}^{N-1} \\omega_N^{xy} \\vert x \\rangle.\n",
+    "$$\n",
+    "\n",
+    "Taking the inner product between any two columns gives us this expression:\n",
+    "\n",
+    "$$\n",
+    "\\langle \\phi_z \\vert \\phi_y \\rangle = \\frac{1}{N} \\sum_{x = 0}^{N-1} \\omega_N^{x (y - z)}\n",
+    "$$\n",
+    "\n",
+    "One way to evaluate sums like this is to use the following formula for the sum of the first $N$ terms of a geometric series.\n",
+    "\n",
+    "$$\n",
+    "1 + \\alpha + \\alpha^2 + \\cdots + \\alpha^{N-1} =\n",
+    "\\begin{cases}\n",
+    "\\frac{\\alpha^N - 1}{\\alpha - 1} & \\text{if } \\alpha\\not=1\\\\[2mm]\n",
+    "N & \\text{if } \\alpha=1\n",
+    "\\end{cases}\n",
+    "$$\n",
+    "\n",
+    "Specifically, we can use this formula when $\\alpha = \\omega_N^{y-z}$.\n",
+    "When $y = z$, we have $\\alpha = 1$, so using the formula and dividing by $N$ gives\n",
+    "\n",
+    "$$\n",
+    "\\langle \\phi_y \\vert \\phi_y \\rangle = 1.\n",
+    "$$\n",
+    "\n",
+    "When $y\\neq z$, we have $\\alpha \\neq 1$, so using the formula reveals this:\n",
+    "\n",
+    "$$\n",
+    "\\langle \\phi_z \\vert \\phi_y \\rangle = \\frac{1}{N} \\frac{\\omega_N^{N(y-z)} - 1}{\\omega_N^{y-z} - 1} \n",
+    "= \\frac{1}{N} \\frac{1 - 1}{\\omega_N^{y-z} - 1} = 0.\n",
+    "$$\n",
+    "\n",
+    "This happens because $\\omega_N^N = e^{2\\pi i} = 1$, so $\\omega_N^{N(y-z)} = 1^{y-z} = 1$, and the numerator becomes $0$ (while the denominator is nonzero because $\\omega_N^{y-z} \\neq 1)$.\n",
+    "\n",
+    "At a more intuitive level, what we're effectively doing in the formula for $\\langle \\phi_z\\vert\\phi_y\\rangle$ is summing a bunch of points that are evenly spread around the unit circle — so that they cancel each other out, leaving $0$ when we sum them.\n",
+    "\n",
+    "So, we have established that $\\{\\vert\\phi_0\\rangle,\\ldots,\\vert\\phi_{N-1}\\rangle\\}$ is an orthonormal set.\n",
+    "\n",
+    "$$\n",
+    "\\langle \\phi_z \\vert \\phi_y \\rangle =\n",
+    "\\begin{cases}\n",
+    "0 & y=z\\\\\n",
+    "1 & y\\not=z\n",
+    "\\end{cases}\n",
+    "$$\n",
+    "\n",
+    "This reveals that $\\mathrm{QFT}_N$ is unitary."
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "22732ff6",
+   "metadata": {},
+   "source": [
+    "#### Controlled-phase gates\n",
+    "\n",
+    "In order to implement the quantum Fourier transform as a quantum circuit, we're going to need to make use of *controlled-phase* gates.\n",
+    "\n",
+    "Recall from Lesson 1 that a *phase operation* is a single-qubit quantum operation of the form\n",
+    "\n",
+    "$$\n",
+    "P_{\\alpha} = \n",
+    "\\begin{pmatrix}\n",
+    "1 & 0\\\\\n",
+    "0 & e^{i\\alpha}\n",
+    "\\end{pmatrix}\n",
+    "$$\n",
+    "\n",
+    "for any real number $\\alpha$.\n",
+    "(In Lesson 1 we used the name $\\theta$ in place of $\\alpha$, but in this lesson we'll reserve the letter $\\theta$ for the parameter in phase estimation.)\n",
+    "\n",
+    "A controlled version of this gate has the following matrix:\n",
+    "\n",
+    "$$\n",
+    "CP_{\\alpha} = \n",
+    "\\begin{pmatrix}\n",
+    "1 & 0 & 0 & 0\\\\\n",
+    "0 & 1 & 0 & 0\\\\\n",
+    "0 & 0 & 1 & 0\\\\\n",
+    "0 & 0 & 0 & e^{i\\alpha}\n",
+    "\\end{pmatrix}\n",
+    "$$\n",
+    "\n",
+    "For this controlled gate, it doesn't actually matter which qubit is the control and which is the target — the two possibilities are equivalent.\n",
+    "We can use any of the following symbols to represent this gate in quantum circuit diagrams:\n",
+    "\n",
+    "![Quantum circuit diagram representation for controlled-phase gates](images/controlled-phase-gates.png)\n",
+    "\n",
+    "For the third form, the label $\\alpha$ is also sometimes placed on the side of the control line or under the lower control when that's convenient.\n",
+    "\n",
+    "Using controlled-phase gates we can perform the following transformation,\n",
+    "where $a$ is a bit and $y \\in \\{0,\\ldots,2^{m-1} - 1\\}$ is a number encoded in binary notation as a string of $m-1$ bits.\n",
+    "\n",
+    "$$\n",
+    "\\vert y \\rangle \\vert a \\rangle \\mapsto \\omega_{2^m}^{ay} \\vert y \\rangle \\vert a \\rangle\n",
+    "$$\n",
+    "\n",
+    "As an example, here's  for how this is done for $m=5$.\n",
+    "\n",
+    "![Quantum circuit diagram for phase injection](images/phase-injection.png)\n",
+    "\n",
+    "This can be naturally generalized for any $m$ — the phase gates always start from $\\frac{\\pi}{2}$ on the most significant bit of $x$ down to $\\pi/2^{m-1}$ on the least significant bit."
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "b00638a6",
+   "metadata": {},
+   "source": [
+    "#### Circuit implementation of the QFT\n",
+    "\n",
+    "Now we'll see how we can implement the quantum Fourier transform with a circuit when the dimension $N = 2^m$ is a power of $2$.\n",
+    "There are, in fact, multiple ways to implement the quantum Fourier transform, but this is arguably the simplest method.\n",
+    "\n",
+    "The implementation is recursive in nature, and so that's how it's most naturally described.\n",
+    "The base case is that the quantum Fourier transform on a single qubit is a Hadamard operation.\n",
+    "\n",
+    "To perform the quantum Fourier transform on $m$ qubits when $m \\geq 2$, we can perform the following steps, whose actions we'll describe for standard basis states of the form $\\vert x \\rangle \\vert a\\rangle$, where $x\\in\\{0,\\ldots,2^{m-1} - 1\\}$ is an integer encoded as $m-1$ bits using binary notation and $a$ is a single bit.\n",
+    "\n",
+    "1. First apply the $2^{m-1}$-dimensional quantum Fourier transform to the bottom/leftmost $m-1$ qubits to obtain \n",
+    "   this state:\n",
+    "\n",
+    "   $$\n",
+    "   \\Bigl(\\mathrm{QFT}_{2^{m-1}} \\vert x \\rangle\\Bigr) \\vert a\\rangle\n",
+    "   = \\frac{1}{\\sqrt{2^{m-1}}} \\sum_{y = 0}^{2^{m-1} - 1} \\omega_{2^{m-1}}^{xy} \\vert y \\rangle \\vert a \\rangle\n",
+    "   $$\n",
+    "\n",
+    "   This is done by recursively applying the method being described for one fewer qubit, using the Hadamard \n",
+    "   operation on a single qubit as the base case.\n",
+    "\n",
+    "2. Use the top/rightmost qubit as a control to inject the phase $\\omega_{2^m}^y$ for each standard basis \n",
+    "   state $\\vert y\\rangle$ of the remaining $m-1$ qubits (as described above) to obtain this state:\n",
+    "\n",
+    "   $$\n",
+    "   \\frac{1}{\\sqrt{2^{m-1}}} \\sum_{y = 0}^{2^{m-1} - 1} \\omega_{2^{m-1}}^{xy}\n",
+    "   \\omega_{2^m}^{ay} \\vert y \\rangle \\vert a \\rangle\n",
+    "   $$\n",
+    "   \n",
+    "3. Perform a Hadamard gate on the top/rightmost qubit to obtain this state:\n",
+    "\n",
+    "   $$\n",
+    "   \\frac{1}{\\sqrt{2^{m}}} \\sum_{y = 0}^{2^{m-1} - 1} \\sum_{b=0}^1 \n",
+    "   (-1)^{ab} \\omega_{2^{m-1}}^{xy} \\omega_{2^m}^{ay} \n",
+    "   \\vert y \\rangle \\vert b \\rangle\n",
+    "   $$\n",
+    "\n",
+    "4. Permute the order of the qubits so that the least significant bit becomes the most significant bit, with \n",
+    "   all others shifted:\n",
+    "\n",
+    "   $$\n",
+    "   \\frac{1}{\\sqrt{2^{m}}} \\sum_{y = 0}^{2^{m-1} - 1} \\sum_{b=0}^1 \n",
+    "   (-1)^{ab} \\omega_{2^{m-1}}^{xy} \\omega_{2^m}^{ay} \n",
+    "   \\vert b \\rangle \\vert y \\rangle \n",
+    "   $$"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "b7315ca9",
+   "metadata": {},
+   "source": [
+    "For example, here's the circuit we obtain for $N = 32 = 2^5$.\n",
+    "In this diagram, the qubits are given names that correspond to the standard basis vectors $\\vert x\\rangle \\vert a\\rangle$ (for the input) and $\\vert b\\rangle \\vert y\\rangle$ (for the output) for clarity.\n",
+    "\n",
+    "![Quantum circuit diagram for the 32-dimensional quantum Fourier transform](images/QFT_32.png)"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "34486329",
+   "metadata": {},
+   "source": [
+    "<br>\n",
+    "\n",
+    "#### Analysis\n",
+    "\n",
+    "The key formula we need to verify that the circuit just described implements the $2^m$-dimensional quantum Fourier transform is this one:\n",
+    "\n",
+    "$$\n",
+    "(-1)^{ab}\n",
+    "\\omega_{2^{m-1}}^{xy}\n",
+    "\\omega_{2^m}^{ay}\n",
+    "=\n",
+    "\\omega_{2^m}^{(2x+ a)(2^{m-1}b + y)}.\n",
+    "$$\n",
+    "\n",
+    "This formula works for any choice of integers $a$, $b$, $x$, and $y$, but we'll only need it for\n",
+    "$a,b\\in\\{0,1\\}$ and $x,y\\in\\{0,\\ldots,2^{m-1}-1\\}$.\n",
+    "We can check the formula by expanding the product in the exponent on the right-hand side,\n",
+    "\n",
+    "$$\n",
+    "  \\omega_{2^m}^{(2x+ a)(2^{m-1}b + y)}\n",
+    "  = \\omega_{2^m}^{2^m xb} \\omega_{2^m}^{2xy} \\omega_{2^m}^{2^{m-1}ab} \\omega_{2^m}^{ay}\n",
+    "  = (-1)^{ab} \\omega_{2^{m-1}}^{xy} \\omega_{2^m}^{ay},\n",
+    "$$\n",
+    "\n",
+    "where the second equality makes use of the observation that\n",
+    "\n",
+    "$$\n",
+    "\\omega_{2^m}^{2^m xb} = \\bigl(\\omega_{2^m}^{2^m}\\bigr)^{xb} = 1^{xb} = 1.\n",
+    "$$\n",
+    "\n",
+    "Now, the $2^m$-dimensional quantum Fourier transform is defined as follows for every $u\\in\\{0,\\ldots,2^m - 1\\}$.\n",
+    "\n",
+    "$$\n",
+    "\\mathrm{QFT}_{2^m} \\vert u\\rangle = \\frac{1}{\\sqrt{2^m}}\n",
+    "\\sum_{v = 0}^{2^m - 1} \\omega_{2^m}^{uv} \\vert v\\rangle\n",
+    "$$\n",
+    "\n",
+    "If we write $u$ and $v$ as\n",
+    "\n",
+    "$$\n",
+    "\\begin{aligned}\n",
+    "u & = 2x + a\\\\\n",
+    "v & = 2^{m-1}b + y\n",
+    "\\end{aligned}\n",
+    "$$\n",
+    "\n",
+    "for $a,b\\in\\{0,1\\}$ and $x,y\\in\\{0,\\ldots,2^{m-1} - 1\\}$, we obtain\n",
+    "\n",
+    "$$\n",
+    "\\begin{aligned}\n",
+    "\\mathrm{QFT}_{2^m} \\vert 2x + a\\rangle\n",
+    "& = \n",
+    "\\frac{1}{\\sqrt{2^m}}\n",
+    "\\sum_{y = 0}^{2^{m-1} - 1}\n",
+    "\\sum_{b=0}^1 \n",
+    "\\omega_{2^m}^{(2x+ a)(2^{m-1}b + y)} \\vert b 2^{m-1} + y\\rangle\\\\[2mm]\n",
+    "& =\n",
+    "\\frac{1}{\\sqrt{2^m}}\n",
+    "\\sum_{y = 0}^{2^{m-1} - 1}\n",
+    "\\sum_{b=0}^1 \n",
+    "(-1)^{ab}\n",
+    "\\omega_{2^{m-1}}^{xy}\n",
+    "\\omega_{2^m}^{ay}\n",
+    "\\vert b 2^{m-1} + y\\rangle.\n",
+    "\\end{aligned}\n",
+    "$$\n",
+    "\n",
+    "Finally, by thinking about the standard basis states $\\vert x \\rangle \\vert a\\rangle$ and $\\vert b \\rangle \\vert y \\rangle$ as binary encodings of integers in the range $\\{0,\\ldots,2^m-1\\}$,\n",
+    "\n",
+    "$$\n",
+    "\\begin{aligned}\n",
+    "\\vert x \\rangle \\vert a\\rangle & = \\vert 2x + a \\rangle\\\\\n",
+    "\\vert b \\rangle \\vert y \\rangle & = \\vert 2^{m-1}b + y\\rangle,\n",
+    "\\end{aligned}\n",
+    "$$\n",
+    "\n",
+    "we see that the circuit above implements the required operation.\n",
+    "\n",
+    "If this method for performing the quantum Fourier transform seems remarkable, it's because it is.\n",
+    "It's essentially the same methodology that underlies the fast Fourier transform — one of the most important and useful classical algorithms ever discovered — in the form of a quantum circuit."
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "11c6351c",
+   "metadata": {},
+   "source": [
+    "#### Computational cost\n",
+    "\n",
+    "Now let's count how many gates are used in the circuit just described.\n",
+    "The controlled-phase gates aren't in the standard gate set that we discussed in the previous lesson, but to begin we'll ignore this and count each of them as a single gate.\n",
+    "\n",
+    "Let's let $s_m$ denote the number of gates we need for each possible choice of $m$.\n",
+    "If $m=1$, the quantum Fourier transform is just a Hadamard operation, so\n",
+    "\n",
+    "$$\n",
+    "s_1 = 1.\n",
+    "$$\n",
+    "\n",
+    "If $m\\geq 2$, then in the circuit above we need $s_{m-1}$ gates for the quantum Fourier transform on $m-1$ qubits, plus $m-1$ controlled-phase gates, plus a Hadamard gate, plus $m-1$ swap gates, so\n",
+    "\n",
+    "$$\n",
+    "s_m = s_{m-1} + (2m - 1).\n",
+    "$$\n",
+    "\n",
+    "We can obtain a closed-form expression by summing:\n",
+    "\n",
+    "$$\n",
+    "s_m = \\sum_{k = 1}^m (2k - 1) = m^2.\n",
+    "$$\n",
+    "\n",
+    "We don't actually need as many swap gates as the method describes — if we rearrange the gates just a bit, we can push all of the swap gates out to the right and reduce the number of swap gates required to $\\lfloor m/2\\rfloor$.\n",
+    "Asymptotically speaking this isn't a major improvement: we still obtain circuits with size $O(m^2)$ for performing $\\mathrm{QFT}_{2^m}$.\n",
+    "\n",
+    "If we wish to implement the quantum Fourier transform using only gates from our standard gate set, we need to either build or approximate each of the controlled-phase gates with gates from our set.\n",
+    "The number required depends on how much accuracy we require, but the total number remains quadratic in the number $m$.\n",
+    "(We can even come up with a pretty good approximation to the quantum Fourier transform with a sub-quadratic number of gates by using the fact that when $\\alpha$ is very small, we have $e^{i\\alpha} \\approx 1$, so the controlled-phase gate $CP_{\\alpha}$ can be very well approximated by doing nothing at all in such cases.)"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "793c8ec5",
+   "metadata": {},
+   "source": [
+    "#### QFTs in Qiskit\n",
+    "\n",
+    "A circuit implementation of the QFT on any number of qubits can be obtained from Qiskit's circuit library.\n",
+    "(Note that for three or more qubits the circuit will differ slightly from the general description above because it incorporates some minor optimizations. In particular, the number of swap gates required can be reduced significantly by effectively pushing them to the end of the circuit and adjusting the controlled-phase gates accordingly.)"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 7,
+   "id": "adbb2d1a",
+   "metadata": {},
+   "outputs": [
+    {
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+      ],
+      "text/plain": [
+       "<Figure size 1541.66x367.889 with 1 Axes>"
+      ]
+     },
+     "execution_count": 7,
+     "metadata": {},
+     "output_type": "execute_result"
+    }
+   ],
+   "source": [
+    "from qiskit.circuit.library import QFT\n",
+    "QFT(4).decompose().draw()"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "25964ba6",
+   "metadata": {},
+   "source": [
+    "### General procedure and analysis\n",
+    "\n",
+    "Now we'll examine the phase-estimation procedure in general.\n",
+    "The idea is to extend the single- and double-qubit versions of phase estimation that we considered above in the natural way, as suggested by the following diagram.\n",
+    "\n",
+    "![Phase estimation procedure](images/phase-estimation-procedure.png)\n",
+    "\n",
+    "Notice that for each new control qubit added on the top, we *double* the number of times the unitary operation $U$ is performed.\n",
+    "Rather than drawing however many copies of the controlled-$U$ operation are needed to do this in the diagram, we've instead raised $U$ to the required powers.\n",
+    "\n",
+    "In general, adding additional control qubits on the top like this will contribute significantly to the size of the circuit: if we have $m$ control qubits, like the diagram depicts, a total of $2^m - 1$ copies of the controlled-$U$ operation are required.\n",
+    "This means that a significant computational cost is incurred as $m$ is increased — but as we will see, it also leads to a significantly more accurate approximation of $\\theta$.\n",
+    "\n",
+    "It is important to note, however, that for *some* choices of $U$ it may be possible to create a circuit that implements the operation $U^k$ for large values of $k$ in a more efficient way than simply repeating $k$ times the circuit for $U$.\n",
+    "We'll see a specific example of this in the context of integer factorization later in the lesson, where the efficient algorithm for *modular exponentiation* discussed in the previous lesson comes to the rescue.\n",
+    "\n",
+    "Now let us analyze the circuit just described.\n",
+    "The state immediately prior to the quantum Fourier transform looks like this:\n",
+    "\n",
+    "$$\n",
+    "\\frac{1}{\\sqrt{2^m}} \\sum_{x = 0}^{2^m - 1} \\bigl( U^x \\vert\\psi\\rangle \\bigr) \\vert x\\rangle\n",
+    "= \\vert\\psi\\rangle \\otimes \\frac{1}{\\sqrt{2^m}} \\sum_{x = 0}^{2^m - 1} e^{2\\pi i x\\theta}  \\vert x\\rangle.\n",
+    "$$\n",
+    "\n",
+    "#### A special case\n",
+    "\n",
+    "Along similar lines to what we did in the $m=2$ case above, we can consider the special case that\n",
+    "$\\theta = y/2^m$ for $y\\in\\{0,\\ldots,2^m-1\\}$, and we see that this state can alternatively be written like this:\n",
+    "\n",
+    "$$\n",
+    "\\vert\\psi\\rangle \\otimes \\frac{1}{\\sqrt{2^m}} \\sum_{x = 0}^{2^m - 1} e^{2\\pi i \\frac{xy}{2^m}}  \\vert x\\rangle\n",
+    "= \\vert\\psi\\rangle \\otimes \\frac{1}{\\sqrt{2^m}} \\sum_{x = 0}^{2^m - 1} \\omega_{2^m}^{xy}  \\vert x\\rangle\n",
+    "= \\vert\\psi\\rangle \\otimes \\mathrm{QFT}_{2^m} \\vert y\\rangle.\n",
+    "$$\n",
+    "\n",
+    "So, when the inverse of the quantum Fourier transform is applied, the state becomes\n",
+    "\n",
+    "$$\n",
+    "\\vert\\psi\\rangle \\vert y\\rangle\n",
+    "$$\n",
+    "\n",
+    "and the measurements reveal $y$ (encoded in binary).\n"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "7e14c809",
+   "metadata": {},
+   "source": [
+    "#### Bounding the probabilities\n",
+    "\n",
+    "For other values of $\\theta$, meaning ones that don't take the form $y/2^m$ for an integer $y$, the measurement outcomes won't be certain, but we can prove certain bounds on the probabilities for different outcomes.\n",
+    "Going forward, let's consider an arbitrary choice of $\\theta$ satisfying $0\\leq \\theta < 1$.\n",
+    "\n",
+    "After the quantum Fourier transform is performed, the state of the circuit is this:\n",
+    "\n",
+    "$$\n",
+    "\\vert \\psi \\rangle \\otimes \n",
+    "\\frac{1}{2^m} \\sum_{y=0}^{2^m - 1} \\sum_{x=0}^{2^m-1} e^{2\\pi i x (\\theta - y/2^m)} \\vert y\\rangle.\n",
+    "$$\n",
+    "\n",
+    "So, when the measurements on the top $m$ qubits are performed, we see each outcome $y$ with probability\n",
+    "\n",
+    "$$\n",
+    "p_y = \n",
+    "\\left\\vert \n",
+    "\\frac{1}{2^m} \\sum_{x=0}^{2^m - 1} e^{2\\pi i x (\\theta - y/2^m)} \\right|^2\n",
+    "$$\n",
+    "\n",
+    "To get a better handle on these probabilities, we'll make use of the same formula that we saw before, for the sum of the initial portion of a geometric series.\n",
+    "\n",
+    "$$\n",
+    "1 + \\alpha + \\alpha^2 + \\cdots + \\alpha^{N-1} =\n",
+    "\\begin{cases}\n",
+    "\\frac{\\alpha^N - 1}{\\alpha - 1} & \\text{if } \\alpha\\not=1\\\\[2mm]\n",
+    "N & \\text{if } \\alpha=1\n",
+    "\\end{cases}\n",
+    "$$\n",
+    "\n",
+    "We can simplify the sum appearing in the formula for $p_y$ by taking $\\alpha = e^{2\\pi i (\\theta - y/2^m)}$.\n",
+    "Here's what we obtain.\n",
+    "\n",
+    "$$\n",
+    "\\sum_{x=0}^{2^m - 1} e^{2\\pi i x (\\theta - y/2^m)}\n",
+    "=\n",
+    "\\begin{cases}\n",
+    "2^m & \\theta = y/2^m\\\\[2mm]\n",
+    "\\frac{e^{2\\pi (2^m \\theta - y)} - 1}{e^{2\\pi (\\theta - y/2^m)} - 1}\n",
+    "& \\theta\\neq y/2^m\n",
+    "\\end{cases}\n",
+    "$$\n",
+    "\n",
+    "So, in the case that $\\theta = y/2^m$, we find that $p_y = 1$ (as we already knew from considering this special case),\n",
+    "and in the case that $\\theta \\neq y/2^m$, we find that\n",
+    "\n",
+    "$$\n",
+    "p_y = \\frac{1}{2^{2m}} \\left\\vert \\frac{e^{2\\pi i (2^m \\theta - y)} - 1}{e^{2\\pi i (\\theta - y/2^m)} - 1}\\right\\vert^2.\n",
+    "$$"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "c8ce0834",
+   "metadata": {},
+   "source": [
+    "We can learn more about these probabilities by thinking about how arc length and chord length on the unit circle are related.\n",
+    "Here's a figure that illustrates the relationships we need for any real number $\\delta\\in \\bigl[ -\\frac{1}{2},\\frac{1}{2}\\bigr]$."
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "07b4b461",
+   "metadata": {},
+   "source": [
+    "![Illustration of the relationship between arc and chord lengths](images/arc-and-chord.png)\n"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "d5cf40a2",
+   "metadata": {},
+   "source": [
+    "First, the chord length (drawn in blue) can't possibly be larger than the arc length (drawn in purple):\n",
+    "\n",
+    "$$\n",
+    "\\bigl\\vert e^{2\\pi i \\delta} - 1\\bigr\\vert \\leq 2\\pi\\vert\\delta\\vert.\n",
+    "$$\n",
+    "\n",
+    "Relating these lengths in the other direction, we see that the ratio of the arc length to the chord length is greatest when $\\delta = \\pm 1/2$, and in this case the ratio is half the circumference of the circle divided by the diameter, which is $\\pi/2$.\n",
+    "Thus, we can write\n",
+    "\n",
+    "$$\n",
+    "4\\vert\\delta\\vert \\leq\n",
+    "\\bigl\\vert e^{2\\pi i \\delta} - 1\\bigr\\vert.\n",
+    "$$"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "d0c45343",
+   "metadata": {},
+   "source": [
+    "An analysis based on this relation reveals the following two facts.\n",
+    "\n",
+    "1. Suppose that $\\theta$ is a real number and $y\\in \\{0,\\ldots,2^m-1\\}$ satisfies\n",
+    "   \n",
+    "   $$\n",
+    "   \\Bigl\\vert \\theta - \\frac{y}{2^m}\\Bigr\\vert \\leq 2^{-(m+1)}.\n",
+    "   $$\n",
+    "   \n",
+    "   This means that $y/2^m$ is either the best $m$-bit approximation to $\\theta$, or it's one of the two best approximations in case $\\theta$ is exactly halfway between $y/2^m$ and either $(y-1)/2^m$ or $(y+1)/2^m$.\n",
+    "\n",
+    "   We'll prove that $p_y$ has to be pretty large in this case.\n",
+    "   By the assumption we're considering, it follows that $\\vert 2^m \\theta - y \\vert \\leq 1/2$, so we can use the second observation above relating arc and chord lengths to conclude that\n",
+    "   \n",
+    "   $$\n",
+    "   \\left\\vert e^{2\\pi (2^m \\theta - y)} - 1\\right\\vert \n",
+    "   \\geq 4 \\vert 2^m \\theta - y \\vert = 4 \\cdot 2^m \\cdot \\Bigl\\vert \\theta - \\frac{y}{2^m}\\Bigr\\vert.\n",
+    "   $$\n",
+    "  \n",
+    "   We can also use the first observation about arc and chord lengths to conclude that\n",
+    "   \n",
+    "   $$\n",
+    "   \\left\\vert e^{2\\pi (\\theta - y/2^m)} - 1\\right\\vert \\leq 2\\pi \\Bigl\\vert \\theta - \\frac{y}{2^m}\\Bigr\\vert.\n",
+    "   $$\n",
+    "   \n",
+    "   Putting these two inequalities to use on $p_y$ reveals\n",
+    "   \n",
+    "   $$\n",
+    "   p_y \\geq \\frac{1}{2^{2m}} \\frac{16 \\cdot 2^{2m}}{4 \\pi^2} = \\frac{4}{\\pi^2} \\approx 0.405.\n",
+    "   $$\n",
+    "   \n",
+    "   (So, our observation that the best outcome occurs with probability greater than $40\\%$ in the $m=2$ version of phase estimation discussed earlier in fact holds for every choice of $m$.)"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "83e02530",
+   "metadata": {},
+   "source": [
+    "2. Now suppose that $y\\in \\{0,\\ldots,2^m-1\\}$ satisfies\n",
+    "\n",
+    "   $$\n",
+    "   2^{-m} \\leq \\Bigl\\vert \\theta - \\frac{y}{2^m}\\Bigr\\vert \\leq \\frac{1}{2}.\n",
+    "   $$\n",
+    "\n",
+    "   This means that there's a better approximation $z/2^m$ to $\\theta$ in between $\\theta$ and $y/2^m$.\n",
+    "      \n",
+    "   This time we'll prove that $p_y$ can't be too big.\n",
+    "   We can start with the simple observation that\n",
+    "   \n",
+    "   $$\n",
+    "   \\left\\vert e^{2\\pi (2^m \\theta - y)} - 1\\right\\vert \\leq 2,\n",
+    "   $$\n",
+    "   \n",
+    "   which follows from the fact that any two points on the unit circle can differ in absolute value by at most $2$.\n",
+    "   \n",
+    "   We can also use the second observation about arc and chord lengths from above, this time working with the denominator of $p_y$ rather than the numerator, to conclude\n",
+    "   \n",
+    "   $$\n",
+    "   \\left\\vert e^{2\\pi (\\theta - y/2^m)} - 1\\right\\vert \\geq  4\\Bigl\\vert \\theta - \\frac{y}{2^m}\\Bigr\\vert\n",
+    "   \\geq 4 \\cdot 2^{-m}.\n",
+    "   $$\n",
+    "   \n",
+    "   Putting the two inequalities together reveals\n",
+    "   \n",
+    "   $$\n",
+    "   p_y \\leq \\frac{1}{2^{2m}} \\frac{4}{16 \\cdot 2^{-2m}} = \\frac{1}{4}.\n",
+    "   $$\n"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "fb142a09",
+   "metadata": {},
+   "source": [
+    "This is quite good, in the sense that very close approximations to $\\theta$ are likely to occur, with probability greater than $40\\%$, whereas approximations off by more than $2^{-m}$ are less likely, with probability at most $25\\%$.\n",
+    "\n",
+    "We may, however, wish to boost our confidence.\n",
+    "One way to do this is to repeat the phase estimation procedure several times to gather statistical evidence about $\\theta$.\n",
+    "Notice that the state $\\vert\\psi\\rangle$ of the bottom collection of qubits is unchanged by the phase estimation procedure, so it can be used to run the procedure as many times as we like.\n",
+    "\n",
+    "Each time we rum the circuit, we get a best $m$-bit approximation to $\\theta$ with probability greater than $40\\%$, while the probability of being off by more than $2^{-m}$ is bounded by $25\\%$.\n",
+    "So, if we run the circuit several times and take the most commonly appearing outcome of the runs, it's exceedingly likely that the outcome that appears most commonly will not be one that occurs at most $25\\%$ of the time.\n",
+    "As a result, we'll be very likely to obtain an approximation $y/2^m$ that's within $1/2^m$ of the value $\\theta$, and indeed the unlikely chance that we're off by more than $1/2^m$ decreases exponentially in the number of times the procedure is run."
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "1eb0ce8d",
+   "metadata": {},
+   "source": [
+    "#### Qiskit implementation\n",
+    "\n",
+    "Here's an implementation of phase estimation in Qiskit.\n",
+    "Try adjusting $\\theta$ and the number of control qubits $m$ to see how the results change."
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 8,
+   "id": "349571e1",
+   "metadata": {},
+   "outputs": [
+    {
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+      ],
+      "text/plain": [
+       "<Figure size 2130.06x451.5 with 1 Axes>"
+      ]
+     },
+     "execution_count": 8,
+     "metadata": {},
+     "output_type": "execute_result"
+    }
+   ],
+   "source": [
+    "from qiskit import QuantumCircuit, QuantumRegister, ClassicalRegister\n",
+    "from qiskit.circuit.library import QFT\n",
+    "\n",
+    "theta = .7\n",
+    "m = 3 # Number of control qubits\n",
+    "\n",
+    "control_register = QuantumRegister(m, name=\"Control\")\n",
+    "target_register = QuantumRegister(1, name=\"|ψ>\")\n",
+    "output_register = ClassicalRegister(m, name=\"Result\")\n",
+    "qc = QuantumCircuit(control_register, target_register, output_register)\n",
+    "\n",
+    "# Prepare the eigenvector |ψ>\n",
+    "qc.x(target_register)\n",
+    "qc.barrier()\n",
+    "\n",
+    "# Perform phase estimation\n",
+    "for index, qubit in enumerate(control_register):\n",
+    "    qc.h(qubit)\n",
+    "    for _ in range(2**index):\n",
+    "        qc.cp(2*pi*theta, qubit, target_register)\n",
+    "qc.barrier()\n",
+    "\n",
+    "# Do inverse quantum Fourier transform\n",
+    "qc.compose(\n",
+    "    QFT(m, inverse=True),\n",
+    "    inplace=True\n",
+    ")\n",
+    "\n",
+    "# Measure everything\n",
+    "qc.measure(range(m), range(m))\n",
+    "qc.draw()"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 9,
+   "id": "61f7f268",
+   "metadata": {},
+   "outputs": [
+    {
+     "data": {
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+      ],
+      "text/plain": [
+       "<Figure size 700x500 with 1 Axes>"
+      ]
+     },
+     "execution_count": 9,
+     "metadata": {},
+     "output_type": "execute_result"
+    }
+   ],
+   "source": [
+    "result = Sampler().run(qc).result()\n",
+    "plot_histogram(result.quasi_dists)"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 10,
+   "id": "b614e0fc",
+   "metadata": {},
+   "outputs": [
+    {
+     "name": "stdout",
+     "output_type": "stream",
+     "text": [
+      "Most probable output: 6\n",
+      "Estimated theta: 0.75\n"
+     ]
+    }
+   ],
+   "source": [
+    "most_probable = max(result.quasi_dists[0],\n",
+    "                    key=result.quasi_dists[0].get)\n",
+    "\n",
+    "print(f\"Most probable output: {most_probable}\")\n",
+    "print(f\"Estimated theta: {most_probable/2**m}\")"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "8c3bd3a7",
+   "metadata": {},
+   "source": [
+    "## Shor's algorithm"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "0074c9ef",
+   "metadata": {},
+   "source": [
+    "Now we'll turn our attention to the integer factorization problem, and see how it can be solved efficiently on a quantum computer using phase estimation.\n",
+    "The algorithm we'll obtain is *Shor's algorithm for integer factorization*.\n",
+    "Shor didn't describe his algorithm specifically in terms of phase estimation, but it is a natural and intuitive way to explain how it works.\n",
+    "\n",
+    "In the two subsections that follow we'll describe the two main parts of Shor's algorithm.\n",
+    "In the first subsection we'll define an intermediate problem known as the *order-finding problem* and see how phase estimation provides a solution to this problem.\n",
+    "In the second subsection we'll explain how an efficient solution to the order-finding problem also gives us an efficient solution to the integer factorization problem.\n",
+    "(When a solution to one problem provides a solution to another problem like this, we say that the second problem *reduces* to the first — so in this case we're reducing integer factorization to order finding.)\n",
+    "This part of Shor's algorithm doesn't make use of quantum computing at all — it's completely classical."
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "b3be9237",
+   "metadata": {},
+   "source": [
+    "### Order finding\n",
+    "\n",
+    "#### Some basic number theory\n",
+    "\n",
+    "To explain the order-finding problem and how it can be solved using phase estimation, it will be very helpful to explain a couple of basic concepts in number theory and introduce some handy notation along the way.\n",
+    "\n",
+    "To begin, for any given positive integer $N$, we'll define a set\n",
+    "\n",
+    "$$\n",
+    "\\mathbb{Z}_N = \\{0,1,\\ldots,N-1\\}.\n",
+    "$$\n",
+    "\n",
+    "For instance, $\\mathbb{Z}_1 = \\{0\\}$, $\\mathbb{Z}_2 = \\{0,1\\}$, $\\mathbb{Z}_3 = \\{0,1,2\\}$,\n",
+    "and so on.\n",
+    "\n",
+    "These are sets of numbers, but we can think of them as more than sets.\n",
+    "In particular, we can think about *arithmetic operations* on $\\mathbb{Z}_N$ such as addition and multiplication — and if we agree to always take our answers modulo $N$, we'll always stay within this set when we perform these operations.\n",
+    "(The two specific operations of addition and multiplication, both taken modulo $N$, turn $\\mathbb{Z}_N$ into a *ring*, which is a fundamentally important type of object in algebra.)\n",
+    "\n",
+    "For example, $3$ and $5$ are elements of $\\mathbb{Z}_7$, and if we multiply them together we get $3\\cdot 5 = 15$, which leaves a remainder of $1$ when divided by $7$.\n",
+    "Sometimes we express this as follows.\n",
+    "\n",
+    "$$\n",
+    "3 \\cdot 5 \\equiv 1 \\; (\\textrm{mod } 7)\n",
+    "$$\n",
+    "\n",
+    "But we can also simply write $3 \\cdot 5 = 1$, provided that it's been made clear that we're working in $\\mathbb{Z}_7$, just to keep our notation as simple and clear as possible.\n",
+    "\n",
+    "As an example, here are the addition and multiplication tables for $\\mathbb{Z}_6$.\n",
+    "\n",
+    "$$\n",
+    "\\begin{array}{c|cccccc}\n",
+    "    + & 0 & 1 & 2 & 3 & 4 & 5 \\\\\\hline\n",
+    "    0 & 0 & 1 & 2 & 3 & 4 & 5 \\\\\n",
+    "    1 & 1 & 2 & 3 & 4 & 5 & 0 \\\\\n",
+    "    2 & 2 & 3 & 4 & 5 & 0 & 1 \\\\\n",
+    "    3 & 4 & 5 & 0 & 1 & 2 & 3 \\\\\n",
+    "    4 & 5 & 0 & 1 & 2 & 3 & 4 \\\\\n",
+    "    5 & 0 & 1 & 2 & 3 & 4 & 5 \\\\\n",
+    "\\end{array}\n",
+    "\\qquad\n",
+    "\\begin{array}{c|cccccc}\n",
+    "\\cdot & 0 & 1 & 2 & 3 & 4 & 5 \\\\\\hline\n",
+    "    0 & 0 & 0 & 0 & 0 & 0 & 0 \\\\\n",
+    "    1 & 0 & 1 & 2 & 3 & 4 & 5 \\\\\n",
+    "    2 & 0 & 2 & 4 & 0 & 2 & 4 \\\\\n",
+    "    3 & 0 & 3 & 0 & 3 & 0 & 3 \\\\\n",
+    "    4 & 0 & 4 & 2 & 0 & 4 & 2 \\\\\n",
+    "    5 & 0 & 5 & 4 & 3 & 2 & 1 \\\\\n",
+    "\\end{array}\n",
+    "$$\n",
+    "\n",
+    "Among the $N$ elements of $\\mathbb{Z}_N$, the elements $a\\in\\mathbb{Z}_N$ that satisfy $\\gcd(a,N) = 1$ are special.\n",
+    "Frequently the set containing these elements is denoted with a star like this:\n",
+    "\n",
+    "$$\n",
+    "\\mathbb{Z}_N^{\\ast} = \\{a\\in \\mathbb{Z}_N : \\gcd(a,N) = 1\\}.\n",
+    "$$\n",
+    "\n",
+    "If we focus our attention on the operation of multiplication, the set $\\mathbb{Z}_N^{\\ast}$ forms a *group* — and specifically an *abelian group* — which is another important type of object in algebra.\n",
+    "It's a basic fact about these sets (and indeed about finite groups in general), that if we pick any element $a\\in\\mathbb{Z}_N^{\\ast}$ and repeatedly multiply $a$ to itself, we'll always eventually get the number $1$.\n",
+    "\n",
+    "For a first example, let's take $N=6$.\n",
+    "We have that $5\\in\\mathbb{Z}_6^{\\ast}$ because $\\gcd(5,6) = 1$, and if we multiply $5$ to itself we get $1$\n",
+    "(as the table above confirms).\n",
+    "\n",
+    "$$\n",
+    "5^2 = 1 \\quad \\text{(working within $\\mathbb{Z}_6$)}\n",
+    "$$\n",
+    "\n",
+    "As a second example, let's take $N = 21$.\n",
+    "If we go through the numbers from $0$ to $20$, these are the ones that have GCD equal to $1$ with $21:$\n",
+    "\n",
+    "$$\n",
+    "\\mathbb{Z}_{21}^{\\ast} = \\{1,2,4,5,8,10,11,13,16,17,19,20\\}.\n",
+    "$$\n",
+    "\n",
+    "For each of these elements, it is possible to raise that number to a positive integer power to get $1$.\n",
+    "Here are the smallest powers for which this works:\n",
+    "\n",
+    "$$\n",
+    "\\begin{aligned}\n",
+    "1^{1} = 1\\\\\n",
+    "2^{6} = 1\\\\\n",
+    "4^{3} = 1\\\\\n",
+    "5^{6} = 1\\\\\n",
+    "8^{2} = 1\\\\\n",
+    "10^{6} = 1\\\\\n",
+    "11^{6} = 1\\\\\n",
+    "13^{2} = 1\\\\\n",
+    "16^{3} = 1\\\\\n",
+    "17^{6} = 1\\\\\n",
+    "19^{6} = 1\\\\\n",
+    "20^{2} = 1\\\\\n",
+    "\\end{aligned}\n",
+    "$$\n",
+    "\n",
+    "Naturally we're working within $\\mathbb{Z}_{21}$ for all of these equations, which we haven't bothered to write — we take it to be implicit to avoid cluttering things up. We'll continue to do that throughout the rest of the lesson.\n",
+    "\n",
+    "You can check each of these equations above, as well as the fact that these are the smallest positive integer powers for which the equations work, using the following code cell (changing the numbers as needed)."
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 11,
+   "id": "64db66b1",
+   "metadata": {},
+   "outputs": [
+    {
+     "name": "stdout",
+     "output_type": "stream",
+     "text": [
+      "k \t a^k \n",
+      "\n",
+      " 1 \t 17\n",
+      " 2 \t 16\n",
+      " 3 \t 20\n",
+      " 4 \t  4\n",
+      " 5 \t  5\n",
+      " 6 \t  1\n",
+      " 7 \t 17\n",
+      " 8 \t 16\n",
+      " 9 \t 20\n",
+      "10 \t  4\n",
+      "11 \t  5\n",
+      "12 \t  1\n"
+     ]
+    }
+   ],
+   "source": [
+    "N = 21\n",
+    "a = 17\n",
+    "max_power = 12\n",
+    "\n",
+    "print(\"k \\t a^k \\n\")\n",
+    "for k in range(1,max_power+1):\n",
+    "    print(\"%2d \\t %2d\" %(k, a**k % N))     # The % operation computes the remainder modulo N"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "53484e4c",
+   "metadata": {},
+   "source": [
+    "Notice that after we get back to $1$, the cycle repeats — which makes sense because multiplying $1$ by $a$ brings us back to $a$, which is where we started.\n",
+    "\n",
+    "Although it isn't essential for the sake of the lesson, we can also check that we never get back to $1$ when $\\gcd(a,N)\\neq 1$ — so we're relying on the fact that $a\\in\\mathbb{Z}_N^{\\ast}$ for this to work."
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 12,
+   "id": "4df0ffff",
+   "metadata": {},
+   "outputs": [
+    {
+     "name": "stdout",
+     "output_type": "stream",
+     "text": [
+      "k \t a^k \n",
+      "\n",
+      " 1 \t 18\n",
+      " 2 \t  9\n",
+      " 3 \t 15\n",
+      " 4 \t 18\n",
+      " 5 \t  9\n",
+      " 6 \t 15\n",
+      " 7 \t 18\n",
+      " 8 \t  9\n",
+      " 9 \t 15\n",
+      "10 \t 18\n",
+      "11 \t  9\n",
+      "12 \t 15\n"
+     ]
+    }
+   ],
+   "source": [
+    "N = 21\n",
+    "a = 18\n",
+    "max_power = 12\n",
+    "\n",
+    "print(\"k \\t a^k \\n\")\n",
+    "for k in range(1,max_power+1):\n",
+    "    print(\"%2d \\t %2d\" %(k, a**k % N))\n",
+    "    "
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "db405808",
+   "metadata": {},
+   "source": [
+    "<br>\n",
+    "\n",
+    "#### Problem statement\n",
+    "\n",
+    "Now we can state the order-finding problem.\n",
+    "\n",
+    "<!-- ::: q-block.note -->\n",
+    "\n",
+    "**Order finding**\n",
+    "\n",
+    "Input: positive integers $N$ and $a$ satisfying $\\gcd(N, a) = 1$\n",
+    "\n",
+    "Output: the smallest positive integer $r$ such that $a^r \\equiv 1 \\pmod N$\n",
+    "\n",
+    "<!-- ::: -->\n",
+    "\n",
+    "Alternatively, in terms of the notation we just introduced above, we're given $a \\in \\mathbb{Z}_N^{\\ast}$, and we're looking for the smallest positive integer $r$ such that $a^r = 1$.\n",
+    "This number $r$ is called the *order* of $a$ modulo $N$."
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "a12335d6",
+   "metadata": {},
+   "source": [
+    "#### Multiplication by an element in $\\mathbb{Z}_N^{\\ast}$\n",
+    "\n",
+    "To connect the order-finding problem to phase estimation, let's think about the operation defined on a system whose classical states correspond to $\\mathbb{Z}_N$, where we multiply by a fixed element $a\\in\\mathbb{Z}_N^{\\ast}$.\n",
+    "\n",
+    "$$\n",
+    "M_a \\vert x\\rangle = \\vert ax \\rangle \\qquad \\text{(for each $x\\in\\mathbb{Z}_N$)}\n",
+    "$$\n",
+    "\n",
+    "To be clear, we're doing the multiplication in $\\mathbb{Z}_N$, so it's implicit that we're taking the remainder modulo $N$ inside of the ket on the right-hand side of the equation.\n",
+    "\n",
+    "For example, if $N = 15$ and $a=2$, we have\n",
+    "\n",
+    "$$\n",
+    "\\begin{aligned}\n",
+    "M_{2} \\vert 0 \\rangle & = \\vert 0\\rangle \\\\[1mm]\n",
+    "M_{2} \\vert 1 \\rangle & = \\vert 2\\rangle \\\\[1mm]\n",
+    "M_{2} \\vert 2 \\rangle & = \\vert 4\\rangle \\\\[1mm]\n",
+    "M_{2} \\vert 3 \\rangle & = \\vert 6\\rangle \\\\[1mm]\n",
+    "M_{2} \\vert 4 \\rangle & = \\vert 8\\rangle \\\\[1mm]\n",
+    "M_{2} \\vert 5 \\rangle & = \\vert 10\\rangle \\\\[1mm]\n",
+    "M_{2} \\vert 6 \\rangle & = \\vert 12\\rangle \\\\[1mm]\n",
+    "M_{2} \\vert 7 \\rangle & = \\vert 14\\rangle \\\\[1mm]\n",
+    "M_{2} \\vert 8 \\rangle & = \\vert 1\\rangle \\\\[1mm]\n",
+    "M_{2} \\vert 9 \\rangle & = \\vert 3\\rangle \\\\[1mm]\n",
+    "M_{2} \\vert 10 \\rangle & = \\vert 5\\rangle \\\\[1mm]\n",
+    "M_{2} \\vert 11 \\rangle & = \\vert 7\\rangle \\\\[1mm]\n",
+    "M_{2} \\vert 12 \\rangle & = \\vert 9\\rangle \\\\[1mm]\n",
+    "M_{2} \\vert 13 \\rangle & = \\vert 11\\rangle \\\\[1mm]\n",
+    "M_{2} \\vert 14 \\rangle & = \\vert 13\\rangle\n",
+    "\\end{aligned}\n",
+    "$$\n",
+    "\n",
+    "So long as $\\gcd(a,N)=1$, this is a unitary operation.\n",
+    "It shuffles the elements of the standard basis\n",
+    "$\\{\\vert 0\\rangle,\\ldots,\\vert N-1\\rangle\\}$, so as a matrix it's a *permutation matrix*.\n",
+    "It's evident from its definition that this operation is deterministic, and a simple way to see that it's invertible is to think about the order $r$ of $a$ modulo $N$, and to recognize that the inverse of $M_a$ is $M_a^{r-1}$.\n",
+    "\n",
+    "$$\n",
+    "M_a^{r-1} M_a = M_a^r = M_{a^r} = M_1 = I\n",
+    "$$"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "bf37dea7",
+   "metadata": {},
+   "source": [
+    "There's another way to think about the inverse that doesn't require any knowledge of $r$ — which, after all, is what we're trying to compute.\n",
+    "For every element $a\\in\\mathbb{Z}_N^{\\ast}$ there's always a unique element $b\\in\\mathbb{Z}_N^{\\ast}$ that satisfies $ab=1$. \n",
+    "We denote this element $b$ by $a^{-1}$, and it can be computed efficiently.\n",
+    "(An extension of Euclid's GCD algorithm does it at cost quadratic in $\\operatorname{lg}(N)$.)\n",
+    "And thus\n",
+    "\n",
+    "$$\n",
+    "M_{a^{-1}} M_a = M_{a^{-1}a} = M_1 = I.\n",
+    "$$\n",
+    "\n",
+    "So, the operation $M_a$ is both deterministic and invertible.\n",
+    "That implies that it's described by a permutation matrix, and it's therefore unitary."
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "6fcc4a86",
+   "metadata": {},
+   "source": [
+    "#### Eigenvectors and eigenvalues of multiplication operations\n",
+    "\n",
+    "Now let's think about the eigenvectors and eigenvalues of the operation $M_a$, assuming that $a\\in\\mathbb{Z}_N^{\\ast}$. \n",
+    "As was just argued, this assumption tells us that $M_a$ is unitary.\n",
+    "\n",
+    "There are $N$ eigenvalues of $M_a$, possibly including the same eigenvalue repeated multiple times, and in general there's some freedom in selecting corresponding eigenvectors — but we won't need to worry about all of the possibilities.\n",
+    "Let's start simple and identify just one eigenvector of $M_a$.\n",
+    "\n",
+    "$$\n",
+    "\\vert \\psi_0 \\rangle = \\frac{\\vert 1 \\rangle + \\vert a \\rangle + \\cdots + \\vert a^{r-1} \\rangle}{\\sqrt{r}}\n",
+    "$$\n",
+    "\n",
+    "The number $r$ is the order of $a$ modulo $N$ — here and throughout the remainder of the lesson.\n",
+    "The eigenvalue associated with this eigenvector is $1$ because it isn't changed when we multiply by $a$.\n",
+    "\n",
+    "$$\n",
+    "M_a \\vert \\psi_0 \\rangle \n",
+    "= \\frac{\\vert a \\rangle + \\cdots + \\vert a^{r-1} \\rangle + \\vert a^r \\rangle}{\\sqrt{r}}\n",
+    "= \\frac{\\vert a \\rangle + \\cdots + \\vert a^{r-1} \\rangle + \\vert 1 \\rangle}{\\sqrt{r}}\n",
+    "= \\vert \\psi_0 \\rangle\n",
+    "$$\n",
+    "\n",
+    "This happens because $a^r = 1$, so each standard basis state $\\vert a^k \\rangle$ gets shifted to $\\vert a^{k+1} \\rangle$ for $k\\leq r-1$, and $\\vert a^{r-1} \\rangle$ gets shifted back to $\\vert 1\\rangle$.\n",
+    "Informally speaking, it's like we're slowly stirring $\\vert \\psi_0 \\rangle$, but it's already completely stirred so nothing changes.\n",
+    "\n",
+    "Here's another example of an eigenvector of $M_a$.\n",
+    "This one happens to be more interesting in the context of order finding and phase estimation.\n",
+    "\n",
+    "$$\n",
+    "\\vert \\psi_1 \\rangle = \\frac{\\vert 1 \\rangle + \\omega_r^{-1} \\vert a \\rangle + \\cdots + \\omega_r^{r-1}\\vert a^{r-1} \\rangle}{\\sqrt{r}}\n",
+    "$$\n",
+    "\n",
+    "Alternatively, we can write this vector using a summation as follows.\n",
+    "\n",
+    "$$\n",
+    "\\vert \\psi_1 \\rangle = \\frac{1}{\\sqrt{r}}\n",
+    "\\sum_{k = 0}^{r-1} \\omega_r^{-k} \\vert a^k \\rangle \n",
+    "$$\n",
+    "\n",
+    "Here we see the complex number $\\omega_r = e^{2\\pi i/r}$ showing up naturally, due to the underlying structure of multiplication by $a$ modulo $N$.\n",
+    "This time the corresponding eigenvalue is $\\omega_r$.\n",
+    "To see this, we can first compute like this:\n",
+    "\n",
+    "$$\n",
+    "M_a \\vert \\psi_1 \\rangle \n",
+    "= \\sum_{k = 0}^{r-1} \\omega_r^{-k} M_a\\vert a^k \\rangle \n",
+    "= \\sum_{k = 0}^{r-1} \\omega_r^{-k} \\vert a^{k+1} \\rangle \n",
+    "= \\sum_{k = 1}^{r} \\omega_r^{-(k - 1)} \\vert a^{k} \\rangle \n",
+    "= \\omega_r \\sum_{k = 1}^{r} \\omega_r^{-k} \\vert a^{k} \\rangle.\n",
+    "$$\n",
+    "\n",
+    "Then, because $\\omega_r^{-r} = 1 = \\omega_r^0$ and $\\vert a^r \\rangle = \\vert 1\\rangle = \\vert a^0\\rangle$, we see that\n",
+    "\n",
+    "$$\n",
+    "\\sum_{k = 1}^{r} \\omega_r^{-k} \\vert a^{k} \\rangle = \\sum_{k = 0}^{r-1} \\omega_r^{-k} \\vert a^k \\rangle\n",
+    "= \\vert\\psi_1\\rangle,\n",
+    "$$\n",
+    "\n",
+    "so $M_a \\vert\\psi_1\\rangle = \\omega_r \\vert\\psi_1\\rangle$.\n",
+    "\n",
+    "Using the same reasoning, we can identify additional eigenvector/eigenvalue pairs for $M_a$.\n",
+    "Indeed, for any choice of $j\\in\\{0,\\ldots,r-1\\}$ we have that\n",
+    "\n",
+    "$$\n",
+    "\\vert \\psi_j \\rangle = \\frac{1}{\\sqrt{r}}\n",
+    "\\sum_{k = 0}^{r-1} \\omega_r^{-jk} \\vert a^k \\rangle \n",
+    "$$\n",
+    "\n",
+    "is an eigenvector of $M_a$ whose corresponding eigenvalue is $\\omega_r^j$.\n",
+    "\n",
+    "$$\n",
+    "M_a \\vert \\psi_j \\rangle = \\omega_r^j \\vert \\psi_j \\rangle\n",
+    "$$\n",
+    "\n",
+    "There are other eigenvectors, such as $\\vert 0 \\rangle$, which has eigenvalue $1$, but we'll only be concerned with the eigenvectors $\\vert\\psi_0\\rangle,\\ldots,\\vert\\psi_{r-1}\\rangle$ that we've just identified."
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "3534900a",
+   "metadata": {},
+   "source": [
+    "#### Applying phase estimation\n",
+    "\n",
+    "To solve the order-finding problem for a given choice of $a\\in\\mathbb{Z}_N^{\\ast}$, we can apply the phase-estimation procedure to the operation $M_a$.\n",
+    "\n",
+    "To do this, we need to implement not only $M_a$ efficiently with a quantum circuit, but also $M_a^2$, $M_a^4$, $M_a^8$, and so on, going as far as needed to obtain a precise enough estimate from the phase estimation procedure.\n",
+    "Here we'll explain how this can be done, and we'll figure out exactly how much precision is needed a bit later.\n",
+    "\n",
+    "Let's start with the operation $M_a$ by itself.\n",
+    "Naturally, because we're working with the quantum circuit model, we'll use binary notation to encode the numbers between $0$ and $N-1$.\n",
+    "The largest number we need to encode is $N-1$, so the number of bits we need is\n",
+    "\n",
+    "$$\n",
+    "n = \\operatorname{lg}(N-1) = \\lfloor \\log(N-1) \\rfloor + 1.\n",
+    "$$\n",
+    "\n",
+    "For example, if $N = 21$ we have $n = \\operatorname{lg}(N-1) = 5$.\n",
+    "Here's what the encoding of elements of $\\mathbb{Z}_{20}$ as binary strings of length $5$ looks like.\n",
+    "\n",
+    "$$\n",
+    "\\begin{gathered}\n",
+    "0  \\mapsto 00000\\\\[1mm]\n",
+    "1  \\mapsto 00001\\\\[1mm]\n",
+    "\\vdots\\\\[1mm]\n",
+    "20 \\mapsto 10100\n",
+    "\\end{gathered}\n",
+    "$$\n",
+    "\n",
+    "And now, here's a precise definition of how $M_a$ is defined as an $n$-qubit operation.\n",
+    "\n",
+    "$$\n",
+    "M_a \\vert x\\rangle = \n",
+    "\\begin{cases}\n",
+    "\\vert ax \\; (\\textrm{mod}\\;N)\\rangle & 0\\leq x < N\\\\[1mm]\n",
+    "\\vert x\\rangle & N\\leq x < 2^n\n",
+    "\\end{cases}\n",
+    "$$\n",
+    "\n",
+    "The point is that although we only care about how $M_a$ works for $\\vert 0\\rangle,\\ldots,\\vert N-1\\rangle$, we do have to specify how it works for the remaining $2^n - N$ standard basis states — and we need to do this in a way that still gives us a unitary operation.\n",
+    "Defining $M_a$ so that it does nothing to the remaining standard basis states accomplishes this.\n",
+    "\n",
+    "Using the algorithms for integer multiplication and division discussed in the previous lesson, together with the methodology for reversible, garbage-free implementations of them, we can build a quantum circuit that performs $M_a$, for any choice of $a\\in\\mathbb{Z}_N^{\\ast}$, at cost $O(n^2)$.\n",
+    "Here's one way that this can be done.\n",
+    "\n",
+    "1. We build a circuit for performing the operation\n",
+    "\n",
+    "   $$\n",
+    "   \\vert x \\rangle \\vert y \\rangle \\mapsto \\vert x \\rangle \\vert y \\oplus f_a(x)\\rangle\n",
+    "   $$\n",
+    "\n",
+    "   where\n",
+    "\n",
+    "   $$\n",
+    "   f_a(x) = \n",
+    "   \\begin{cases}\n",
+    "   ax \\; (\\textrm{mod}\\;N) & 0\\leq x < N\\\\[1mm]\n",
+    "   x & N\\leq x < 2^n\n",
+    "   \\end{cases}\n",
+    "   $$\n",
+    "\n",
+    "   using the method described in the previous lesson.\n",
+    "   This gives us a circuit of size $O(n^2)$.\n",
+    "\n",
+    "2. We swap the two $n$-qubit systems using $n$ swap gates to swap the qubits individually.\n",
+    "\n",
+    "3. Along similar lines to the first step, we can build a circuit for the operation\n",
+    "\n",
+    "   $$\n",
+    "   \\vert x \\rangle \\vert y \\rangle \\mapsto \\vert x \\rangle \\bigl\\vert y \\oplus f_{a^{-1}}(x)\\bigr\\rangle\n",
+    "   $$\n",
+    "\n",
+    "   where $a^{-1}$ is the inverse of $a$ in $\\mathbb{Z}_N^{\\ast}$.\n",
+    "\n",
+    "By initializing the bottom $n$ qubits and composing the three steps, we obtain this transformation:\n",
+    "\n",
+    "$$\n",
+    "\\vert x \\rangle \\vert 0^n \\rangle \n",
+    "\\stackrel{\\text{step 1}}{\\mapsto} \n",
+    "\\vert x \\rangle \\vert f_a(x)\\rangle\n",
+    "\\stackrel{\\text{step 2}}{\\mapsto}\n",
+    "\\vert f_a(x)\\rangle \\vert x \\rangle\n",
+    "\\stackrel{\\text{step 3}}{\\mapsto}\n",
+    "\\vert f_a(x)\\rangle \\bigl\\vert x \\oplus f_{a^{-1}}(f_a(x)) \\bigr\\rangle\n",
+    "= \\vert f_a(x)\\rangle\\vert 0^n \\rangle \n",
+    "$$\n",
+    "\n",
+    "The total cost of the circuit we obtain is $O(n^2)$.\n",
+    "\n",
+    "To perform $M_a^2$, $M_a^4$, $M_a^8$, and so on, we can use exactly the same method, except that we replace $a$ with $a^2$, $a^4$, $a^8$, and so on, as elements of $\\mathbb{Z}_N^{\\ast}$.\n",
+    "That is, for any power $k$ we choose, we can create a circuit for $M_a^k$ not by iterating $k$ times the circuit for $M_a$, but instead by computing $b = a^k \\in \\mathbb{Z}_N^{\\ast}$ and then using the circuit for $M_b$.\n",
+    "\n",
+    "The computation of powers $a^k \\in \\mathbb{Z}_N$ is the *modular exponentiation* problem mentioned in the previous lesson.\n",
+    "This computation can be done *classically*, using the algorithm for modular exponentiation mentioned in the previous lesson (often called the *power algorithm* in computational number theory).\n",
+    "This time we don't need to implement this algorithm reversibly with a quantum circuit, we just need to do it classically.\n",
+    "\n",
+    "And we're fortunate that this is possible.\n",
+    "We're effectively offloading the problem of iterating $M_a$ a huge number of times, which can be exponential in the number $m$ we choose in phase estimation, to an efficient classical computation.\n",
+    "In terms of the quantum circuit we're running, the cost of $M_a$ iterated $k$ times is simply the cost of $M_b$, for $b = a^k$ — and so the cost is $O(n^2)$.\n",
+    "\n",
+    "For an arbitrary choice of a quantum circuit in the phase estimation problem this won't be possible, resulting in a cost for phase estimation that's *exponential* in the number $m$ we use in phase estimation. \n",
+    "By the power of computational number theory, the cost in the case at hand is *linear* in $m$."
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "c804ee0d",
+   "metadata": {},
+   "source": [
+    "#### A convenient eigenvector/eigenvalue pair\n",
+    "\n",
+    "To understand how we can solve the order-finding problem using phase estimation, let's start by supposing that\n",
+    "we run the phase estimation procedure on the operation $M_a$ using the eigenvector $\\vert\\psi_1\\rangle$.\n",
+    "Getting our hands on this eigenvector isn't easy, as it turns out, so this won't be the end of the story — but it's helpful to start here.\n",
+    "\n",
+    "The eigenvalue of $M_a$ corresponding to the eigenvector $\\vert \\psi_1\\rangle$ is\n",
+    "\n",
+    "$$\n",
+    "\\omega_r = e^{2\\pi i \\frac{1}{r}}.\n",
+    "$$\n",
+    "\n",
+    "That is, $\\omega_r = e^{2\\pi i \\theta}$ for $\\theta = 1/r$.\n",
+    "So, if we run the phase estimation procedure on $M_a$ using the eigenvector $\\vert\\psi_1\\rangle$, we'll get an approximation to $1/r$.\n",
+    "By computing the reciprocal we'll be able to learn $r$ — provided that our approximation is good enough.\n",
+    "\n",
+    "To be more precise, when we run the phase-estimation procedure using $m$ control qubits, what we obtain is a number\n",
+    "$y\\in\\{0,\\ldots,2^m-1\\}$, and we take $y/2^m$ as a guess for $\\theta$, which is $1/r$ in the case at hand.\n",
+    "To figure out what $r$ is from this approximation, the natural thing to do is to compute the reciprocal of our approximation and round to the nearest integer.\n",
+    "\n",
+    "$$\n",
+    "\\left\\lfloor \\frac{2^m}{y} + \\frac{1}{2} \\right\\rfloor\n",
+    "$$\n",
+    "\n",
+    "For example, let's suppose $r = 6$ and we perform phase estimation on $M_a$ with the eigenvector $\\vert\\psi_1\\rangle$ using $m = 5$ control bits.\n",
+    "The best $5$-bit approximation to $1/r = 1/6$ is $5/32$, and we have a pretty good chance (about $68\\%$ in this case) to obtain the outcome $y=5$ from phase estimation.\n",
+    "We have\n",
+    "\n",
+    "$$\n",
+    "\\frac{2^m}{y} = \\frac{32}{5} = 6.4\n",
+    "$$\n",
+    "\n",
+    "and rounding to the nearest integer gives $6$, which is the correct answer.\n",
+    "\n",
+    "On the other hand, if we don't use enough precision we may not get the right answer.\n",
+    "For instance, if we take $m = 4$ control qubits in phase estimation, we might obtain the best $4$-bit approximation to $1/r = 1/6$, which is $3/16$.\n",
+    "Taking the reciprocal yields\n",
+    "\n",
+    "$$\n",
+    "\\frac{2^m}{y} = \\frac{16}{3} = 5.333 \\cdots\n",
+    "$$\n",
+    "\n",
+    "and rounding to the nearest integer gives an incorrect answer of $5$.\n",
+    "\n",
+    "How much precision do we need to get the right answer?\n",
+    "We know that the order $r$ is an integer, and intuitively speaking what we need is enough precision to distinguish $1/r$ from nearby possibilities, including $1/(r+1)$ and $1/(r-1)$.\n",
+    "The closest number to $1/r$ that we need to be concerned with is $1/(r+1)$, and the distance between these two numbers is\n",
+    "\n",
+    "$$\n",
+    "\\frac{1}{r} - \\frac{1}{r+1} = \\frac{1}{r(r+1)}.\n",
+    "$$\n",
+    "\n",
+    "So, if we want to make sure that we don't mistake $1/r$ for $1/(r+1)$, it suffices to use enough precision to guarantee that a best approximation $y/2^m$ to $1/r$ is closer to $1/r$ than it is to $1/(r+1)$.\n",
+    "If we use enough precision so that\n",
+    "\n",
+    "$$\n",
+    "\\left\\vert\n",
+    "\\frac{y}{2^m} - \\frac{1}{r}\n",
+    "\\right\\vert\n",
+    "< \\frac{1}{2 r (r+1)},\n",
+    "$$\n",
+    "\n",
+    "so that the error is less than half of the distance between $1/r$ and $1/(r+1)$, then $y/2^m$ will be closer to $1/r$ than to any other possibility, including $1/(r+1)$ and $1/(r-1)$.\n",
+    "\n",
+    "We can double-check this as follows.\n",
+    "Suppose that \n",
+    "\n",
+    "$$\n",
+    "\\frac{y}{2^m} = \\frac{1}{r} + \\varepsilon\n",
+    "$$\n",
+    "\n",
+    "for $\\varepsilon$ satisfying\n",
+    "\n",
+    "$$\n",
+    "\\vert\\varepsilon\\vert < \\frac{1}{2 r (r+1)}.\n",
+    "$$\n",
+    "\n",
+    "When we take the reciprocal we obtain\n",
+    "\n",
+    "$$\n",
+    "\\frac{2^m}{y} = \\frac{1}{\\frac{1}{r} + \\varepsilon} = \\frac{r}{1+\\varepsilon r} = r - \\frac{\\varepsilon r^2}{1+\\varepsilon r}.\n",
+    "$$\n",
+    "\n",
+    "By maximizing in the numerator and minimizing in the denominator, we can bound how far away we are from $r$ as follows.\n",
+    "\n",
+    "$$\n",
+    "\\left\\vert\n",
+    "\\frac{\\varepsilon r^2}{1+\\varepsilon r}\n",
+    "\\right\\vert\n",
+    "\\leq \\frac{ \\frac{r^2}{2 r(r+1)}}{1 - \\frac{r}{2r(r+1)}}\n",
+    "%= \\frac{r^2}{2 r (r+1) - r}\n",
+    "= \\frac{r}{2 r + 1}\n",
+    "< \\frac{1}{2}\n",
+    "$$\n",
+    "\n",
+    "We're less than $1/2$ away from $r$, so as expected we'll get $r$ when we round.\n",
+    "\n",
+    "Unfortunately, because we don't yet know what $r$ is, we can't use it to tell us how much accuracy we need.\n",
+    "What we can do instead is to use the fact that $r$ must be smaller than $N$ to ensure that we use enough precision.\n",
+    "In particular, if we use enough accuracy to guarantee that the best approximation $y/2^m$ to $1/r$ satisfies\n",
+    "\n",
+    "$$\n",
+    "\\left\\vert \\frac{y}{2^m} - \\frac{1}{r} \\right\\vert \\leq \\frac{1}{2N^2},\n",
+    "$$\n",
+    "\n",
+    "then we'll have enough precision to correctly determine $r$ when we take the reciprocal.\n",
+    "Taking $m = 2\\operatorname{lg}(N)+1$ ensures that we have a high chance to obtain an estimation with this precision using the method described previously.\n",
+    "(Taking $m = 2\\operatorname{lg}(N)$ is good enough if we're comfortable with a lower-bound of 40% on the probability of success.)"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "8f67a3de",
+   "metadata": {},
+   "source": [
+    "#### Other eigenvector/eigenvalue pairs\n",
+    "\n",
+    "As we just saw, if we had the eigenvector $\\vert \\psi_1 \\rangle$ of $M_a$, we would be able to learn $r$ through phase estimation, so long as we use enough control qubits to get sufficient precision to do this.\n",
+    "Unfortunately it's not easy to get our hands on the eigenvector $\\vert\\psi_1\\rangle$, so we need to figure out how to proceed.\n",
+    "\n",
+    "Let's suppose we proceed just like we did above, except with the eigenvector $\\vert\\psi_k\\rangle$ in place of $\\vert\\psi_1\\rangle$, for any choice of $k\\in\\{0,\\ldots,r-1\\}$ that we choose to think about.\n",
+    "The result we get from the phase estimation procedure will be an approximation\n",
+    "\n",
+    "$$\n",
+    "\\frac{y}{2^m} \\approx \\frac{k}{r}.\n",
+    "$$\n",
+    "\n",
+    "Working under the assumption that we don't know either $k$ or $r$, this might or might not allow us to identify $r$.\n",
+    "For example, if $k = 0$ we'll get an approximation $y/2^m$ to $0$, which unfortunately tells us nothing.\n",
+    "This, however, is an unusual case; for other values of $k$, we'll at least be able to learn something about $r$.\n",
+    "\n",
+    "We can use an algorithm known as the *continued fraction algorithm* to turn our approximation $y/2^m$ into nearby fractions — including $k/r$ if the approximation is good enough.\n",
+    "We won't explain the continued fraction algorithm here.\n",
+    "Instead, here's a statement of a known fact about this algorithm.\n",
+    "\n",
+    "\n",
+    "<!-- ::: q-block.note -->\n",
+    "\n",
+    "**Fact**\n",
+    "\n",
+    "Given an integer $N\\geq 2$ and a real number $\\alpha\\in(0,1)$, there is at most choice of integers $u,v\\in\\{0,\\ldots,N-1\\}$ with $v\\neq 0$ and $\\gcd(u,v)=1$ satisfying $\\vert \\alpha - u/v\\vert < \\frac{1}{2N^2}$.\n",
+    "\n",
+    "Given $\\alpha$ and $N$, the *continued fraction algorithm* finds $u$ and $v$ (or reports that they don't exist). \n",
+    "\n",
+    "This algorithm can be implemented as a Boolean circuit having size $O((\\operatorname{lg}(N))^3)$.\n",
+    "\n",
+    "<!-- ::: -->\n",
+    "\n",
+    "If we have a very close approximation $y/2^m$ to $r/k$, and we run the continued fraction algorithm for $N$ and $\\alpha = y/2^m$, we'll get $u$ and $v$, as they're described in the fact.\n",
+    "A careful reading of the fact allows us to conclude that\n",
+    "\n",
+    "$$\n",
+    "\\frac{u}{v} = \\frac{k}{r}.\n",
+    "$$\n",
+    "\n",
+    "So we don't necessarily learn $k$ and $r$, we only learn $k/r$ in lowest terms.\n",
+    "\n",
+    "For example, and as we've already noticed, we're not going to learn anything from $k=0$.\n",
+    "But that's the only value of $k$ where that happens.\n",
+    "When $k$ is nonzero, it might have common factors with $r$ — but the number $v$ we obtain from the continued fraction algorithm must divide $r$.\n",
+    "\n",
+    "It's far from obvious, but it is a known fact that if we have the ability to learn $u$ and $v$ for $u/v = k/r$ for $k\\in\\{0,\\ldots,r-1\\}$ chosen *uniformly at random*, then we're very likely to recover $r$ after just a few samples.\n",
+    "In particular, if our guess for $r$ is the *least common multiple* of all the values for $v$ that we observe, we'll be right with high probability.\n",
+    "Some values of $k$ aren't good because they share common factors with $r$, and those common factors are hidden to us when we learn $u$ and $v$.\n",
+    "But *random* choices of $k$ aren't likely to hide factors of $r$ for long, and the probability that we don't guess $r$ correctly drops exponentially in the number of samples."
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "697531be",
+   "metadata": {},
+   "source": [
+    "#### Proceeding without an eigenvector\n",
+    "\n",
+    "So far we haven't addressed the issue of how we get our hands on an eigenvector $\\vert\\psi_k\\rangle$ of $M_a$ to run the phase estimation procedure on.\n",
+    "As it turns out, we don't need to create them.\n",
+    "What we will do instead is to run the phase estimation procedure on the state $\\vert 1\\rangle$, by which we mean the $n$-bit binary encoding of the number $1$, in place of an eigenvector $\\vert\\psi\\rangle$ of $M_a$.\n",
+    "\n",
+    "So far, we've talked about running the phase estimation procedure on a particular eigenvector, but nothing prevents us from running the procedure on an input state that isn't an eigenvector of $M_a$, and that's what we're doing here with the state $\\vert 1\\rangle$.\n",
+    "(This isn't an eigenvector of $M_a$ for $a\\in\\mathbb{Z}_N^{\\ast}$ unless $a=1$, which is a choice for $a$ that we'll avoid.)\n",
+    "\n",
+    "The following equation helps to explain why we choose the state $\\vert 1\\rangle$ in place of an eigenvector.\n",
+    "\n",
+    "$$\n",
+    "\\vert 1\\rangle = \\frac{1}{\\sqrt{r}} \\sum_{k = 0}^{r-1} \\vert \\psi_k\\rangle\n",
+    "$$\n",
+    "\n",
+    "This can be verified by taking the inner product of the right-hand side with standard basis states and using formulas mentioned previously.\n",
+    "\n",
+    "In greater detail, let's imagine that we run the phase estimation procedure with the state $\\vert 1\\rangle$ in place of one of the eigenvectors $\\vert\\psi_k\\rangle$.\n",
+    "After the quantum Fourier transform is performed, this leaves us with the state\n",
+    "\n",
+    "$$\n",
+    "\\frac{1}{\\sqrt{r}} \\sum_{k = 0}^{r-1} \\vert \\psi_k\\rangle \\vert \\gamma_k\\rangle,\n",
+    "$$\n",
+    "\n",
+    "where\n",
+    "\n",
+    "$$\n",
+    "\\vert\\gamma_k\\rangle = \n",
+    "\\frac{1}{2^m} \\sum_{y=0}^{2^m - 1} \\sum_{x=0}^{2^m-1} e^{2\\pi i x (k/r - y/2^m)} \\vert y\\rangle\n",
+    "$$\n",
+    "\n",
+    "represents the state of the top $m$ qubits after the inverse of the quantum Fourier transform is performed.\n",
+    "When the top $m$ qubits are measured, we therefore obtain an approximation $y/2^m$ to the value $k/r$ where $k\\in\\{0,\\ldots,r-1\\}$ is chosen uniformly at random.\n",
+    "\n",
+    "As we've already discussed, this allows us to learn $r$ with a high degree of confidence after several independent runs, which was our goal."
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "342f425e",
+   "metadata": {},
+   "source": [
+    "#### Total cost\n",
+    "\n",
+    "The cost to implement each controlled-unitary $M_a^k$ is $O(n^2)$.\n",
+    "There are $m$ controlled-unitary operations, so the total cost for the controlled-unitary operations is $O(n^3)$.\n",
+    "In addition, we have $m$ Hadamard gates (which contribute $O(n)$ to the cost), and the quantum Fourier transform contributes $O(n^2)$ to the cost.\n",
+    "Thus, the cost of the controlled-unitary operations dominates the cost of the entire procedure — which is therefore $O(n^3)$.\n",
+    "\n",
+    "In addition to the quantum circuit itself, there are a few classical computations that need to be performed along the way.\n",
+    "This includes computing the powers $a^k$ in $\\mathbb{Z}_N$ for $k = 2, 4, 8, \\ldots, 2^{m-1}$, which are needed to create the controlled-unitary gates, as well as the continued fraction algorithm that converts approximations of $\\theta$ into fractions.\n",
+    "In both cases, these computations can be performed by Boolean circuits having cost $O(n^3)$.\n",
+    "\n",
+    "As is typical, all of these bounds can be improved using asymptotically fast algorithms; these bounds assume we're using standard algorithms for basic arithmetic operations."
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "5259f80f",
+   "metadata": {},
+   "source": [
+    "### Factoring by order-finding\n",
+    "\n",
+    "The very last thing we need to discuss is how solving the order-finding problem helps us to factor.\n",
+    "This part is completely classical — it has nothing specifically to do with quantum computing.\n",
+    "\n",
+    "Here's the basic idea.\n",
+    "We want to factorize the number $N$, and we can do this *recursively*.\n",
+    "Specifically, we can focus on the task of *splitting* $N$, which means finding any two integers $b,c\\geq 2$ for which $N = bc$. \n",
+    "This isn't possible if $N$ is a prime number, but we can efficiently test to see if $N$ is prime using a primality testing algorithm first, and if $N$ isn't prime we'll try to split it.\n",
+    "Once we split $N$, we can simply recurse on $b$ and $c$ until all of our factors are prime and we obtain the prime factorization of $N$.\n",
+    "\n",
+    "Splitting even integers is easy: we just output $2$ and $N/2$.\n",
+    "\n",
+    "It's also easy to split perfect powers, meaning numbers of the form $N = s^j$ for integers $s,j\\geq 2$, just by \n",
+    "approximating the roots $N^{1/2}$, $N^{1/3}$, $N^{1/4}$, etc., and checking nearby integers as suspects for $s$.\n",
+    "We don't need to go further than $\\log(N)$ steps into this sequence, because at that point the root drops below $2$ and won't reveal additional candidates.\n",
+    "\n",
+    "It's good that we can do both of these things because order-finding won't help us for even numbers or for *prime* powers, where the number $s$ happens to be prime.\n",
+    "\n",
+    "If $N$ is odd and not a prime power, order-finding allows us to split $N$.\n",
+    "\n",
+    "<!-- ::: q-block.note -->\n",
+    "\n",
+    "Input an odd, composite integer N that is not a prime power.\n",
+    "\n",
+    "Iterate the following steps:\n",
+    "\n",
+    "1. Randomly choose $a\\in\\{2,\\ldots,N-1\\}$. \n",
+    "2. Compute $d=\\gcd(a,N)$.\n",
+    "3. If $d > 1$ then output $b = d$ and $c = N/d$ and stop. Otherwise continue to the next step knowing that $a\\in\\mathbb{Z}_N^{\\ast}$.\n",
+    "4. Let $r$ be the order of $a$ modulo $N$. (Here's where we need order-finding.) \n",
+    "5. If $r$ is even: \\\n",
+    "   5.1 Compute $x = a^{r/2} - 1$ (modulo $N$) \\\n",
+    "   5.2 Compute $d = \\gcd(x,N)$. \\\n",
+    "   5.3 If $d>1$ then output $b=d$ and $c = N/d$ and stop.\n",
+    "6. If this point is reached, the iteration has failed to find a factor of $N$.\n",
+    "\n",
+    "<!-- ::: -->\n",
+    "\n",
+    "An iteration of this algorithm may fail to find a factor of $N$.\n",
+    "Specifically, this happens in two situations:\n",
+    "- The order of $a$ modulo $N$ is odd.\n",
+    "- The order of $a$ modulo $N$ is even and $\\gcd\\bigl(a^{r/2} - 1, N\\bigr) = 1$.\n",
+    "\n",
+    "Using basic number theory it can be proved that, for a random choice of $a$, with probability at least $1/2$ neither of these events happens.\n",
+    "In fact, the probability is at most $2^{-(m-1)}$ for $m$ being the number of distinct prime factors of $N$.\n",
+    "This is why the assumption that $N$ is not a prime power is important.\n",
+    "The assumption that $N$ is odd is also required for this to be true, which is why the (easy) case that $N$ is even has to be handled separately.\n",
+    "\n",
+    "So, if we repeat the process $t$ times, randomly choosing $a$ each time, we'll succeed in splitting $N$ with probability at least $1 - 2^{-t}$.\n",
+    "\n",
+    "We won't go through this analysis in detail, but here's the basic idea.\n",
+    "If we have a choice of $a$ for which the order $r$ of $a$ modulo $N$ is even, then it makes sense to consider the\n",
+    "numbers\n",
+    "\n",
+    "$$\n",
+    "a^{r/2} - 1\\; (\\textrm{mod}\\; N) \\quad \\text{and} \\quad a^{r/2} + 1\\; (\\textrm{mod}\\; N).\n",
+    "$$\n",
+    "\n",
+    "Using the formula $Z^2 - 1 = (Z+1)(Z-1)$, we conclude that\n",
+    "\n",
+    "$$\n",
+    "\\bigl(a^{r/2} - 1\\bigr) \\bigl(a^{r/2} + 1\\bigr) = a^r - 1.\n",
+    "$$\n",
+    "\n",
+    "We know that $a^r \\; (\\textrm{mod}\\; N) = 1$ by the definition of the order, which is another way of saying that\n",
+    "$N$ evenly divides $a^r - 1$.\n",
+    "So, $N$ evenly divides the number\n",
+    "\n",
+    "$$\n",
+    "\\bigl(a^{r/2} - 1\\bigr) \\bigl(a^{r/2} + 1\\bigr),\n",
+    "$$\n",
+    "\n",
+    "which means that every prime factor of $N$ must divide either $a^{r/2} - 1$ or $a^{r/2} + 1$ (or both).\n",
+    "For a randomly selected $a$, we're likely to have prime factors of $N$ dividing both terms, which allows us to split $N$ by computing the GCD."
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "e31d3e27",
+   "metadata": {},
+   "source": [
+    "### Implementation in Qiskit\n",
+    "\n",
+    "Here we'll implement Shor's algorithm in Qiskit to factor number $15$.\n",
+    "First we'll hard code a controlled-multiplication operation for a given element $a\\in\\mathbb{Z}_{15}^{\\ast}$."
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 13,
+   "id": "708b8242",
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "def c_amod15(a):\n",
+    "    \"\"\"\n",
+    "    Controlled multiplication by a mod 15.\n",
+    "    This is hard-coded for simplicity.\n",
+    "    \"\"\"\n",
+    "    if a not in [2,4,7,8,11,13]:\n",
+    "        raise ValueError(\"'a' must not have common factors with 15\")\n",
+    "    U = QuantumCircuit(4)\n",
+    "    if a in [2,13]:\n",
+    "        U.swap(2,3)\n",
+    "        U.swap(1,2)\n",
+    "        U.swap(0,1)\n",
+    "    if a in [7,8]:\n",
+    "        U.swap(0,1)\n",
+    "        U.swap(1,2)\n",
+    "        U.swap(2,3)\n",
+    "    if a in [4, 11]:\n",
+    "        U.swap(1,3)\n",
+    "        U.swap(0,2)\n",
+    "    if a in [7,11,13]:\n",
+    "        for q in range(4):\n",
+    "            U.x(q)\n",
+    "    U = U.to_gate()\n",
+    "    U.name = f\"{a} mod 15\"\n",
+    "    c_U = U.control()\n",
+    "    return c_U"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "952577bc",
+   "metadata": {},
+   "source": [
+    "Here's the phase estimation procedure from earlier implemented as a function."
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 14,
+   "id": "09cc6893",
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "def phase_estimation(\n",
+    "        controlled_operation: QuantumCircuit,\n",
+    "        psi_prep: QuantumCircuit,\n",
+    "        precision: int\n",
+    "    ):\n",
+    "    \"\"\"\n",
+    "    Carry out phase estimation on a simulator.\n",
+    "    Args:\n",
+    "        controlled_operation: The operation to perform phase estimation on,\n",
+    "                              controlled by one qubit.\n",
+    "        psi_prep: Circuit to prepare |ψ>\n",
+    "        precision: Number of counting qubits to use\n",
+    "    Returns:\n",
+    "        float: Best guess for phase of U|ψ>\n",
+    "    \"\"\"\n",
+    "    control_register = QuantumRegister(precision)\n",
+    "    output_register = ClassicalRegister(precision)\n",
+    "    \n",
+    "    target_register = QuantumRegister(psi_prep.num_qubits)\n",
+    "    qc = QuantumCircuit(control_register, target_register, output_register)\n",
+    "\n",
+    "    # Prepare |ψ>\n",
+    "    qc.compose(psi_prep,\n",
+    "               qubits=target_register,\n",
+    "               inplace=True)\n",
+    "\n",
+    "    # Do phase estimation\n",
+    "    for index, qubit in enumerate(control_register):\n",
+    "        qc.h(qubit)\n",
+    "        for _ in range(2**index):\n",
+    "            qc.compose(\n",
+    "                controlled_operation,\n",
+    "                qubits=[qubit] + list(target_register),\n",
+    "                inplace=True\n",
+    "            )\n",
+    "\n",
+    "    qc.compose(\n",
+    "        QFT(precision, inverse=True),\n",
+    "        qubits=control_register,\n",
+    "        inplace=True\n",
+    "    )\n",
+    "\n",
+    "    qc.measure(control_register, output_register)\n",
+    "    \n",
+    "    measurement = Sampler().run(qc, shots=1).result().quasi_dists[0].popitem()[0]\n",
+    "    return measurement / 2**precision"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "d9c82f40",
+   "metadata": {},
+   "source": [
+    "We can't easily prepare eigenvectors of the multiplication by $a\\in\\mathbb{Z}_{15}^{\\ast}$ operation, so we use $\\vert 1\\rangle$ as suggested above."
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 15,
+   "id": "0acc851d",
+   "metadata": {},
+   "outputs": [
+    {
+     "data": {
+      "text/plain": [
+       "<qiskit.circuit.instructionset.InstructionSet at 0x128faf790>"
+      ]
+     },
+     "execution_count": 15,
+     "metadata": {},
+     "output_type": "execute_result"
+    }
+   ],
+   "source": [
+    "psi_prep = QuantumCircuit(4)\n",
+    "psi_prep.x(0)"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "35181600",
+   "metadata": {},
+   "source": [
+    "And finally we can run the circuit to try to find a nontrivial factor of $15$."
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 16,
+   "id": "1d1e7c7b",
+   "metadata": {},
+   "outputs": [
+    {
+     "name": "stdout",
+     "output_type": "stream",
+     "text": [
+      "\n",
+      "Attempt 1\n"
+     ]
+    },
+    {
+     "name": "stdout",
+     "output_type": "stream",
+     "text": [
+      "\n",
+      "Attempt 2\n"
+     ]
+    },
+    {
+     "name": "stdout",
+     "output_type": "stream",
+     "text": [
+      "Non-trivial factor found: 3\n"
+     ]
+    }
+   ],
+   "source": [
+    "from fractions import Fraction\n",
+    "from math import gcd\n",
+    "\n",
+    "a = 8\n",
+    "N = 15\n",
+    "\n",
+    "FACTOR_FOUND = False\n",
+    "ATTEMPT = 0\n",
+    "while not FACTOR_FOUND:\n",
+    "    ATTEMPT += 1\n",
+    "    print(f\"\\nAttempt {ATTEMPT}\")\n",
+    "    \n",
+    "    phase = phase_estimation(\n",
+    "        c_amod15(a),\n",
+    "        psi_prep,\n",
+    "        precision=8\n",
+    "    )\n",
+    "    frac = Fraction(phase).limit_denominator(N)\n",
+    "    r = frac.denominator\n",
+    "    if phase != 0:\n",
+    "        # Guess for a factor is gcd(x^{r/2} - 1 , 15)\n",
+    "        guess = gcd(a**(r//2)-1, N)\n",
+    "        if guess not in [1,N] and (N % guess) == 0:\n",
+    "            # Guess is a factor!\n",
+    "            print(f\"Non-trivial factor found: {guess}\")\n",
+    "            FACTOR_FOUND = True       "
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 17,
+   "id": "37933870",
+   "metadata": {},
+   "outputs": [
+    {
+     "data": {
+      "text/html": [
+       "<h3>Version Information</h3><table><tr><th>Qiskit Software</th><th>Version</th></tr><tr><td><code>qiskit-terra</code></td><td>0.24.1</td></tr><tr><td><code>qiskit-aer</code></td><td>0.12.1</td></tr><tr><td><code>qiskit-ibmq-provider</code></td><td>0.20.2</td></tr><tr><td><code>qiskit</code></td><td>0.43.2</td></tr><tr><td><code>qiskit-nature</code></td><td>0.6.0</td></tr><tr><td><code>qiskit-finance</code></td><td>0.3.4</td></tr><tr><td><code>qiskit-optimization</code></td><td>0.5.0</td></tr><tr><td><code>qiskit-machine-learning</code></td><td>0.6.1</td></tr><tr><th>System information</th></tr><tr><td>Python version</td><td>3.10.11</td></tr><tr><td>Python compiler</td><td>Clang 14.0.0 (clang-1400.0.29.202)</td></tr><tr><td>Python build</td><td>main, Apr  7 2023 07:31:31</td></tr><tr><td>OS</td><td>Darwin</td></tr><tr><td>CPUs</td><td>8</td></tr><tr><td>Memory (Gb)</td><td>32.0</td></tr><tr><td colspan='2'>Tue Jul 18 12:32:18 2023 BST</td></tr></table>"
+      ],
+      "text/plain": [
+       "<IPython.core.display.HTML object>"
+      ]
+     },
+     "metadata": {},
+     "output_type": "display_data"
+    }
+   ],
+   "source": [
+    "# pylint: disable=unused-import\n",
+    "import qiskit.tools.jupyter\n",
+    "%qiskit_version_table"
+   ]
+  }
+ ],
+ "metadata": {
+  "kernelspec": {
+   "display_name": "Python 3",
+   "language": "python",
+   "name": "python3"
+  },
+  "language_info": {
+   "codemirror_mode": {
+    "name": "ipython",
+    "version": 3
+   },
+   "file_extension": ".py",
+   "mimetype": "text/x-python",
+   "name": "python",
+   "nbconvert_exporter": "python",
+   "pygments_lexer": "ipython3",
+   "version": "3.9"
+  }
+ },
+ "nbformat": 4,
+ "nbformat_minor": 5
+}
diff --git a/notebooks/v2/ch-applications/qaoa.ipynb b/notebooks/v2/ch-applications/qaoa.ipynb
index 43fc83568..b66b7c5d7 100644
--- a/notebooks/v2/ch-applications/qaoa.ipynb
+++ b/notebooks/v2/ch-applications/qaoa.ipynb
@@ -1,6 +1,7 @@
 {
  "cells": [
   {
+   "attachments": {},
    "cell_type": "markdown",
    "metadata": {
     "tags": [
@@ -12,17 +13,19 @@
    ]
   },
   {
+   "attachments": {},
    "cell_type": "markdown",
    "metadata": {},
    "source": [
     "In this tutorial, we will:\n",
-    "- introduce combinatorial optimization problems,\n",
-    "- explain approximate optimization algorithms,\n",
-    "- explain how the Quantum Approximate Optimization Algorithm (QAOA) works,\n",
-    "- and run a simple example on a simulator or a real quantum system."
+    "- introduce combinatorial optimization problems;\n",
+    "- explain approximate optimization algorithms;\n",
+    "- explain how the Quantum Approximate Optimization Algorithm (QAOA) works; and\n",
+    "- run a simple example on a simulator or a real quantum system."
    ]
   },
   {
+   "attachments": {},
    "cell_type": "markdown",
    "metadata": {},
    "source": [
@@ -38,6 +41,7 @@
    ]
   },
   {
+   "attachments": {},
    "cell_type": "markdown",
    "metadata": {},
    "source": [
@@ -45,6 +49,7 @@
    ]
   },
   {
+   "attachments": {},
    "cell_type": "markdown",
    "metadata": {},
    "source": [
@@ -53,12 +58,13 @@
    ]
   },
   {
+   "attachments": {},
    "cell_type": "markdown",
    "metadata": {},
    "source": [
     "## QAOA\n",
     "\n",
-    "QAOA (Quantum Approximate Optimization Algorithm) introduced by Farhi et al.[1] is a quantum algorithm that attempts to \n",
+    "QAOA (Quantum Approximate Optimization Algorithm), introduced by Farhi et al. [1], is a quantum algorithm that attempts to \n",
     "solve such combinatorial problems.<br> \n",
     "\n",
     "It is a variational algorithm that uses a unitary $U(\\boldsymbol{\\beta}, \\boldsymbol{\\gamma})$ characterized by the parameters \n",
@@ -80,8 +86,8 @@
     "\n",
     "where $\\lvert \\psi_0 \\rangle$ is a suitable initial state. \n",
     "\n",
-    "We will demonstrate these steps using the Max-Cut problem discussed above. For that we would first define\n",
-    "the underlying graph of the problem shown above."
+    "We will demonstrate these steps using the Max-Cut problem discussed above. For that, we would first define\n",
+    "the underlying graph of the problem as shown above."
    ]
   },
   {
@@ -111,6 +117,7 @@
    ]
   },
   {
+   "attachments": {},
    "cell_type": "markdown",
    "metadata": {},
    "source": [
@@ -146,6 +153,7 @@
    ]
   },
   {
+   "attachments": {},
    "cell_type": "markdown",
    "metadata": {},
    "source": [
@@ -193,6 +201,7 @@
    ]
   },
   {
+   "attachments": {},
    "cell_type": "markdown",
    "metadata": {},
    "source": [
@@ -229,6 +238,7 @@
    ]
   },
   {
+   "attachments": {},
    "cell_type": "markdown",
    "metadata": {},
    "source": [
@@ -238,7 +248,7 @@
     "\n",
     "$$\\lvert \\psi_0 \\rangle = \\bigg(\\frac{1}{\\sqrt{2}}\\big(\\lvert 0 \\rangle + \\lvert 1 \\rangle\\big)\\bigg)^{\\otimes n}$$\n",
     "\n",
-    "Such a state, when number of qubits is 4 ($n=4$), can be prepared by applying Hadamard gates starting from an all zero state as shown in \n",
+    "Such a state, when the number of qubits is 4 ($n=4$), can be prepared by applying Hadamard gates starting from an all-zero state as shown in \n",
     "the circuit below. "
    ]
   },
@@ -270,12 +280,13 @@
    ]
   },
   {
+   "attachments": {},
    "cell_type": "markdown",
    "metadata": {},
    "source": [
     "### The QAOA circuit\n",
     "\n",
-    "So far we have seen that the preparation of a quantum state during QAOA is composed of three elements\n",
+    "So far, we have seen that the preparation of a quantum state during QAOA is composed of three elements:\n",
     "- Preparing an initial state\n",
     "- Applying the unitary `{latex} U(H_P) = e^{-i \\gamma H_P}` corresponding to the problem Hamiltonian\n",
     "- Then, applying the mixing unitary `{latex} U(H_B) = e^{-i \\beta H_B}`\n",
@@ -312,6 +323,7 @@
    ]
   },
   {
+   "attachments": {},
    "cell_type": "markdown",
    "metadata": {},
    "source": [
@@ -461,6 +473,7 @@
    ]
   },
   {
+   "attachments": {},
    "cell_type": "markdown",
    "metadata": {},
    "source": [
@@ -468,6 +481,7 @@
    ]
   },
   {
+   "attachments": {},
    "cell_type": "markdown",
    "metadata": {},
    "source": [
@@ -506,6 +520,7 @@
    ]
   },
   {
+   "attachments": {},
    "cell_type": "markdown",
    "metadata": {},
    "source": [
@@ -513,6 +528,7 @@
    ]
   },
   {
+   "attachments": {},
    "cell_type": "markdown",
    "metadata": {},
    "source": [
@@ -520,12 +536,13 @@
    ]
   },
   {
+   "attachments": {},
    "cell_type": "markdown",
    "metadata": {},
    "source": [
     "### 1. Constructing Problem Hamiltonian <a id='Constructing Hamiltonian'></a>\n",
     "\n",
-    "Any maximization problem can be cast in terms of a minimization problem and vice versa. Hence the general form a combinatorial optimization problem is given by\n",
+    "Any maximization problem can be cast in terms of a minimization problem and vice versa. Hence, the general form a combinatorial optimization problem is given by\n",
     "\n",
     "\n",
     "\n",
@@ -552,12 +569,13 @@
    ]
   },
   {
+   "attachments": {},
    "cell_type": "markdown",
    "metadata": {},
    "source": [
     "#### 1.1 Diagonal Hamiltonians\n",
     "\n",
-    "This cost function can be mapped to a Hamiltonian that is diagonal in the computational basis. Given the cost-function $C$ this Hamiltonian is then written as\n",
+    "This cost function can be mapped to a Hamiltonian that is diagonal in the computational basis. Given the cost function, $C$, this Hamiltonian is then written as\n",
     "\n",
     "\n",
     "\n",
@@ -595,6 +613,7 @@
    ]
   },
   {
+   "attachments": {},
    "cell_type": "markdown",
    "metadata": {},
    "source": [
@@ -605,7 +624,7 @@
     "\n",
     "#### 2.1 (weighted) $MAXCUT$\n",
     "\n",
-    "Consider an $n$-node non-directed graph *G = (V, E)* where *|V| = n* with edge weights $w_{ij}>0$, `{latex} w_{ij}=w_{ji}`, for $(i,j)\\in E$. A cut is defined as a partition of the original set V into two subsets. The cost function to be optimized is in this case the sum of weights of edges connecting points in the two different subsets, *crossing* the cut. By assigning $x_i=0$ or $x_i=1$ to each node $i$, one tries to maximize the global profit function (here and in the following summations run over indices 1,2,...,n)\n",
+    "Consider an $n$-node non-directed graph *G = (V, E)* where *|V| = n* with edge weights $w_{ij}>0$, `{latex} w_{ij}=w_{ji}`, for $(i,j)\\in E$. A cut is defined as a partition of the original set V into two subsets. The cost function to be optimized is, in this case, the sum of weights of edges connecting points in the two different subsets, *crossing* the cut. By assigning $x_i=0$ or $x_i=1$ to each node $i$, one tries to maximize the global profit function (here and in the following summations run over indices 1,2,...,n)\n",
     "\n",
     "\n",
     "\n",
@@ -648,16 +667,17 @@
     "\n",
     "for a bit string $\\textbf{x} \\in \\{0,1\\}^{133}$. The clause can only be satisfied by setting the bits $x_1 = 0$,$x_3 = 0$ and $x_{132} = 1$. The $\\text{3-SAT}$ problem now asks whether there is a bit string that satisfies all the $m$ clauses or whether no such string exists. This decision problem is the prime example of a problem that is $NP$-complete.  \n",
     "\n",
-    "The closely related optimization problem $\\text{MAX 3-SAT}$ asks to find the bit string $\\textbf{x}$ that satisfies the maximal number of clauses in $C(\\textbf{x})$. This can of course be turned again in to a decision problem if we ask where there exists a bit string that satisfies more than $\\tilde{m}$ of the $m$ clauses, which is again $NP$-complete."
+    "The closely related optimization problem $\\text{MAX 3-SAT}$ asks to find the bit string $\\textbf{x}$ that satisfies the maximal number of clauses in $C(\\textbf{x})$. Of course, this can be turned again in to a decision problem if we ask where there exists a bit string that satisfies more than $\\tilde{m}$ of the $m$ clauses, which is again $NP$-complete."
    ]
   },
   {
+   "attachments": {},
    "cell_type": "markdown",
    "metadata": {},
    "source": [
     "### 3. Approximate optimization algorithms <a id='approximateOPT'></a>\n",
     "\n",
-    "Both the previously considered problems $MAXCUT$ and $\\text{MAX 3-SAT}$ are actually known to be a NP-hard problems [3](#references). In fact it turns out that many combinatorial optimization problems are computationally hard to solve in general. In light of this fact, we can't expect to find a provably efficient algorithm, i.e. an algorithm with polynomial runtime in the problem size, that solves these problems. This also applies to quantum algorithms. There are two main approaches to dealing with such problems. First approach is approximation algorithms that are guaranteed to find solution of specified quality in polynomial time. The second approach are heuristic algorithms that don't have a polynomial runtime guarantee but appear to perform well on some instances of such problems. \n",
+    "Both the previously considered problems $MAXCUT$ and $\\text{MAX 3-SAT}$ are known to be a NP-hard problems [3](#references). In fact, it turns out that many combinatorial optimization problems are computationally hard to solve in general. In light of this, we can't expect to find a provably efficient algorithm, i.e., an algorithm with polynomial runtime in the problem size, that solves these problems; this also applies to quantum algorithms. There are two main approaches to dealing with such problems. First approach is approximation algorithms that are guaranteed to find solution of specified quality in polynomial time. The second approach, heuristic algorithms, don't have a polynomial runtime guarantee, but appear to perform well on some instances of such problems. \n",
     "\n",
     "Approximate optimization algorithms are efficient and provide a provable guarantee on how close the approximate solution is to the actual optimum of the problem. The guarantee typically comes in the form of an approximation ratio, $\\alpha \\leq 1$. A probabilistic approximate optimization algorithm guarantees that it produces a bit-string $\\textbf{x}^* \\in \\{0,1\\}^n$ so that *with high probability* we have that with a positive `{latex} C_\\text{max} = \\max_{\\textbf{x}}C(\\textbf{x})` \n",
     "\n",
@@ -667,18 +687,19 @@
     "\n",
     "\n",
     "\n",
-    "For the $MAXCUT$ problem there is a famous approximate algorithm due to Goemans and Williamson [2](#references) . This algorithm is based on an SDP relaxation of the original problem combined with a probabilistic rounding technique that yields an with high probability approximate solution $\\textbf{x}^*$ that has an approximation ratio of $\\alpha \\approx 0.878$. This approximation ratio is actually believed to optimal so we do not expect to see an improvement by using a quantum algorithm."
+    "For the $MAXCUT$ problem, there is a famous approximate algorithm due to Goemans and Williamson [2](#references). This algorithm, based on an SDP relaxation of the original problem, is combined with a probabilistic rounding technique that yields a high probability approximate solution $\\textbf{x}^*$ that has an approximation ratio of $\\alpha \\approx 0.878$. This approximation ratio is believed to be optimal. So, we do not expect to see an improvement by using a quantum algorithm."
    ]
   },
   {
+   "attachments": {},
    "cell_type": "markdown",
    "metadata": {},
    "source": [
     "<!-- vale QiskitTextbook.SentenceLengthErr = NO -->\n",
     "\n",
     "### 4. The QAOA algorithm <a id=\"QAOA\"></a>\n",
-    "The Quantum approximate optimization algorithm (QAOA) by Farhi, Goldstone and Gutmann [1](#references) is an example of a heuristic algorithm. Unlike Goemans-Williamson algorithm, QAOA does not come with performance guarantees.\n",
-    "QAOA takes the approach of classical approximate algorithms and looks for a quantum analogue that will likewise produce a classical bit string $x^*$ that with high probability is expected to have a good approximation ratio $\\alpha$. Before discussing the details, let us first present the general idea of this approach. \n",
+    "The Quantum Approximate Optimization Algorithm (QAOA) by Farhi, Goldstone and Gutmann [1](#references) is an example of a heuristic algorithm. Unlike Goemans-Williamson algorithm, QAOA does not come with performance guarantees.\n",
+    "QAOA takes the approach of classical approximate algorithms and looks for a quantum analogue that will likewise produce a classical bit string $x^*$. The outputted bit string is expected, with a high probability, to have a good approximation ratio $\\alpha$. Before discussing the details, let us first present the general idea of this approach. \n",
     "\n",
     "#### 4.1 Overview:\n",
     "\n",
@@ -808,6 +829,7 @@
    ]
   },
   {
+   "attachments": {},
    "cell_type": "markdown",
    "metadata": {},
    "source": [
@@ -827,6 +849,7 @@
    ]
   },
   {
+   "attachments": {},
    "cell_type": "markdown",
    "metadata": {},
    "source": [
diff --git a/notebooks/v2/ch-machine-learning/machine-learning-qiskit-pytorch.ipynb b/notebooks/v2/ch-machine-learning/machine-learning-qiskit-pytorch.ipynb
index 758a16539..4e7833482 100644
--- a/notebooks/v2/ch-machine-learning/machine-learning-qiskit-pytorch.ipynb
+++ b/notebooks/v2/ch-machine-learning/machine-learning-qiskit-pytorch.ipynb
@@ -160,10 +160,7 @@
     "    def run(self, thetas):\n",
     "        t_qc = transpile(self._circuit,\n",
     "                         self.backend)\n",
-    "        qobj = assemble(t_qc,\n",
-    "                        shots=self.shots,\n",
-    "                        parameter_binds = [{self.theta: theta} for theta in thetas])\n",
-    "        job = self.backend.run(qobj)\n",
+    "        job = self.backend.run(t_qc)\n",
     "        result = job.result().get_counts()\n",
     "        \n",
     "        counts = np.array(list(result.values()))\n",
@@ -425,7 +422,11 @@
   {
    "cell_type": "code",
    "execution_count": 9,
-   "metadata": {},
+   "metadata": {
+    "tags": [
+     "ignore-warning"
+    ]
+   },
    "outputs": [
     {
      "name": "stderr",
diff --git a/notebooks/v2/ch-states/atoms-computation.ipynb b/notebooks/v2/ch-states/atoms-computation.ipynb
index 60cbd6ac6..18c257f8c 100644
--- a/notebooks/v2/ch-states/atoms-computation.ipynb
+++ b/notebooks/v2/ch-states/atoms-computation.ipynb
@@ -258,7 +258,7 @@
    "source": [
     "The reason for running many times and showing the result as a histogram is because quantum computers may have some randomness in their results. In this case, since we aren’t doing anything quantum, we get just the ```00000000``` result with certainty.\n",
     "\n",
-    "Note that this result comes from a quantum simulator, which is a standard computer calculating what an ideal quantum computer would do. Simulations are only possible for small numbers of qubits (~30 qubits), but they are nevertheless a very useful tool when designing your first quantum circuits. To run on a real device you simply need to replace ```Aer.get_backend('aer_simulator')``` with the backend object of the device you want to use. "
+    "Note that this result comes from a quantum simulator, which is a standard computer calculating what an ideal quantum computer would do. Simulations are only possible for small numbers of qubits (~30 qubits), but they are nevertheless a very useful tool when designing your first quantum circuits. To run on a real device you simply need to replace `Aer.get_backend('aer_simulator')` with the backend object of the device you want to use. "
    ]
   },
   {
diff --git a/notebooks/v2/ch-states/introduction.ipynb b/notebooks/v2/ch-states/introduction.ipynb
index 0e89e0fde..884820288 100644
--- a/notebooks/v2/ch-states/introduction.ipynb
+++ b/notebooks/v2/ch-states/introduction.ipynb
@@ -74,7 +74,7 @@
    "cell_type": "markdown",
    "metadata": {},
    "source": [
-    "This chapter will be most effective for readers who are already familiar with vectors and matrices. Those who aren't familiar will likely be fine too, though it might be useful to consult our [Introduction to Linear Algebra for Quantum Computing](/course/ch-appendix/an-introduction-to-linear-algebra-for-quantum-computing) from time to time.\n",
+    "This chapter will be most effective for readers who are already familiar with vectors and matrices. Those who aren't familiar will likely be fine too, though it might be useful to consult our [Introduction to Linear Algebra for Quantum Computing](/course/ch-appendix/linear_algebra) from time to time.\n",
     "\n",
     "Since we will be using Qiskit, our Python-based framework for quantum computing, it would also be useful to know the basics of Python. Those who need a primer can consult the [Introduction to Python and Jupyter notebooks](/course/ch-prerequisites/introduction-to-python-and-jupyter-notebooks)."
    ]
diff --git a/notebooks/v2/ch-states/representing-qubit-states.ipynb b/notebooks/v2/ch-states/representing-qubit-states.ipynb
index f32fb3ca8..0969f5695 100644
--- a/notebooks/v2/ch-states/representing-qubit-states.ipynb
+++ b/notebooks/v2/ch-states/representing-qubit-states.ipynb
@@ -19,7 +19,7 @@
     "\n",
     "In quantum computers, our basic variable is the _qubit:_ a quantum variant of the bit. These have exactly the same restrictions as normal bits do: they can store only a single binary piece of information, and can only ever give us an output of [[0 or 1|-1 or 1|0|1|-1]]. However, they can also be manipulated in ways that can only be described by quantum mechanics. This gives us new gates to play with, allowing us to find new ways to design algorithms.\n",
     "\n",
-    "To fully understand these new gates, we first need to understand how to write down qubit states. For this we will use the mathematics of vectors, matrices, and complex numbers. Though we will introduce these concepts as we go, it would be best if you are comfortable with them already. If you need a more in-depth explanation or a refresher, you can find the guide [here](/course/ch-appendix/an-introduction-to-linear-algebra-for-quantum-computing).\n"
+    "To fully understand these new gates, we first need to understand how to write down qubit states. For this we will use the mathematics of vectors, matrices, and complex numbers. Though we will introduce these concepts as we go, it would be best if you are comfortable with them already. If you need a more in-depth explanation or a refresher, you can find the guide [here](/course/ch-appendix/linear_algebra).\n"
    ]
   },
   {
diff --git a/notebooks/v2/ch-states/single-qubit-gates.ipynb b/notebooks/v2/ch-states/single-qubit-gates.ipynb
index 8e73e6c58..e14b6b58f 100644
--- a/notebooks/v2/ch-states/single-qubit-gates.ipynb
+++ b/notebooks/v2/ch-states/single-qubit-gates.ipynb
@@ -298,7 +298,7 @@
     "\n",
     "$$ |\\circlearrowleft\\rangle, \\quad |\\circlearrowright\\rangle$$\n",
     "\n",
-    "We leave it as an exercise to calculate these. There are in fact an infinite number of bases; to form one, we simply need two orthogonal vectors. The eigenvectors of both Hermitian and unitary matrices form a basis for the vector space. Due to this property, we can be sure that the eigenstates of the X-gate and the Y-gate indeed form a basis for 1-qubit states (read more about this in the [linear algebra page](/course/ch-appendix/an-introduction-to-linear-algebra-for-quantum-computing) in the appendix)\n",
+    "We leave it as an exercise to calculate these. There are in fact an infinite number of bases; to form one, we simply need two orthogonal vectors. The eigenvectors of both Hermitian and unitary matrices form a basis for the vector space. Due to this property, we can be sure that the eigenstates of the X-gate and the Y-gate indeed form a basis for 1-qubit states (read more about this in the [linear algebra page](/course/ch-appendix/linear_algebra) in the appendix)\n",
     "\n",
     "### Quick Exercises\n",
     "1. Verify that $|+\\rangle$ and $|-\\rangle$ are in fact eigenstates of the X-gate.\n",
diff --git a/notebooks/v2/intro/entangled-states.ipynb b/notebooks/v2/intro/entangled-states.ipynb
index fbd35c049..6fb836aff 100644
--- a/notebooks/v2/intro/entangled-states.ipynb
+++ b/notebooks/v2/intro/entangled-states.ipynb
@@ -2126,7 +2126,7 @@
     "1. If z measurements are made on both qubits, they never both output `0`;\n",
     "2. If an x measurement of one qubit outputs `1`, a z measurement of the other will output `0` (because the `11` result never occurs in these cases).\n",
     "\n",
-    "These correspond exactly to the observations made for the colored shapes in the classical scenario. If we apply the same logic as we did in that case, we can analyse the case where both Alice and Bob make an x measurement, and they both get the result `1`. Bob would then be able to apply observation 2 and infer that Alice would certainly have gotten a `0` if she had instead done a z measurement. Alice can do the same, inferring that Bob would have gotten a `1` if he'd have made a z measurement. So together they conclude that, if they had both made z measurements, they would both have gotten the output `0`. Since observation 1 shows us that such a result would be impossible, Alice and Bob could predict that the `11` result will never occur.\n",
+    "These correspond exactly to the observations made for the colored shapes in the classical scenario. If we apply the same logic as we did in that case, we can analyse the case where both Alice and Bob make an x measurement, and they both get the result `1`. Bob would then be able to apply observation 2 and infer that Alice would certainly have gotten a `0` if she had instead done a z measurement. Alice can do the same, inferring that Bob would have gotten a `0` if he had made a z measurement. So together they conclude that, if they had both made z measurements, they would both have gotten the output `0`. Since observation 1 shows us that such a result would be impossible, Alice and Bob could predict that the `11` result will never occur.\n",
     "\n",
     "So let's see if they would be right."
    ]
diff --git a/notebooks/v2/intro/why-quantum-computing.ipynb b/notebooks/v2/intro/why-quantum-computing.ipynb
index 94e810d1d..8fd864bd1 100644
--- a/notebooks/v2/intro/why-quantum-computing.ipynb
+++ b/notebooks/v2/intro/why-quantum-computing.ipynb
@@ -21,6 +21,7 @@
    ]
   },
   {
+   "attachments": {},
    "cell_type": "markdown",
    "metadata": {
     "formulas": {
@@ -95,7 +96,7 @@
     "\n",
     "![graph of constant and linear running times vs input sizes for different running times](images/why-qc/graph-linear-constant.svg)\n",
     "\n",
-    "Here is one final example that is very particularly interesting to us. Let’s say I have a secret number (such as a PIN), and the problem is to guess it. In this case, the size of the problem is the length of the number.  \n",
+    "Here is one final example that is particularly interesting to us. Let’s say I have a secret number (such as a PIN), and the problem is to guess it. In this case, the size of the problem is the length of the number.  \n",
     "\n",
     "Let’s say the only way we can check if our answer is correct is by punching it into a keypad. Since we have no information about what that number might be, the best algorithm to find this secret number uses a ‘brute-force’ method, which means it does nothing clever and simply tries every possible number.\n",
     "\n",
diff --git a/notebooks/v2/quantum-hardware/density-matrix.ipynb b/notebooks/v2/quantum-hardware/density-matrix.ipynb
index e009b7e7d..e97479571 100644
--- a/notebooks/v2/quantum-hardware/density-matrix.ipynb
+++ b/notebooks/v2/quantum-hardware/density-matrix.ipynb
@@ -15,7 +15,7 @@
    "cell_type": "markdown",
    "metadata": {},
    "source": [
-    "Throughout the majority of the Qiskit textbook, we have been representing the state of our qubits using the state-vector notation introduced in the [Representing Qubit States](../ch-states/representing-qubit-states) section. Working with this representation is convenient when dealing with states that can always be expressed as a [linear combination](../ch-appendix/an-introduction-to-linear-algebra-for-quantum-computing#an-6-146) of basis states, each with an associated probability amplitude. However, in quantum computation and quantum communication, there are many practical scenarios in which the state of our qubits cannot be written down as linear combinations in a given basis, but instead must be expressed in terms of ensembles (statistical mixtures) of multiple states, each with an associated probability of occurrence.\n",
+    "Throughout the majority of the Qiskit textbook, we have been representing the state of our qubits using the state-vector notation introduced in the [Representing Qubit States](../ch-states/representing-qubit-states) section. Working with this representation is convenient when dealing with states that can always be expressed as a [linear combination](../ch-appendix/linear_algebra#linear_algebra-6-146) of basis states, each with an associated probability amplitude. However, in quantum computation and quantum communication, there are many practical scenarios in which the state of our qubits cannot be written down as linear combinations in a given basis, but instead must be expressed in terms of ensembles (statistical mixtures) of multiple states, each with an associated probability of occurrence.\n",
     "\n",
     "For example, consider the case where Alice wants to send Bob the state $| + \\rangle$. Imagine that, due to noise in their communication channel, there is a chance that the relative phase of the state could flip with some probability $p$. Therefore, Bob could end up with the \"flipped\" state $| - \\rangle$ (with probability $p$), or with the desired state $| + \\rangle$ (with probability $1 - p$). Notice that Bob’s state is either $| + \\rangle$ **or** $| - \\rangle$, but not a quantum superposition of the two. Thus, if we want to know what will happen to Bob’s qubit after he, for instance, applies some gates and performs a measurement, we will have to consider both of these cases separately. This might not be too difficult a task when dealing with only two states, but in situations where the number of possible states is larger, keeping track of how each of these states evolves individually can become impractical. As we will see throughout this section, this is where the density matrix representation comes in handy.\n",
     "\n",
@@ -63,10 +63,10 @@
     "\n",
     "$$ \n",
     "\\begin{aligned}\n",
-    "& \\rho = \\begin{bmatrix} \\alpha_0 \\\\ \\alpha_1 \\\\ \\vdots \\\\ \\alpha_N \\end{bmatrix} \\begin{bmatrix} \\alpha_0^* & \\alpha_1^* & \\dots & \\alpha_N^* \\end{bmatrix} \n",
+    "& \\rho = \\begin{bmatrix} \\alpha_0 \\\\ \\alpha_1 \\\\ \\vdots \\\\ \\alpha_{N-1} \\end{bmatrix} \\begin{bmatrix} \\alpha_0^* & \\alpha_1^* & \\dots & \\alpha_{N-1}^* \\end{bmatrix} \n",
     "\\\\\n",
     "\\\\\n",
-    "& \\rho = \\begin{bmatrix} |\\alpha_0|^2 & \\alpha_0 \\alpha_1^* & \\dots & \\alpha_0 \\alpha_N^* \\\\ \\alpha_1 \\alpha_0^* & |\\alpha_1|^2 & \\dots & \\alpha_1 \\alpha_N^* \\\\ \\vdots & \\vdots & \\ddots & \\vdots \\\\ \\alpha_N \\alpha_0^* & \\alpha_N \\alpha_1^* & \\dots & |\\alpha_N|^2 \\end{bmatrix}\n",
+    "& \\rho = \\begin{bmatrix} |\\alpha_0|^2 & \\alpha_0 \\alpha_1^* & \\dots & \\alpha_0 \\alpha_{N-1}^* \\\\ \\alpha_1 \\alpha_0^* & |\\alpha_1|^2 & \\dots & \\alpha_1 \\alpha_{N-1}^* \\\\ \\vdots & \\vdots & \\ddots & \\vdots \\\\ \\alpha_{N-1} \\alpha_0^* & \\alpha_{N-1} \\alpha_1^* & \\dots & |\\alpha_{N-1}|^2 \\end{bmatrix}\n",
     "\\end{aligned}\n",
     "$$\n",
     "\n",
@@ -109,7 +109,7 @@
    "cell_type": "markdown",
    "metadata": {},
    "source": [
-    "Let's once again consider the entangled pure state $| \\psi_{AB} \\rangle $. We can prepare this state by applying a Hadamard gate to the first qubit, and an CNOT between the first and second qubits:"
+    "Let's once again consider the entangled pure state $| \\psi_{AB} \\rangle $. We can prepare this state by applying a Hadamard gate to the first qubit, and a CNOT between the first and second qubits:"
    ]
   },
   {
@@ -176,7 +176,7 @@
    "cell_type": "markdown",
    "metadata": {},
    "source": [
-    "Similarly, we can use the `DensityMatrix.from_instruction()` class method to obtain density matrix representation for this same state:"
+    "Similarly, we can use the `DensityMatrix.from_instruction()` class method to obtain the density matrix representation for this same state:"
    ]
   },
   {
@@ -265,7 +265,7 @@
    "source": [
     "## 2. Mixed States <a id=\"mixed\"></a>\n",
     "\n",
-    "Mixed states are those that consist of statistical ensembles of different quantum states. This means that, unlike pure states, mixed states cannot be represented as linear superpositions of normalized state vectors. Lets first take a look at a simple example to explain what we mean by this. \n",
+    "Mixed states are those that consist of statistical ensembles of different quantum states. This means that, unlike pure states, mixed states cannot be represented as linear superpositions of normalized state vectors. Let's first take a look at a simple example to explain what we mean by this. \n",
     "\n",
     "Consider, once again, the two-qubit entangled state:\n",
     "\n",
@@ -299,9 +299,9 @@
     "\n",
     "$$ \\left \\{ p_0, p_1 \\right \\} = \\left \\{ 1/2, 1/2 \\right \\} $$\n",
     "\n",
-    "What this representation shows is that, upon measurement of $q_1$, $\\psi_B$ will be in either state $| 0_B \\rangle $ `or` state $| 1_B \\rangle $, each with a classical probability of occurrence of $ 1/2$. Notice we avoided using the ket notation for state $\\psi_B$ since we reserve kets only for state vectors that can be written as linear combinations in an orthonormal basis, which $\\psi_B$ is not.\n",
+    "What this representation shows is that, upon measurement of $q_1$, $\\psi_B$ will be in either state $| 0_B \\rangle $ **or** state $| 1_B \\rangle $, each with a classical probability of occurrence of $ 1/2$. Notice we avoided using the ket notation for state $\\psi_B$ since we reserve kets only for state vectors that can be written as linear combinations in an orthonormal basis, which $\\psi_B$ is not.\n",
     "\n",
-    "In general, a mixed state consisting of an ensemble of n pure states can be expressed in the form of a list of outcome elements:\n",
+    "In general, a mixed state consisting of an ensemble of $n$ pure states can be expressed in the form of a list of outcome elements:\n",
     "\n",
     "$$\n",
     "\\left \\{ |\\psi_j \\rangle \\right \\}_{j = 1}^n = \\left \\{ | \\psi_1 \\rangle, | \\psi_2 \\rangle, \\dots, | \\psi_n \\rangle \\right \\},\n",
@@ -313,7 +313,7 @@
     "\\left \\{ p_j \\right \\}_{j = 1}^n = \\left \\{ p_1, p_2, \\dots, p_n \\right \\}\n",
     "$$\n",
     "\n",
-    "Here, $p_j$ corresponds to the <i>classical</i> probability of the system being in state $|\\psi_j \\rangle$, and the total number of possible states, $n$, need not be equal to the dimension of the underlying [Hilbert space](/course/ch-appendix/an-introduction-to-linear-algebra-for-quantum-computing).\n",
+    "Here, $p_j$ corresponds to the <i>classical</i> probability of the system being in state $|\\psi_j \\rangle$, and the total number of possible states, $n$, need not be equal to the dimension of the underlying [Hilbert space](/course/ch-appendix/linear_algebra).\n",
     "\n",
     "Although this way of expressing $\\psi_B$ (or any general mixed state) is perfectly valid, it turns out to be somewhat inconvenient. Since a mixed state can consist of a myriad of pure states, it can be difficult to track how the whole ensemble evolves when, for example, gates are applied to it. It is here that we turn to the density matrix representation.\n",
     "\n",
@@ -348,7 +348,7 @@
     "\n",
     "$$ | \\psi_1 \\rangle = \\frac{1}{\\sqrt{2}} \\left( | 0 \\rangle  + | 1 \\rangle  \\right) $$\n",
     "\n",
-    "The remaining 20% of the times, the pulse applied to rotate the state is either too short or too long by $\\frac{\\pi}{6}$ radians about the x-axis. This means that when we use this Hadamard gate, we could end up with following two undesired outcome states:\n",
+    "The remaining 20% of the times, the pulse applied to rotate the state is either too short or too long by $\\frac{\\pi}{6}$ radians about the $x$-axis. This means that when we use this Hadamard gate, we could end up with following two undesired outcome states:\n",
     "\n",
     "$$ | \\psi_2 \\rangle = \\frac{\\sqrt{3}}{2}| 0 \\rangle  + \\frac{1}{2} | 1 \\rangle, \\quad \\quad | \\psi_3 \\rangle = \\frac{1}{2} | 0 \\rangle  + \\frac{\\sqrt{3}}{2} | 1 \\rangle $$\n",
     "\n",
@@ -356,13 +356,13 @@
     "\n",
     "<img src=\"images/noisyH_states.png\" width=\"750\"/>\n",
     "\n",
-    "Since we do not know the outcome of our qubit everytime we prepare it, we can represent it as a mixed state of the form:\n",
+    "Since we do not know the outcome of our qubit every time we prepare it, we can represent it as a mixed state of the form:\n",
     "\n",
     "$$ \\rho_H = \\frac{4}{5} | \\psi_1 \\rangle \\langle \\psi_1 | + \\frac{1}{10} | \\psi_2 \\rangle \\langle \\psi_2 | + \\frac{1}{10} | \\psi_3 \\rangle \\langle \\psi_3 | $$\n",
     "\n",
     "Here, the factors $\\frac{4}{5}, \\frac{1}{10}$ and $\\frac{1}{10}$ correspond to the classical probabilities of obtaining the states `{latex} | \\psi_1 \\rangle, | \\psi_2 \\rangle` and $| \\psi_3 \\rangle$, respectively. \n",
     "\n",
-    "By replacing each of these three possible state vectors into $\\rho$, we can find the density matrix that represents this mixture:\n",
+    "By substituting each of these three possible state vectors into $\\rho_H$, we can find the density matrix that represents this mixture:\n",
     "\n",
     "$$\n",
     "\\begin{aligned}\n",
@@ -437,7 +437,7 @@
     "\n",
     "### Unitary Evolution <a id=\"evolution\"></a>\n",
     "\n",
-    "A very natural question to ask at this point is: How do mixed states evolve under [unitary operations](../ch-gates/proving-universality.ipynb#2.2-Unitary-and-Hermitian-matrices-)? It can be shown, with little effort, that if an initial arbitrary state $|\\psi_j \\rangle$ with probability of occurrence $p_j$ evolves into the state $\\hat U|\\psi_j \\rangle$ after a unitary evolution, then the evolution of a density matrix, consisting of an essemble of normalized states `{latex} \\{|\\psi_j \\rangle \\}_{j = 1}^{n}` with probabilities `{latex} \\{p_j\\}_{j = 1}^{n}`, is given by:\n",
+    "A very natural question to ask at this point is: How do mixed states evolve under [unitary operations](../ch-gates/proving-universality.ipynb#2.2-Unitary-and-Hermitian-matrices-)? It can be shown, with little effort, that if an initial arbitrary state $|\\psi_j \\rangle$ with probability of occurrence $p_j$ evolves into the state $\\hat U|\\psi_j \\rangle$ after a unitary evolution, then the evolution of a density matrix, consisting of an ensemble of normalized states `{latex} \\{|\\psi_j \\rangle \\}_{j = 1}^{n}` with probabilities `{latex} \\{p_j\\}_{j = 1}^{n}`, is given by:\n",
     "\n",
     "$$ \n",
     "\\rho = \\sum_{j} p_j |\\psi_j \\rangle \\langle \\psi_j | \\enspace \\xrightarrow[]{\\enspace \\hat U \\enspace} \\enspace \\rho' = \\sum_{j} p_j \\hat U |\\psi_j \\rangle \\langle \\psi_j | \\hat U^{\\dagger} = \\hat U \\rho \\hat U^{\\dagger}\n",
@@ -457,7 +457,7 @@
     "\\end{aligned}\n",
     "$$\n",
     "\n",
-    "If we are interested in knowing what will be the resulting state $\\rho_{0}'$ after a [Y Pauli gate](../ch-states/single-qubit-gates.ipynb#ynzgatez) is applied, we then have:\n",
+    "If we are interested in knowing what the resulting state $\\rho_{0}'$ will be after a [Pauli Y gate](../ch-states/single-qubit-gates.ipynb#ynzgatez) is applied, we then have:\n",
     "\n",
     "$$\n",
     "\\begin{aligned}\n",
@@ -615,7 +615,7 @@
     "\\rho = \\begin{bmatrix} \\rho_{11} & \\rho_{12} & \\dots & \\rho_{1m} \\\\ \\rho_{21} & \\rho_{22} & \\dots & \\rho_{2m} \\\\ \\vdots & \\vdots & \\ddots & \\vdots \\\\ \\rho_{m1} & \\rho_{m2} & \\dots & \\rho_{mm} \\end{bmatrix}\n",
     "$$\n",
     "\n",
-    "It's worth noticing that, in $\\rho_{uv}$, the diagonal terms $\\rho_{kk}$ actually correspond to the probability of finding the system in a particular basis state $|\\phi_k \\rangle$:\n",
+    "It's worth noticing that, in $\\rho_{uv}$, the diagonal terms $\\rho_{kk}$ actually correspond to the probability of finding the system in a particular basis state $|\\phi_k \\rangle$ upon measurement:\n",
     "\n",
     "$$\n",
     "\\rho_{kk} = \\sum_{j} p_j \\alpha_{k}^{(j)} \\left ( \\alpha_{k}^{(j)} \\right )^* = \\sum_{j} p_j |\\alpha_{k}^{(j)} |^{2}\n",
@@ -792,7 +792,7 @@
     "\\text{Tr}_B \\left (| \\xi_u \\rangle \\langle \\xi_v | \\otimes | \\chi_u \\rangle \\langle \\chi_v | \\right ) \\equiv | \\xi_u \\rangle \\langle \\xi_v | \\text{ Tr} \\left ( | \\chi_u \\rangle \\langle \\chi_v | \\right )\n",
     "$$\n",
     "\n",
-    "$| \\xi_u \\rangle $ and $| \\xi_v \\rangle $ are arbitrary states in the subspace of $A$, and $| \\chi_u \\rangle $ and $| \\chi_v \\rangle $ arbitrary states in the subspace of $B$. $\\text{Tr}$ is the standard trace operation, which for two arbitrary states `{latex} \\text{Tr} \\left ( | \\chi_u \\rangle \\langle \\chi_v | \\right ) = \\langle \\chi_v |\\chi_u \\rangle `. Similarly, we can calculate the reduced density matrix of subsystem $B$ using the partial trace over $A$:\n",
+    "$| \\xi_u \\rangle $ and $| \\xi_v \\rangle $ are arbitrary states in the subspace of $A$, and $| \\chi_u \\rangle $ and $| \\chi_v \\rangle $ arbitrary states in the subspace of $B$. $\\text{Tr}$ is the standard trace operation, which for two arbitrary states satisfies `{latex} \\text{Tr} \\left ( | \\chi_u \\rangle \\langle \\chi_v | \\right ) = \\langle \\chi_v |\\chi_u \\rangle `. Similarly, we can calculate the reduced density matrix of subsystem $B$ using the partial trace over $A$:\n",
     "\n",
     "$$\n",
     "\\text{Tr}_A \\left (| \\xi_u \\rangle \\langle \\xi_v | \\otimes | \\chi_u \\rangle \\langle \\chi_v | \\right ) \\equiv \\text{Tr} \\left ( | \\xi_u \\rangle \\langle \\xi_v | \\right ) | \\chi_u \\rangle \\langle \\chi_v | \n",
@@ -886,8 +886,8 @@
     }
    ],
    "source": [
-    "rho_B = qi.partial_trace(rho_AB,[0])\n",
-    "rho_A = qi.partial_trace(rho_AB,[1])\n",
+    "rho_B = qi.partial_trace(rho_AB,[1])\n",
+    "rho_A = qi.partial_trace(rho_AB,[0])\n",
     "\n",
     "display(rho_B.draw('latex', prefix=\" \\\\rho_{B} = \"),\n",
     "        rho_A.draw('latex', prefix=\" \\\\rho_{A} = \"))"
@@ -948,7 +948,7 @@
     "\\hat \\sigma_z = \\begin{bmatrix} 1 & 0 \\\\ 0 & -1 \\end{bmatrix}\n",
     "$$\n",
     "\n",
-    "The $r_x$, $r_y$ and $r_z$ coefficients correspond to the components of what is known as the Bloch vector $\\vec r$. This is in fact the vector that represents our mixed state inside the Bloch sphere. Replacing the matrices into the expression for $\\rho$ and simplifying:\n",
+    "The $r_x$, $r_y$ and $r_z$ coefficients correspond to the components of what is known as the Bloch vector $\\vec r$. This is in fact the vector that represents our mixed state inside the Bloch sphere. Substituting the matrices into the expression for $\\rho$ and simplifying:\n",
     "\n",
     "$$\n",
     "\\rho = \\frac{1}{2} \\begin{bmatrix} 1 + r_z & r_x - i r_y \\\\ r_x + i r_y & 1 - r_z \\end{bmatrix} \n",
@@ -964,7 +964,7 @@
     "\n",
     "$$ r_x = 2 \\left(\\frac{\\sqrt{3}}{20} + \\frac{2}{5} \\right), \\quad r_y = 0, \\quad r_z = 0 $$\n",
     "\n",
-    "This means that, in the bloch sphere, $\\rho_H$ is represented by a vector $\\vec r$ extending from the origin in the positive x direction, with length $ r_x \\approx 0.973 $.\n",
+    "This means that, in the bloch sphere, $\\rho_H$ is represented by a vector $\\vec r$ extending from the origin in the positive $x$ direction, with length $ r_x \\approx 0.973 $.\n",
     "\n",
     "In Qiskit, we can use the `plot_bloch_multivector()` function to plot the density matrix of our mixed state in the Bloch sphere:"
    ]
@@ -1168,7 +1168,7 @@
     "\n",
     "$$ r_x \\approx 0.86602, \\quad r_y = 0, \\quad r_z = 0 $$\n",
     "\n",
-    "So, if now we use the `plot_bloch_multivector()` function in Qiskit to plot the composite state $\\rho_{CD}$, we see we get two Bloch vectors that correctly represent each of these reduced density matrices $\\rho_C$ and $\\rho_D$:"
+    "So, if we now use the `plot_bloch_multivector()` function in Qiskit to plot the composite state $\\rho_{CD}$, we see we get two Bloch vectors that correctly represent each of these reduced density matrices $\\rho_C$ and $\\rho_D$:"
    ]
   },
   {
diff --git a/notebooks/v2/quantum-machine-learning/introduction.ipynb b/notebooks/v2/quantum-machine-learning/introduction.ipynb
index 71309377c..ee72f4615 100644
--- a/notebooks/v2/quantum-machine-learning/introduction.ipynb
+++ b/notebooks/v2/quantum-machine-learning/introduction.ipynb
@@ -1,6 +1,7 @@
 {
  "cells": [
   {
+   "attachments": {},
    "cell_type": "markdown",
    "metadata": {
     "gloss": {
@@ -13,10 +14,11 @@
    "source": [
     "# Introduction\n",
     "\n",
-    "Machine learning has established itself as a successful interdisciplinary field which seeks to find patterns in data. Throwing in quantum computing gives rise to interesting areas of research that aim to use the principles of quantum mechanics to augment machine learning, or vice-versa. In this page, we aim to give you a glimpse into the exciting and rapidly changing field of near-term quantum machine learning."
+    "Machine learning has established itself as a successful interdisciplinary field which seeks to find patterns in data. Throwing in quantum computing gives rise to interesting areas of research that aim to use the principles of quantum mechanics to augment machine learning, or vice-versa. On this page, we aim to give you a glimpse into the exciting and rapidly changing field of near-term quantum machine learning."
    ]
   },
   {
+   "attachments": {},
    "cell_type": "markdown",
    "metadata": {},
    "source": [
@@ -52,7 +54,7 @@
     "\n",
     "<!-- ::: column(width=350) -->\n",
     "\n",
-    "Given a collection of unlabeled data $(x_i)$, we aim to learn some structure of the data; for example, grouping a set of viewers based their movie viewing history in order to recommend new movies.\n",
+    "Given a collection of unlabeled data $(x_i)$, we aim to learn some structure of the data; for example, grouping a set of viewers based on their movie viewing history in order to recommend new movies.\n",
     "\n",
     "<!-- ::: -->\n",
     "\n",
@@ -76,6 +78,7 @@
    ]
   },
   {
+   "attachments": {},
    "cell_type": "markdown",
    "metadata": {},
    "source": [
@@ -83,7 +86,7 @@
     "\n",
     "There are four different approaches to combining quantum computing and machine learning, differentiated by whether the data is classical _(**C**)_ or quantum _(**Q**)_, or whether the algorithm runs on a classical _(**C**)_ or quantum _(**Q**)_ computer. This is illustrated below from Reference 1.\n",
     "\n",
-    "In this context, a quantum dataset consists of observations from a natural or artificial quantum system, such as measurements of qubit interactions, while a classical dataset consists of observations from a classical system, such as times series, text, or images.\n",
+    "In this context, a quantum dataset consists of observations from a natural or artificial quantum system, such as measurements of qubit interactions, while a classical dataset consists of observations from a classical system, such as time series, text, or images.\n",
     "\n",
     "<!-- ::: q-block.tabs.qml-approaches -->\n",
     "\n",
@@ -122,6 +125,7 @@
    ]
   },
   {
+   "attachments": {},
    "cell_type": "markdown",
    "metadata": {},
    "source": [
@@ -170,17 +174,19 @@
    ]
   },
   {
+   "attachments": {},
    "cell_type": "markdown",
    "metadata": {},
    "source": [
     "There are two distinct categories of **QC** algorithms: those that require quantum random access memory ([qRAM](gloss:qram)), where data can be accessed in superposition, and those that don't. The various proposed qRAM-based QML algorithms, e.g. [qPCA](https://doi.org/10.1038/nphys3029), [qSVM](https://doi.org/10.1103/PhysRevLett.113.130503) and [qClustering](https://arxiv.org/abs/1307.0411), boast exponential speedups compared to their classical algorithms, however there are currently no viable hardware candidates for realizing qRAM. \n",
     "\n",
-    "Recently, most of the focus of **CQ** approaches to machine learning have been on near-term algorithms that can be executed on the current quantum devices. Classical machine learning techniques have made great strides in the past decade, enabled in large part by the availability of sufficiently powerful hardware. Maybe the existence of quantum hardware might enable further advances in the field.\n",
+    "Recently, most of the focus of **CQ** approaches to machine learning has been on near-term algorithms that can be executed on the current quantum devices. Classical machine learning techniques have made great strides in the past decade, enabled in large part by the availability of sufficiently powerful hardware. Maybe the existence of quantum hardware might enable further advances in the field.\n",
     "\n",
     "Note that this is a very dynamic area of current investigation by multiple research teams worldwide, and there are still many open questions, as well as confusing terminology and notation. We will try and be as consistent as possible in this course, but also point out when concepts have been referred to by different names, or how different concepts relate to each other."
    ]
   },
   {
+   "attachments": {},
    "cell_type": "markdown",
    "metadata": {},
    "source": [
diff --git a/notebooks/v2/quantum-machine-learning/qgan.ipynb b/notebooks/v2/quantum-machine-learning/qgan.ipynb
index d198087ce..1bdce66a0 100644
--- a/notebooks/v2/quantum-machine-learning/qgan.ipynb
+++ b/notebooks/v2/quantum-machine-learning/qgan.ipynb
@@ -1,6 +1,7 @@
 {
  "cells": [
   {
+   "attachments": {},
    "cell_type": "markdown",
    "id": "35533bd4",
    "metadata": {
@@ -122,23 +123,24 @@
    ]
   },
   {
+   "attachments": {},
    "cell_type": "markdown",
    "id": "dc364349",
    "metadata": {},
    "source": [
     "## Introduction\n",
     "\n",
-    "Generative [adversarial](gloss:adversarial) networks (GANs) [[1]](https://arxiv.org/abs/1406.2661) have swiftly risen to prominence as one of the most widely-adopted methods for unsupervised learning, with showcased abilities in photo-realistic image generation. Given the success of classical GANs, a natural question is whether this success will translate into a quantum computing GAN. In this page, we explore the theory behind quantum generative adversarial networks (QGANs), as well as the practice of implementing one in Qiskit to learn a [Gaussian distribution](gloss:gaussian-distribution). Lastly, we end off with a discussion around the potential use cases of quantum generative adversarial networks and link to relevant research for those who want to read further.\n",
+    "Generative [adversarial](gloss:adversarial) networks (GANs) [[1]](https://arxiv.org/abs/1406.2661) have swiftly risen to prominence as one of the most widely-adopted methods for unsupervised learning, with showcased abilities in photo-realistic image generation. Given the success of classical GANs, a natural question is whether this success will translate into a quantum computing GAN. In this page, we explore the theory behind quantum generative adversarial networks (QGANs), as well as the practice of implementing one in Qiskit to learn a [Gaussian distribution](gloss:gaussian-distribution). Lastly, we end off with a discussion around the potential use cases of quantum generative adversarial networks and links to relevant research for those who want to read further.\n",
     "\n",
     "## Classical generative models (theory)\n",
     "\n",
     "### Generative models\n",
     "\n",
-    "Until recently, the success of supervised learning models has completely overshadowed their generative counterparts. So much so, that the popularity of these supervised models might make it difficult to even conceptualize another approach to machine learning.\n",
+    "Until recently, the success of supervised learning models has completely overshadowed their generative counterparts. So much so that the popularity of these supervised models might make it difficult to even conceptualize another approach to machine learning.\n",
     "\n",
-    "The supervised approach, which feels intuitive to us by now, tries to make accurate predictions on new data, demonstrating that it has learned some underlying relations of the dataset. Generative models are different. Instead of focusing on key relationships between input data and labels, they learn to model the underlying distribution holistically, allowing it to generate new data samples. It's the difference between telling apart cats and dogs, and generating completely new images of cats and dogs. The latter is a richer, but also more difficult task.\n",
+    "The supervised approach, which feels intuitive to us by now, tries to make accurate predictions on new data, demonstrating that it has learned some underlying relations of the dataset. Generative models are different. Instead of focusing on key relationships between input data and labels, they learn to model the underlying distribution holistically, allowing it to generate new data samples. It's the difference between telling apart cats and dogs, and generating completely new images of cats and dogs. The latter is a richer but more difficult task.\n",
     "\n",
-    "Why is it more difficult? Adequately discriminating between given data can often be achieved through picking up on a few tell-tale features (like whiskers don't belong on eyes) which help form the strong decision boundaries in the high dimensional space. Consequently, machine learning researchers take great interest in generative models as these learning tasks seem to stab at a deeper notion of learning—trying to reproduce the underlying *creator* function.\n",
+    "Why is it more difficult? Adequately discriminating between given data can often be achieved through picking up on a few tell-tale features (like whiskers don't belong on eyes) to help form strong decision boundaries in the high dimensional space. Consequently, machine learning researchers take great interest in generative models as these learning tasks seem to stab at a deeper notion of learning—trying to reproduce the underlying *creator* function.\n",
     "\n",
     "So given a pool of training data, the goal of a generative model is to learn/reproduce the probability distribution that generated them. A particularly eye-catching application of GANs is generating [high-resolution visuals](https://thispersondoesnotexist.com/) or [composing music](https://magenta.tensorflow.org/gansynth). Below is a generated image of a fake cat.\n",
     "\n",
@@ -186,6 +188,7 @@
    ]
   },
   {
+   "attachments": {},
    "cell_type": "markdown",
    "id": "cdb1f887",
    "metadata": {
@@ -264,6 +267,7 @@
    ]
   },
   {
+   "attachments": {},
    "cell_type": "markdown",
    "id": "b13a292e",
    "metadata": {
@@ -279,7 +283,7 @@
     "\n",
     "There are many analogous concepts, specifically with the adversarial training, between GANs and QGANs. Most importantly, the training structure of GANs largely persists to QGANs. We alternately train the generator & discriminator circuit parameters, while freezing the other's parameters. Through this, the quantum generator learns the target quantum state by proxy of the discriminator's signals, similar to GANs. It's proven that the [stable equilibrium](https://en.wikipedia.org/wiki/Nash_equilibrium) of the quantum adversarial game also occurs when the generator produces data identical to a target distribution [[3]](https://journals.aps.org/prl/abstract/10.1103/PhysRevLett.121.040502).\n",
     "\n",
-    "The objective with a fully quantum QGAN is for the quantum generator to reproduce a desired state $|\\psi\\rangle$, using of course an adversarial training strategy. Similar to other variational quantum algorithms, the quantum generator moves towards this through an iterative update to its parameters directed by a classical optimizer. However, in the case of a QGAN, the generator's cost function landscape itself evolves and becomes better as the discriminator improves at recognizing real samples. Let's look at the general circuit schematics of what we will be building.\n",
+    "The objective of a fully quantum QGAN is for the quantum generator to reproduce a desired state $|\\psi\\rangle$, of course using an adversarial training strategy. Similar to other variational quantum algorithms, the quantum generator moves towards this through an iterative update to its parameters directed by a classical optimizer. However, in the case of a QGAN, the generator's cost function landscape evolves and becomes better as the discriminator improves at recognizing real samples. Let's look at the general circuit schematics of what we will be building.\n",
     "\n",
     "![real_rig](images/qgan/real_rig.svg)\n",
     "\n",
@@ -289,13 +293,14 @@
    ]
   },
   {
+   "attachments": {},
    "cell_type": "markdown",
    "id": "c293984e",
    "metadata": {},
    "source": [
     "The two circuits illustrated above show the generator concatenated with the discriminator and the unitary loading the target data concatenated with the discriminator.\n",
     "\n",
-    "The discriminator acts on the output from the generator, respectively, the target data unitary, as well as an additional qubit $|0\\rangle$. Finally, the discriminator classification takes place by measuring the last qubit. If the outcome corresponds to $|0\\rangle$ respectively $|1\\rangle$ the input data is being classified as real or fake respectively.\n",
+    "The discriminator acts on the output from the generator, respectively, the target data unitary, as well as an additional qubit $|0\\rangle$. Finally, the discriminator classification takes place by measuring the last qubit. If the outcome corresponds to $|0\\rangle$ or $|1\\rangle$, the input data is being classified as real or fake respectively.\n",
     "\n",
     "Looking at the first circuit diagram shows us how real data samples are fed into the discriminator. Since we are dealing with a fully quantum QGAN, we need to somehow encode this data of ours into a quantum state to feed into the discriminator. To that end, we prepare the real data through a unitary $\\class{u-sub-r}{U_R}$ such that we define:\n",
     "\n",
@@ -363,6 +368,7 @@
    ]
   },
   {
+   "attachments": {},
    "cell_type": "markdown",
    "id": "951faa09",
    "metadata": {
@@ -391,6 +397,7 @@
    ]
   },
   {
+   "attachments": {},
    "cell_type": "markdown",
    "id": "e2b5d880",
    "metadata": {},
@@ -417,6 +424,7 @@
    ]
   },
   {
+   "attachments": {},
    "cell_type": "markdown",
    "id": "8efc46f6",
    "metadata": {},
@@ -444,6 +452,7 @@
    ]
   },
   {
+   "attachments": {},
    "cell_type": "markdown",
    "id": "c7804cea",
    "metadata": {},
@@ -486,6 +495,7 @@
    ]
   },
   {
+   "attachments": {},
    "cell_type": "markdown",
    "id": "9af80d19",
    "metadata": {},
@@ -538,6 +548,7 @@
    ]
   },
   {
+   "attachments": {},
    "cell_type": "markdown",
    "id": "3315ed20",
    "metadata": {},
@@ -578,6 +589,7 @@
    ]
   },
   {
+   "attachments": {},
    "cell_type": "markdown",
    "id": "4c492e03",
    "metadata": {},
@@ -617,6 +629,7 @@
    ]
   },
   {
+   "attachments": {},
    "cell_type": "markdown",
    "id": "30613b6d",
    "metadata": {},
@@ -684,6 +697,7 @@
    ]
   },
   {
+   "attachments": {},
    "cell_type": "markdown",
    "id": "179962d2",
    "metadata": {},
@@ -715,6 +729,7 @@
    ]
   },
   {
+   "attachments": {},
    "cell_type": "markdown",
    "id": "6fe9bf55",
    "metadata": {
@@ -766,6 +781,7 @@
    ]
   },
   {
+   "attachments": {},
    "cell_type": "markdown",
    "id": "6537d4bf",
    "metadata": {},
@@ -796,6 +812,7 @@
    ]
   },
   {
+   "attachments": {},
    "cell_type": "markdown",
    "id": "d5659e5d",
    "metadata": {},
@@ -847,6 +864,7 @@
    ]
   },
   {
+   "attachments": {},
    "cell_type": "markdown",
    "id": "6e18b4bd",
    "metadata": {
@@ -871,6 +889,7 @@
    ]
   },
   {
+   "attachments": {},
    "cell_type": "markdown",
    "id": "f922c9d1",
    "metadata": {},
@@ -1029,6 +1048,7 @@
    ]
   },
   {
+   "attachments": {},
    "cell_type": "markdown",
    "id": "c8740474",
    "metadata": {
@@ -1113,6 +1133,7 @@
    ]
   },
   {
+   "attachments": {},
    "cell_type": "markdown",
    "id": "1bfb97a9",
    "metadata": {},
@@ -1125,6 +1146,7 @@
    ]
   },
   {
+   "attachments": {},
    "cell_type": "markdown",
    "id": "530fc1f1",
    "metadata": {
@@ -1174,6 +1196,7 @@
    ]
   },
   {
+   "attachments": {},
    "cell_type": "markdown",
    "id": "eb8fe912-2b4a-46a8-a1f7-b38e8350db5d",
    "metadata": {},
@@ -1215,6 +1238,7 @@
    ]
   },
   {
+   "attachments": {},
    "cell_type": "markdown",
    "id": "5fad5646",
    "metadata": {},
@@ -1267,6 +1291,7 @@
    ]
   },
   {
+   "attachments": {},
    "cell_type": "markdown",
    "id": "505b825d",
    "metadata": {},
@@ -1286,6 +1311,7 @@
    ]
   },
   {
+   "attachments": {},
    "cell_type": "markdown",
    "id": "f18b385b-d700-46a0-b3f4-f6c3a39dd1eb",
    "metadata": {},
@@ -1348,6 +1374,7 @@
    ]
   },
   {
+   "attachments": {},
    "cell_type": "markdown",
    "id": "c60f6b9e",
    "metadata": {},
@@ -1420,6 +1447,7 @@
    ]
   },
   {
+   "attachments": {},
    "cell_type": "markdown",
    "id": "8dcb4bb7",
    "metadata": {},
@@ -1448,6 +1476,7 @@
    ]
   },
   {
+   "attachments": {},
    "cell_type": "markdown",
    "id": "9951c8da",
    "metadata": {},
@@ -1622,6 +1651,7 @@
    ]
   },
   {
+   "attachments": {},
    "cell_type": "markdown",
    "id": "8b6e28ff-799e-4423-a492-f5640f554fd4",
    "metadata": {},
@@ -1703,6 +1733,7 @@
    ]
   },
   {
+   "attachments": {},
    "cell_type": "markdown",
    "id": "5a46142e",
    "metadata": {},
@@ -1711,6 +1742,7 @@
    ]
   },
   {
+   "attachments": {},
    "cell_type": "markdown",
    "id": "680ea277",
    "metadata": {},
@@ -1737,6 +1769,7 @@
    ]
   },
   {
+   "attachments": {},
    "cell_type": "markdown",
    "id": "eb8c2c33",
    "metadata": {
@@ -1766,6 +1799,7 @@
    ]
   },
   {
+   "attachments": {},
    "cell_type": "markdown",
    "id": "97580477",
    "metadata": {
diff --git a/notebooks/v2/quantum-machine-learning/unsupervised.ipynb b/notebooks/v2/quantum-machine-learning/unsupervised.ipynb
index 421b592d0..b56216bd8 100644
--- a/notebooks/v2/quantum-machine-learning/unsupervised.ipynb
+++ b/notebooks/v2/quantum-machine-learning/unsupervised.ipynb
@@ -1,6 +1,7 @@
 {
  "cells": [
   {
+   "attachments": {},
    "cell_type": "markdown",
    "metadata": {},
    "source": [
@@ -8,6 +9,7 @@
    ]
   },
   {
+   "attachments": {},
    "cell_type": "markdown",
    "metadata": {},
    "source": [
@@ -23,7 +25,7 @@
     "\n",
     "![](images/qgan/gan_general_flow.svg)\n",
     "\n",
-    "Given an a set of data (either classical or quantum) from the probability distribution $P_\\text{real}$, generative adversarial networks train a generator, $G_\\theta$, and discriminator, $D_\\phi$, simultaneously to create samples from $P_G$ that are indistinguishable from $P_\\text{real}$. The GAN optimization problem is formalized as:\n",
+    "Given a set of data (either classical or quantum) from the probability distribution $P_\\text{real}$, generative adversarial networks train a generator, $G_\\theta$, and discriminator, $D_\\phi$, simultaneously to create samples from $P_G$ that are indistinguishable from $P_\\text{real}$. The GAN optimization problem is formalized as:\n",
     "\n",
     "$$\\min_G \\max_D \\vec{E}_{x \\sim P_\\text{real}} \\left [ \\log D(x) \\right ] + \\vec{E}_{z \\sim P_G} [\\log ( 1- D(G(z)))] $$\n",
     "\n",
@@ -33,6 +35,7 @@
    ]
   },
   {
+   "attachments": {},
    "cell_type": "markdown",
    "metadata": {},
    "source": [
diff --git a/notebooks/v2/quantum-machine-learning/vqc.ipynb b/notebooks/v2/quantum-machine-learning/vqc.ipynb
index e165d653c..16489a4f3 100644
--- a/notebooks/v2/quantum-machine-learning/vqc.ipynb
+++ b/notebooks/v2/quantum-machine-learning/vqc.ipynb
@@ -1,6 +1,7 @@
 {
  "cells": [
   {
+   "attachments": {},
    "cell_type": "markdown",
    "metadata": {
     "formulas": {
@@ -53,12 +54,13 @@
    ]
   },
   {
+   "attachments": {},
    "cell_type": "markdown",
    "metadata": {},
    "source": [
     "## Variational algorithms\n",
     "\n",
-    "Variational algorithms were introduced in 2014, with the variational [eigensolver](gloss:eigensolver) in Reference 1 and the quantum approximate optimization algorithm in Reference 2. They are [near-term algorithm](gloss:near-term-algorithm)s, that can be executed on current quantum computers in concert with classical computers. \n",
+    "Variational algorithms were introduced in 2014, with the variational [eigensolver](gloss:eigensolver) in Reference 1 and the quantum approximate optimization algorithm in Reference 2. A variational algorithm is a [near-term algorithm](gloss:near-term-algorithm) that can be executed on current quantum computers in concert with classical computers. \n",
     "\n",
     "Using a parameterized quantum circuit, or ansatz, $U(\\theta)$, we prepare a state $|\\psi(\\theta) \\rangle = U(\\theta) |0 \\rangle$, and measure the expectation value using a quantum computer. We define a cost function $C(\\theta)$, that determines how good $\\theta$ is for the problem we are trying to solve. We then use a classical computer to calculate the cost function and provide updated circuit parameters using an optimization algorithm. The goal of the algorithm is to find the circuit parameters $\\theta$ for the parameterized quantum circuit $U(\\theta)$ that minimizes the cost function $C(\\theta)$. \n",
     "\n",
@@ -66,6 +68,7 @@
    ]
   },
   {
+   "attachments": {},
    "cell_type": "markdown",
    "metadata": {},
    "source": [
@@ -83,6 +86,7 @@
    ]
   },
   {
+   "attachments": {},
    "cell_type": "markdown",
    "metadata": {},
    "source": [
@@ -118,6 +122,7 @@
    ]
   },
   {
+   "attachments": {},
    "cell_type": "markdown",
    "metadata": {},
    "source": [
@@ -154,6 +159,7 @@
    ]
   },
   {
+   "attachments": {},
    "cell_type": "markdown",
    "metadata": {},
    "source": [
@@ -184,6 +190,7 @@
    ]
   },
   {
+   "attachments": {},
    "cell_type": "markdown",
    "metadata": {},
    "source": [
@@ -203,6 +210,7 @@
    ]
   },
   {
+   "attachments": {},
    "cell_type": "markdown",
    "metadata": {},
    "source": [
@@ -227,6 +235,7 @@
    ]
   },
   {
+   "attachments": {},
    "cell_type": "markdown",
    "metadata": {},
    "source": [
@@ -259,6 +268,7 @@
    ]
   },
   {
+   "attachments": {},
    "cell_type": "markdown",
    "metadata": {},
    "source": [
@@ -308,10 +318,11 @@
    ]
   },
   {
+   "attachments": {},
    "cell_type": "markdown",
    "metadata": {},
    "source": [
-    "8. We set up our classical optimizer, using [`SPSA`](https://qiskit.org/documentation/stubs/qiskit.algorithms.optimizers.SPSA.html) (as per Reference 3), initialize our variational circuit parameters for reproducibility, and optimize our cost function modifying the variational circuit parameters, using our 40 training data points. Note that the optimization will take a while to run. "
+    "8. We set up our classical optimizer, using [`SPSA`](https://qiskit.org/documentation/stubs/qiskit.algorithms.optimizers.SPSA.html) (as per Reference 3), initialize our variational circuit parameters for reproducibility, and optimize our cost function using our 40 training data points. This optimization process modifies the variational circuit parameters based on the output of the cost function. Note that the optimization will take a while to run. "
    ]
   },
   {
@@ -377,6 +388,7 @@
    ]
   },
   {
+   "attachments": {},
    "cell_type": "markdown",
    "metadata": {},
    "source": [
@@ -485,6 +497,7 @@
    ]
   },
   {
+   "attachments": {},
    "cell_type": "markdown",
    "metadata": {},
    "source": [
@@ -492,6 +505,7 @@
    ]
   },
   {
+   "attachments": {},
    "cell_type": "markdown",
    "metadata": {},
    "source": [
@@ -517,6 +531,7 @@
    ]
   },
   {
+   "attachments": {},
    "cell_type": "markdown",
    "metadata": {},
    "source": [
@@ -585,6 +600,7 @@
    ]
   },
   {
+   "attachments": {},
    "cell_type": "markdown",
    "metadata": {},
    "source": [
@@ -668,6 +684,7 @@
    ]
   },
   {
+   "attachments": {},
    "cell_type": "markdown",
    "metadata": {},
    "source": [
@@ -681,7 +698,7 @@
     "\n",
     "Our optimal circuit parameters, $\\vec{\\theta}^*$ are found when we find the minimum of the loss function, $f(\\vec{\\theta})$. However, there isn't a simple relationship between the loss function and the circuit parameters.\n",
     "\n",
-    "In fact, the loss landscape can be quite complicated, as shown in hills and valleys of the example below. The optimization method navigates us around the loss landscape, searching for the minimum, as shown by the black points and lines. we can see that two of the three searches end up in a local landscape minimum, rather than a global one. \n",
+    "As a result, the loss landscape can be complicated, as shown in the hills and valleys of the example below. The optimization method navigates us around the loss landscape, searching for the minimum, as shown by the black points and lines. We observe that two-thirds of the searches end up in a local landscape minimum, rather than a global one. \n",
     "\n",
     "![](images/vqc/loss-landscape.png)\n",
     "\n",
@@ -697,6 +714,7 @@
    ]
   },
   {
+   "attachments": {},
    "cell_type": "markdown",
    "metadata": {},
    "source": [
@@ -705,7 +723,7 @@
     "1. Alberto Peruzzo, Jarrod McClean, Peter Shadbolt, Man-Hong Yung, Xiao-Qi Zhou, Peter J. Love, Alán Aspuru-Guzik and Jeremy L. O'Brien, *A variational eigenvalue solver on a quantum processor*, Nature Communications, 5:4213 (2014), [doi.org:10.1038/ncomms5213](https://doi.org/10.1038/ncomms5213), [arXiv:1304.3061](https://arxiv.org/abs/1304.3061).\n",
     "1. Edward Farhi, Jeffrey Goldstone and Sam Gutmann, *A Quantum Approximate Optimization Algorithm* (2014), [arXiv:1411.4028](https://arxiv.org/abs/1411.4028).\n",
     "1. Vojtech Havlicek, Antonio D. Córcoles, Kristan Temme, Aram W. Harrow, Abhinav  Kandala, Jerry M. Chow and Jay M. Gambetta, *Supervised learning with quantum enhanced feature spaces*, Nature 567, 209-212 (2019), [doi.org:10.1038/s41586-019-0980-2](https://doi.org/10.1038/s41586-019-0980-2), [arXiv:1804.11326](https://arxiv.org/abs/1804.11326).\n",
-    "1. Jarrod R. McClean, Sergio Boixo, Vadim N. Smelyanskiy, Ryan Babbush and Hartmut Neven, *Barren plateaus in quantum neural network training landscapes*, Nature Communications volume 9, Article number: 4812 (2018), [doi.org:10.1038/s41467-018-07090-4](https://www.nature.com/articles/s41467-018-07090-4) [arXiv:1803.1117](https://arxiv.org/abs/1803.11173)"
+    "1. Jarrod R. McClean, Sergio Boixo, Vadim N. Smelyanskiy, Ryan Babbush and Hartmut Neven, *Barren plateaus in quantum neural network training landscapes*, Nature Communications volume 9, Article number: 4812 (2018), [doi.org:10.1038/s41467-018-07090-4](https://www.nature.com/articles/s41467-018-07090-4) [arXiv:1803.1117](https://arxiv.org/abs/1803.11173)."
    ]
   },
   {
diff --git a/notebooks/v2/toc.yaml b/notebooks/v2/toc.yaml
index 9245e9f2f..b397f2eac 100644
--- a/notebooks/v2/toc.yaml
+++ b/notebooks/v2/toc.yaml
@@ -169,7 +169,15 @@
     - title: Quantum query algorithms
       id: query-algorithms
       uuid: f236ed68-91f2-470a-8152-29f09e60a30c
-      url: /algorithms/quantum-query-algorithms
+      url: /algorithms/01.query-algorithms/quantum-query-algorithms
+    - title: Algorithmic foundations
+      id: algorithmic-foundations
+      uuid: 00b26932-7d64-436c-86a9-56091c0419ac
+      url: /algorithms/02.algorithmic-foundations/algorithmic-foundations
+    - title: Phase estimation
+      id: phase-estimation
+      uuid: b49f2ca0-240d-417a-8f51-a6288d504ca3
+      url: /algorithms/03.phase-estimation/phase-estimation-and-factoring
 
 - title: Introduction to Quantum Computing
   id: introduction-course
diff --git a/package-lock.json b/package-lock.json
index 8369711d0..804e5595c 100644
--- a/package-lock.json
+++ b/package-lock.json
@@ -13,24 +13,24 @@
         "@carbon/colors": "^10.37.1",
         "@carbon/icons-vue": "~10.48.0",
         "@carbon/themes": "^11.15.0",
-        "@carbon/vue": "^2.45.0",
+        "@carbon/vue": "^2.45.1",
         "@ckeditor/ckeditor5-build-classic": "^34.2.0",
         "@ckeditor/ckeditor5-vue": "^4.0.1",
-        "@codemirror/lang-python": "^6.1.2",
+        "@codemirror/lang-python": "^6.1.3",
         "@github/clipboard-copy-element": "^1.1.2",
         "@ibm/plex": "^6.1.1",
         "@jupyter-widgets/html-manager": "^0.20.5",
-        "@jupyter-widgets/jupyterlab-manager": "^3.1.4",
+        "@jupyter-widgets/jupyterlab-manager": "^3.1.5",
         "@jupyterlab/mathjax2": "^3.5.2",
         "@jupyterlab/outputarea": "^3.5.2",
         "@jupyterlab/rendermime": "^3.5.2",
         "@jupyterlab/services": "^6.5.2",
         "@lumino/widgets": "^1.36.0",
         "@mathigon/boost": "^1.1.1",
-        "@mathigon/euclid": "^1.1.10",
+        "@mathigon/euclid": "^1.1.12",
         "@mathigon/studio": "0.1.18",
         "@qiskit-community/qiskit-vue": "^3.4.0",
-        "@qiskit/web-components": "^0.10.3",
+        "@qiskit/web-components": "^0.15.3",
         "@webcomponents/webcomponentsjs": "^2.6.0",
         "carbon-components": "~10.58.3",
         "carbon-web-components": "^1.21.0",
@@ -59,11 +59,11 @@
       "devDependencies": {
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-      "integrity": "sha512-Km2mWHKKY5GzRg6i1j5OxOHQtuvVsgskLfigG25yTtbyfRGn/GNvIbRyOf1PSCKJ2aT/58TiuUsuOU5UToVViw==",
+      "version": "2.0.22",
+      "resolved": "https://registry.npmjs.org/nodemon/-/nodemon-2.0.22.tgz",
+      "integrity": "sha512-B8YqaKMmyuCO7BowF1Z1/mkPqLk6cs/l63Ojtd6otKjMx47Dq1utxfRxcavH1I7VSaL8n5BUaoutadnsX3AAVQ==",
       "dev": true,
       "requires": {
         "chokidar": "^3.5.2",
@@ -41798,9 +42029,9 @@
           "dev": true
         },
         "semver": {
-          "version": "5.7.1",
-          "resolved": "https://registry.npmjs.org/semver/-/semver-5.7.1.tgz",
-          "integrity": "sha512-sauaDf/PZdVgrLTNYHRtpXa1iRiKcaebiKQ1BJdpQlWH2lCvexQdX55snPFyK7QzpudqbCI0qXFfOasHdyNDGQ==",
+          "version": "5.7.2",
+          "resolved": "https://registry.npmjs.org/semver/-/semver-5.7.2.tgz",
+          "integrity": "sha512-cBznnQ9KjJqU67B52RMC65CMarK2600WFnbkcaiwWq3xy/5haFJlshgnpjovMVJ+Hff49d8GEn0b87C5pDQ10g==",
           "dev": true
         },
         "supports-color": {
@@ -41846,9 +42077,9 @@
           }
         },
         "semver": {
-          "version": "7.3.7",
-          "resolved": "https://registry.npmjs.org/semver/-/semver-7.3.7.tgz",
-          "integrity": "sha512-QlYTucUYOews+WeEujDoEGziz4K6c47V/Bd+LjSSYcA94p+DmINdf7ncaUinThfvZyu13lN9OY1XDxt8C0Tw0g==",
+          "version": "7.5.4",
+          "resolved": "https://registry.npmjs.org/semver/-/semver-7.5.4.tgz",
+          "integrity": "sha512-1bCSESV6Pv+i21Hvpxp3Dx+pSD8lIPt8uVjRrxAUt/nbswYc+tK6Y2btiULjd4+fnq15PX+nqQDC7Oft7WkwcA==",
           "dev": true,
           "requires": {
             "lru-cache": "^6.0.0"
@@ -42023,9 +42254,9 @@
           }
         },
         "semver": {
-          "version": "5.7.1",
-          "resolved": "https://registry.npmjs.org/semver/-/semver-5.7.1.tgz",
-          "integrity": "sha512-sauaDf/PZdVgrLTNYHRtpXa1iRiKcaebiKQ1BJdpQlWH2lCvexQdX55snPFyK7QzpudqbCI0qXFfOasHdyNDGQ==",
+          "version": "5.7.2",
+          "resolved": "https://registry.npmjs.org/semver/-/semver-5.7.2.tgz",
+          "integrity": "sha512-cBznnQ9KjJqU67B52RMC65CMarK2600WFnbkcaiwWq3xy/5haFJlshgnpjovMVJ+Hff49d8GEn0b87C5pDQ10g==",
           "dev": true
         },
         "shebang-command": {
@@ -42860,9 +43091,9 @@
           }
         },
         "semver": {
-          "version": "5.7.1",
-          "resolved": "https://registry.npmjs.org/semver/-/semver-5.7.1.tgz",
-          "integrity": "sha512-sauaDf/PZdVgrLTNYHRtpXa1iRiKcaebiKQ1BJdpQlWH2lCvexQdX55snPFyK7QzpudqbCI0qXFfOasHdyNDGQ=="
+          "version": "5.7.2",
+          "resolved": "https://registry.npmjs.org/semver/-/semver-5.7.2.tgz",
+          "integrity": "sha512-cBznnQ9KjJqU67B52RMC65CMarK2600WFnbkcaiwWq3xy/5haFJlshgnpjovMVJ+Hff49d8GEn0b87C5pDQ10g=="
         },
         "shebang-command": {
           "version": "1.2.0",
@@ -45025,9 +45256,9 @@
       },
       "dependencies": {
         "semver": {
-          "version": "5.7.1",
-          "resolved": "https://registry.npmjs.org/semver/-/semver-5.7.1.tgz",
-          "integrity": "sha512-sauaDf/PZdVgrLTNYHRtpXa1iRiKcaebiKQ1BJdpQlWH2lCvexQdX55snPFyK7QzpudqbCI0qXFfOasHdyNDGQ==",
+          "version": "5.7.2",
+          "resolved": "https://registry.npmjs.org/semver/-/semver-5.7.2.tgz",
+          "integrity": "sha512-cBznnQ9KjJqU67B52RMC65CMarK2600WFnbkcaiwWq3xy/5haFJlshgnpjovMVJ+Hff49d8GEn0b87C5pDQ10g==",
           "optional": true
         }
       }
@@ -45063,9 +45294,9 @@
           }
         },
         "semver": {
-          "version": "5.7.1",
-          "resolved": "https://registry.npmjs.org/semver/-/semver-5.7.1.tgz",
-          "integrity": "sha512-sauaDf/PZdVgrLTNYHRtpXa1iRiKcaebiKQ1BJdpQlWH2lCvexQdX55snPFyK7QzpudqbCI0qXFfOasHdyNDGQ==",
+          "version": "5.7.2",
+          "resolved": "https://registry.npmjs.org/semver/-/semver-5.7.2.tgz",
+          "integrity": "sha512-cBznnQ9KjJqU67B52RMC65CMarK2600WFnbkcaiwWq3xy/5haFJlshgnpjovMVJ+Hff49d8GEn0b87C5pDQ10g==",
           "optional": true
         }
       }
@@ -45101,9 +45332,9 @@
           }
         },
         "semver": {
-          "version": "5.7.1",
-          "resolved": "https://registry.npmjs.org/semver/-/semver-5.7.1.tgz",
-          "integrity": "sha512-sauaDf/PZdVgrLTNYHRtpXa1iRiKcaebiKQ1BJdpQlWH2lCvexQdX55snPFyK7QzpudqbCI0qXFfOasHdyNDGQ==",
+          "version": "5.7.2",
+          "resolved": "https://registry.npmjs.org/semver/-/semver-5.7.2.tgz",
+          "integrity": "sha512-cBznnQ9KjJqU67B52RMC65CMarK2600WFnbkcaiwWq3xy/5haFJlshgnpjovMVJ+Hff49d8GEn0b87C5pDQ10g==",
           "dev": true
         },
         "type-fest": {
@@ -45963,9 +46194,9 @@
           }
         },
         "semver": {
-          "version": "7.3.7",
-          "resolved": "https://registry.npmjs.org/semver/-/semver-7.3.7.tgz",
-          "integrity": "sha512-QlYTucUYOews+WeEujDoEGziz4K6c47V/Bd+LjSSYcA94p+DmINdf7ncaUinThfvZyu13lN9OY1XDxt8C0Tw0g==",
+          "version": "7.5.4",
+          "resolved": "https://registry.npmjs.org/semver/-/semver-7.5.4.tgz",
+          "integrity": "sha512-1bCSESV6Pv+i21Hvpxp3Dx+pSD8lIPt8uVjRrxAUt/nbswYc+tK6Y2btiULjd4+fnq15PX+nqQDC7Oft7WkwcA==",
           "dev": true,
           "requires": {
             "lru-cache": "^6.0.0"
@@ -46073,9 +46304,9 @@
       }
     },
     "semver": {
-      "version": "6.3.0",
-      "resolved": "https://registry.npmjs.org/semver/-/semver-6.3.0.tgz",
-      "integrity": "sha512-b39TBaTSfV6yBrapU89p5fKekE2m/NwnDocOVruQFS1/veMgdzuPcnOM34M6CwxW8jH/lxEa5rBoDeUwu5HHTw=="
+      "version": "6.3.1",
+      "resolved": "https://registry.npmjs.org/semver/-/semver-6.3.1.tgz",
+      "integrity": "sha512-BR7VvDCVHO+q2xBEWskxS6DJE1qRnb7DxzUrogb71CWoSficBxYsiAGd+Kl0mmq/MprG9yArRkyrQxTO6XjMzA=="
     },
     "send": {
       "version": "0.17.2",
@@ -47668,9 +47899,9 @@
           }
         },
         "semver": {
-          "version": "5.7.1",
-          "resolved": "https://registry.npmjs.org/semver/-/semver-5.7.1.tgz",
-          "integrity": "sha512-sauaDf/PZdVgrLTNYHRtpXa1iRiKcaebiKQ1BJdpQlWH2lCvexQdX55snPFyK7QzpudqbCI0qXFfOasHdyNDGQ==",
+          "version": "5.7.2",
+          "resolved": "https://registry.npmjs.org/semver/-/semver-5.7.2.tgz",
+          "integrity": "sha512-cBznnQ9KjJqU67B52RMC65CMarK2600WFnbkcaiwWq3xy/5haFJlshgnpjovMVJ+Hff49d8GEn0b87C5pDQ10g==",
           "dev": true
         }
       }
@@ -48110,9 +48341,9 @@
       }
     },
     "tslib": {
-      "version": "2.4.1",
-      "resolved": "https://registry.npmjs.org/tslib/-/tslib-2.4.1.tgz",
-      "integrity": "sha512-tGyy4dAjRIEwI7BzsB0lynWgOpfqjUdq91XXAlIWD2OwKBH7oCl/GZG/HT4BOHrTlPMOASlMQ7veyTqpmRcrNA=="
+      "version": "2.6.2",
+      "resolved": "https://registry.npmjs.org/tslib/-/tslib-2.6.2.tgz",
+      "integrity": "sha512-AEYxH93jGFPn/a2iVAwW87VuUIkR1FVUKB77NwMF7nBTDkDrrT/Hpt/IrCJ0QXhW27jTBDcf5ZY7w6RiqTMw2Q=="
     },
     "tslint": {
       "version": "5.20.1",
@@ -48214,9 +48445,9 @@
           }
         },
         "semver": {
-          "version": "5.7.1",
-          "resolved": "https://registry.npmjs.org/semver/-/semver-5.7.1.tgz",
-          "integrity": "sha512-sauaDf/PZdVgrLTNYHRtpXa1iRiKcaebiKQ1BJdpQlWH2lCvexQdX55snPFyK7QzpudqbCI0qXFfOasHdyNDGQ==",
+          "version": "5.7.2",
+          "resolved": "https://registry.npmjs.org/semver/-/semver-5.7.2.tgz",
+          "integrity": "sha512-cBznnQ9KjJqU67B52RMC65CMarK2600WFnbkcaiwWq3xy/5haFJlshgnpjovMVJ+Hff49d8GEn0b87C5pDQ10g==",
           "dev": true
         },
         "supports-color": {
@@ -48835,9 +49066,9 @@
           }
         },
         "semver": {
-          "version": "7.3.7",
-          "resolved": "https://registry.npmjs.org/semver/-/semver-7.3.7.tgz",
-          "integrity": "sha512-QlYTucUYOews+WeEujDoEGziz4K6c47V/Bd+LjSSYcA94p+DmINdf7ncaUinThfvZyu13lN9OY1XDxt8C0Tw0g==",
+          "version": "7.5.4",
+          "resolved": "https://registry.npmjs.org/semver/-/semver-7.5.4.tgz",
+          "integrity": "sha512-1bCSESV6Pv+i21Hvpxp3Dx+pSD8lIPt8uVjRrxAUt/nbswYc+tK6Y2btiULjd4+fnq15PX+nqQDC7Oft7WkwcA==",
           "dev": true,
           "requires": {
             "lru-cache": "^6.0.0"
@@ -50208,9 +50439,9 @@
       }
     },
     "word-wrap": {
-      "version": "1.2.3",
-      "resolved": "https://registry.npmjs.org/word-wrap/-/word-wrap-1.2.3.tgz",
-      "integrity": "sha512-Hz/mrNwitNRh/HUAtM/VT/5VH+ygD6DV7mYKZAtHOrbs8U7lvPS6xf7EJKMF0uW1KJCl0H701g3ZGus+muE5vQ=="
+      "version": "1.2.4",
+      "resolved": "https://registry.npmjs.org/word-wrap/-/word-wrap-1.2.4.tgz",
+      "integrity": "sha512-2V81OA4ugVo5pRo46hAoD2ivUJx8jXmWXfUkY4KFNw0hEptvN0QfH3K4nHiwzGeKl5rFKedV48QVoqYavy4YpA=="
     },
     "worker-farm": {
       "version": "1.7.0",
diff --git a/package.json b/package.json
index ef7d7f59f..d3e9ad895 100644
--- a/package.json
+++ b/package.json
@@ -49,24 +49,24 @@
     "@carbon/colors": "^10.37.1",
     "@carbon/icons-vue": "~10.48.0",
     "@carbon/themes": "^11.15.0",
-    "@carbon/vue": "^2.45.0",
+    "@carbon/vue": "^2.45.1",
     "@ckeditor/ckeditor5-build-classic": "^34.2.0",
     "@ckeditor/ckeditor5-vue": "^4.0.1",
-    "@codemirror/lang-python": "^6.1.2",
+    "@codemirror/lang-python": "^6.1.3",
     "@github/clipboard-copy-element": "^1.1.2",
     "@ibm/plex": "^6.1.1",
     "@jupyter-widgets/html-manager": "^0.20.5",
-    "@jupyter-widgets/jupyterlab-manager": "^3.1.4",
+    "@jupyter-widgets/jupyterlab-manager": "^3.1.5",
     "@jupyterlab/mathjax2": "^3.5.2",
     "@jupyterlab/outputarea": "^3.5.2",
     "@jupyterlab/rendermime": "^3.5.2",
     "@jupyterlab/services": "^6.5.2",
     "@lumino/widgets": "^1.36.0",
     "@mathigon/boost": "^1.1.1",
-    "@mathigon/euclid": "^1.1.10",
+    "@mathigon/euclid": "^1.1.12",
     "@mathigon/studio": "0.1.18",
     "@qiskit-community/qiskit-vue": "^3.4.0",
-    "@qiskit/web-components": "^0.10.3",
+    "@qiskit/web-components": "^0.15.3",
     "@webcomponents/webcomponentsjs": "^2.6.0",
     "carbon-components": "~10.58.3",
     "carbon-web-components": "^1.21.0",
@@ -95,11 +95,11 @@
   "devDependencies": {
     "@nuxtjs/eslint-config-typescript": "^10.0.0",
     "@types/bcryptjs": "2.4.2",
-    "@types/codemirror": "^5.60.6",
+    "@types/codemirror": "^5.60.8",
     "@types/express-flash": "0.0.2",
     "@types/fs-extra": "^9.0.13",
     "@types/jsdom": "^16.2.15",
-    "@types/lodash": "^4.14.195",
+    "@types/lodash": "^4.14.197",
     "@types/markdown-it": "^12.2.3",
     "@types/node-fetch": "^2.6.4",
     "@types/pug": "2.0.6",
@@ -118,7 +118,7 @@
     "jsdom": "^19.0.0",
     "nano-staged": "^0.8.0",
     "node-sass": "^6.0.1",
-    "nodemon": "^2.0.20",
+    "nodemon": "^2.0.22",
     "npm-run-all": "^4.1.5",
     "rimraf": "^3.0.2",
     "sass-loader": "^10.2.1",
diff --git a/scripts/content_checks/missing_nb_check.sh b/scripts/content_checks/missing_nb_check.sh
deleted file mode 100755
index 09f6f7f46..000000000
--- a/scripts/content_checks/missing_nb_check.sh
+++ /dev/null
@@ -1,36 +0,0 @@
-#!/usr/bin/env bash
-
-# this script creates a `notebooks.txt` containing list of notebooks that
-# are present in qiskit-community/qiskit-textbook but are not in
-# Qiskit/platypus.
-#
-# this script is run from the `Check for Textbook notebooks` GitHub action
-# which should contain a copy of the qiskit-textbook and platypus repos
-
-
-# diff the notebooks directory of the two repos and add all notebooks found
-# in qiskit-textbook but not found in platypus into a `notebooks.txt` file
-diff -aqr platypus/notebooks qiskit-textbook/content | sort | grep 'Only in qiskit-textbook' | grep -E '\.ipynb$' > notebooks.txt
-
-# count the number of lines (notebooks found) in qiskit-textbook but not in
-# platypus
-NUM_NB=$(cat notebooks.txt | wc -l)
-
-echo 'found $NUM_NB new notebooks'
-
-# set GitHub action variable to be checked by next step/task to determine
-# if an issue should be created
-echo '::set-output name=NUM_NB::$NUM_NB'
-
-# replace all newline characters with `\n` so the list used in a JSON object
-NB_LIST=$(sed -e :a -e '$!N;s/\n/\\n/;ta' notebooks.txt)
-
-# create the JSON that will be used as payload for the create issue REST API
-echo '{' > issue.txt
-echo '  "title": "Missing notebooks",' >> issue.txt
-echo '  "labels": ["bug"],' >> issue.txt
-echo '  "assignees": ["vabarbosa", "frankharkins"],' >> issue.txt
-echo '  "body": "'$NB_LIST'"' >> issue.txt
-echo '}' >> issue.txt
-
-exit 0
diff --git a/server/app.ts b/server/app.ts
index 6b9a578d0..729f76114 100644
--- a/server/app.ts
+++ b/server/app.ts
@@ -14,7 +14,7 @@ import { LOCALES, translate } from '@mathigon/studio/server/utilities/i18n'
 import { generateMockData } from './populate-database'
 
 import {
-  CONFIG, NOTATIONS, TEXTBOOK_HOME, LATEST_TEXTBOOK_VERSION, TRANSLATIONS, UNIVERSAL_NOTATIONS,
+  CONFIG, NOTATIONS, TEXTBOOK_HOME, LATEST_TEXTBOOK_VERSION, TRANSLATIONS, UNIVERSAL_NOTATIONS, LEARNING_REDIRECTS,
   findNextSection, findPrevSection, getSectionIndex, isLearningPath,
   updateGlossary, loadLocaleRawFile, tocFilterByType, removeVersionPrefix
 } from './utilities'
@@ -138,7 +138,8 @@ const start = () => {
     .accounts()
     .redirects({
       '/': TEXTBOOK_HOME,
-      '/textbook': TEXTBOOK_HOME
+      '/textbook': TEXTBOOK_HOME,
+      ...LEARNING_REDIRECTS
     })
     .get('/locales/:locale', (req, res) => {
       const translations = TRANSLATIONS[req.params.locale || 'en'] || {}
diff --git a/server/templates/error.pug b/server/templates/error.pug
index 2b522081f..cb56e21ee 100644
--- a/server/templates/error.pug
+++ b/server/templates/error.pug
@@ -12,6 +12,7 @@ html(lang=`${req.locale.id}` dir=dir)
   body
     div.qiskit__navbar#qiskit-navbar
       q-the-menu
+    q-one-xp-banner
 
     .body
       article.c-textbook__page.c-textbook__page__error
diff --git a/server/templates/signIn.pug b/server/templates/signIn.pug
index defaaa155..499afedb5 100644
--- a/server/templates/signIn.pug
+++ b/server/templates/signIn.pug
@@ -8,6 +8,8 @@ html(lang=lang dir=dir)
   body#app
     div.qiskit__navbar#qiskit-navbar
       q-the-menu
+    q-one-xp-banner
+      
 
     .body
       q-sign-in
diff --git a/server/templates/textbook.pug b/server/templates/textbook.pug
index f17fc04e6..0f6c6919a 100644
--- a/server/templates/textbook.pug
+++ b/server/templates/textbook.pug
@@ -14,6 +14,8 @@ html(lang=course.locale dir=dir)
       q-the-menu
 
     main.body
+      q-one-xp-banner
+
       x-course.c-textbook(id=course.id data-section=section.id data-goals=section.goals data-reveal='true')
         include sidebar
 
diff --git a/server/templates/userAccount.pug b/server/templates/userAccount.pug
index d6113cffb..d07b4482b 100644
--- a/server/templates/userAccount.pug
+++ b/server/templates/userAccount.pug
@@ -8,6 +8,7 @@ html(lang=lang dir=dir)
   body
     div.qiskit__navbar#qiskit-navbar
       q-the-menu
+    q-one-xp-banner
 
     .body
       q-user-account-layout(first-name=userData.firstName, last-name=userData.lastName email=userData.email user-id=userData.userId)
diff --git a/server/utilities.ts b/server/utilities.ts
index 7e4fddc3c..12a332df8 100644
--- a/server/utilities.ts
+++ b/server/utilities.ts
@@ -24,7 +24,18 @@ import {
 import { IS_PRODUCTION } from './configuration'
 
 const TEXTBOOK_HOME = 'https://qiskit.org/learn'
+const LEARNING_HOME = 'https://learning.quantum-computing.ibm.com'
+
+const LEARNING_REDIRECTS = {
+  '/course/algorithm-design': `${LEARNING_HOME}/course/variational-algorithm-design`,
+  '/course/algorithm-design/*': `${LEARNING_HOME}/course/variational-algorithm-design`,
+  '/course/basics': `${LEARNING_HOME}/course/basics-of-quantum-information`,
+  '/course/basics/*': `${LEARNING_HOME}/course/basics-of-quantum-information`,
+  '/course/algorithms': `${LEARNING_HOME}/course/fundamentals-of-quantum-algorithms`,
+  '/course/algorithms/*': `${LEARNING_HOME}/course/fundamentals-of-quantum-algorithms`
+}
 
+// NOTE: if changing this also update the same variable in 'converter/common.ts'
 const LATEST_TEXTBOOK_VERSION = 'v2'
 
 const DEFAULT_SHARED_FOLDER = path.join(CONTENT_DIR, 'shared')
@@ -181,6 +192,7 @@ export {
   TRANSLATIONS,
   UNIVERSAL_NOTATIONS,
   TOC,
+  LEARNING_REDIRECTS,
   findNextSection,
   findPrevSection,
   getSectionIndex,
diff --git a/translations/es/toc.yaml b/translations/es/toc.yaml
index ad6be4d53..df28e4288 100644
--- a/translations/es/toc.yaml
+++ b/translations/es/toc.yaml
@@ -1,92 +1,5 @@
 
 
-#---- Below was added by machine ---
-
-- sections:
-  - id: introduction
-    url: v2/ch-states/introduction
-    uuid: afcc559e-b519-11ec-b909-0242ac120002
-  - id: the-atoms-of-computation
-    url: v2/ch-states/atoms-computation
-    uuid: b31c20ee-b519-11ec-b909-0242ac120002
-  - id: representing-qubit-states
-    url: v2/ch-states/representing-qubit-states
-    uuid: b81717ca-b519-11ec-b909-0242ac120002
-  - id: single-qubit-gates
-    url: v2/ch-states/single-qubit-gates
-    uuid: bd4334a4-b519-11ec-b909-0242ac120002
-  - id: the-case-for-quantum-computers
-    url: v2/ch-states/case-for-quantum
-    uuid: c0bbff62-b519-11ec-b909-0242ac120002
-  title: "Estados cu\xE1nticos y Qubits"
-  url: /v2_ch-states
-- sections:
-  - id: introduction
-    url: v2/ch-gates/introduction
-    uuid: c50f4c90-b519-11ec-b909-0242ac120002
-  - id: multiple-qubits-and-entangled-states
-    url: v2/ch-gates/multiple-qubits-entangled-states
-    uuid: c7cfe502-b519-11ec-b909-0242ac120002
-  - id: basic-circuit-identities
-    url: v2/ch-gates/more-circuit-identities
-    uuid: ce8ce75a-b519-11ec-b909-0242ac120002
-  - id: proving-universality
-    url: v2/ch-gates/proving-universality
-    uuid: d15d3548-b519-11ec-b909-0242ac120002
-  - id: classical-computation-on-a-quantum-computer
-    url: v2/ch-gates/oracles
-    uuid: d4f6e424-b519-11ec-b909-0242ac120002
-  title: "M\xFAltiples Qubits y entrelazamiento"
-  url: /v2_ch-gates
-- sections:
-  - id: quantum-circuits
-    url: v2/ch-algorithms/defining-quantum-circuits
-    uuid: d90cbfb6-b519-11ec-b909-0242ac120002
-  - id: deutsch-jozsa-algorithm
-    url: v2/ch-algorithms/deutsch-jozsa
-    uuid: dc9d3aa2-b519-11ec-b909-0242ac120002
-  - id: bernstein-vazirani-algorithm
-    url: v2/ch-algorithms/bernstein-vazirani
-    uuid: df6e5a40-b519-11ec-b909-0242ac120002
-  title: "Protocolos cu\xE1nticos y algoritmos cu\xE1nticos"
-  url: /v2_ch-algorithms
-- sections:
-  - id: simulating-molecules-using-vqe
-    url: v2/ch-applications/vqe-molecules
-    uuid: 041df288-b51a-11ec-b909-0242ac120002
-  - id: solving-combinatorial-optimization-problems-using-qaoa
-    url: v2/ch-applications/qaoa
-    uuid: 06ff7ada-b51a-11ec-b909-0242ac120002
-  title: "Algoritmos cu\xE1nticos para aplicaciones"
-  url: /v2_ch-applications
-- sections:
-  - id: lab-9-quantum-simulation-as-a-search-algorithm
-    url: v2/ch-labs/Lab09_QuantumSimulationSearchAlgorithm
-    uuid: 59a076f4-b51a-11ec-b909-0242ac120002
-  title: "Laboratorios de computaci\xF3n cu\xE1ntica"
-  url: /v2_ch-labs
-- sections:
-  - id: estimating-pi-using-quantum-phase-estimation-algorithm
-    url: v2/ch-demos/piday-code
-    uuid: 69b9707c-b51a-11ec-b909-0242ac120002
-  title: Juegos y demostraciones
-  url: /v2_ch-demos
-- sections:
-  - id: phase-kickback
-    url: v2/algorithms/phase-kickback
-    uuid: 976e9da6-6c33-4e2f-b7eb-f9588f700f07
-  title: "Algoritmos cu\xE1nticos"
-  type: learning-path
-  url: /v2_algorithms
-- sections:
-  - id: why-quantum-computing
-    url: v2/intro/why-quantum-computing
-    uuid: bdb1d662-e0f6-428a-8830-befe6b47f320
-  title: "Introducci\xF3n a la Computaci\xF3n Cu\xE1ntica"
-  type: learning-path
-  url: /v2_introduction
-
-
 #---- Below was added by machine ---
 
 - sections:
diff --git a/translations/ja/algorithm-design/optimization/optimization.ipynb b/translations/ja/algorithm-design/optimization/optimization.ipynb
new file mode 100644
index 000000000..fad5e1af5
--- /dev/null
+++ b/translations/ja/algorithm-design/optimization/optimization.ipynb
@@ -0,0 +1,256 @@
+{
+  "cells": [
+    {
+      "attachments": {
+      },
+      "cell_type": "markdown",
+      "metadata": {
+      },
+      "source": [
+        "## 最適化ループ\n",
+        "\n",
+        "このレッスンでは、*オプティマイザー*を使用して、ansatz のパラメータ化された量子状態を繰り返し探索する方法を学習します。\n",
+        "\n",
+        "- 最適化ループのブートストラップ\n",
+        "- ローカルおよびグローバルオプティマイザーを使用する際のトレードオフの理解\n",
+        "- 「不毛な台地」の探索と回避の方法\n",
+        "\n",
+        "大まかに言えば、オプティマイザーは探索空間を探索する中心です。オプティマイザーは、コスト関数の評価を使用して、変分ループで次のパラメーターセットを選択し、安定した状態に達するまで処理を繰り返します。この段階で、パラメーター値 $\\vec\\theta^*$ の最適なセットが返されます。\n",
+        "\n",
+        "![最適化ワークフロー](images/optimization_workflow.png)"
+      ]
+    },
+    {
+      "attachments": {
+      },
+      "cell_type": "markdown",
+      "metadata": {
+      },
+      "source": [
+        "## ローカルおよびグローバルオプティマイザー\n",
+        "\n",
+        "### ローカルオプティマイザー\n",
+        "\n",
+        "ローカルオプティマイザーは、初期点 $C(\\vec{\\theta_0})$ から始まるコスト関数を最小化する点を探索し、連続する繰り返しにおいて現在評価している領域で観測したものに基づいて別の点に移動します。これらのアルゴリズムの収束は通常高速ですが、初期点に大きく依存する可能性があります。ローカルオプティマイザーは、現在評価している領域を超えた領域を見ることはできず、極小値に対してとりわけ脆い可能性があり、極小値を見つけると、より望ましい評価結果である他の状態を無視して収束したとみなしてしまいます。\n",
+        "\n",
+        "### グローバルオプティマイザー\n",
+        "\n",
+        "グローバルオプティマイザーは、ドメインのいくつかの領域 (つまりローカルではない) でコスト関数を最小化する点を探索し、オプティマイザーによって決定された $\\Theta_i := \\{{\\vec\\theta_{i,j} | j \\in \\mathcal{J}_\\text{opt}^i} \\}$ パラメータベクトルで反復的に (繰り返し $i$ ) 評価します。これにより極小値の影響を受けにくくなり、ある程度初期化に依存しなくなる一方で、提示される解への収束も大幅に遅くなります。\n",
+        "\n",
+        "### ブートストラップ最適化\n",
+        "\n",
+        "*ブートストラップ*、または事前の最適化に基づいてパラメーター $\\vec\\theta$  の初期値を設定すると、オプティマイザーがより速く解に収束するのに役立ちます。 これを初期点 $\\vec\\theta_0$ と呼び、 $|\\psi(\\vec\\theta_0)\\rangle = U_V(\\vec\\theta_0)|\\rho\\rangle$  を初期状態と呼びます。 この初期状態は参照状態 $|\\rho\\rangle$ とは異なります。前者は最適化ループ中に設定された初期パラメーターに焦点を当てているのに対し、後者は既知の「参照」解の使用に焦点を当てているためです。 $U_V(\\vec\\theta_0) \\equiv I$ (つまり、恒等演算) の場合、これらは一致する可能性があります。\n",
+        "\n",
+        "ローカルオプティマイザーが最適でない極小値に収束する場合、最適化をグローバルにブートストラップし、ローカルで収束を洗練することを試みることができます。これには2つの変分ワークロードを設定する必要がありますが、オプティマイザーがローカルオプティマイザーだけよりも最適な解を見つけることができるようになります。"
+      ]
+    },
+    {
+      "attachments": {
+      },
+      "cell_type": "markdown",
+      "metadata": {
+        "gloss": {
+          "hyperparameter": {
+            "text": "ハイパーパラメータとは、アルゴリズムを制御するために使用するパラメータです。「ハイパー」は、アルゴリズムが見つけようとするパラメータ（θ）と区別されています。",
+            "title": "ハイパーパラメータ"
+          },
+          "local-minimum": {
+            "text": "極小値は、$theta$ の値の小さな範囲に対する、関数の最低点です。一方、最小値は、関数のどこにおいても（つまり、どのような $theta$ の値に対しても）最低の点です。<a href='https://en.wikipedia.org/wiki/Maxima_and_minima'>Read more</a>.",
+            "title": "最小値"
+          }
+        }
+      },
+      "source": [
+        "## 勾配ありと勾配なしのオプティマイザー\n",
+        "\n",
+        "### 勾配あり\n",
+        "\n",
+        "コスト関数 $C(\\vec\\theta)$ において、初期点から関数 $\\vec{\\nabla} C(\\vec\\theta)$ の勾配にアクセスできる場合、関数を最小化する最も簡単な方法は、パラメータを関数の最急降下の方向に更新することです。 つまり、パラメータを $\\vec\\theta_{n+1} = \\vec\\theta_n - \\eta \\vec{\\nabla} C(\\vec\\theta)$ として更新します。ここで、 $\\eta$ は学習率 (更新のサイズを制御する小さな正の[ハイパーパラメータ](gloss:hyperparameter) ) です。 これをコスト関数の極小値 $C({\\vec\\theta^*})$ に収束するまで続けます。[`qiskit.algorithms`](https://qiskit.org/documentation/stubs/qiskit.algorithms.gradients.html) は、勾配を計算するためのいくつかの異なる方法を提供します。詳細については、[こちら](https://learn.qiskit.org/course/machine-learning/training-quantum-circuits#training-2-0)をご覧ください。"
+      ]
+    },
+    {
+      "cell_type": "code",
+      "execution_count": null,
+      "metadata": {
+      },
+      "outputs": [
+
+      ],
+      "source": [
+        "from qiskit.circuit.library import TwoLocal\n",
+        "from qiskit.quantum_info import SparsePauliOp\n",
+        "from qiskit import QuantumCircuit\n",
+        "from qiskit_ibm_runtime import QiskitRuntimeService, Estimator\n",
+        "import numpy as np\n",
+        "\n",
+        "# Add your token below\n",
+        "service = QiskitRuntimeService(\n",
+        "    channel=\"ibm_quantum\",\n",
+        ")\n",
+        "\n",
+        "def cost_function_vqe(theta):\n",
+        "    observable = SparsePauliOp.from_list([(\"II\", 2), (\"XX\", -2), (\"YY\", 3), (\"ZZ\", -3)])\n",
+        "    reference_circuit = QuantumCircuit(2)\n",
+        "    reference_circuit.x(0)\n",
+        "\n",
+        "    variational_form = TwoLocal(\n",
+        "        2,\n",
+        "        rotation_blocks=[\"rz\", \"ry\"],\n",
+        "        entanglement_blocks=\"cx\",\n",
+        "        entanglement=\"linear\",\n",
+        "        reps=1,\n",
+        "    )\n",
+        "    ansatz = reference_circuit.compose(variational_form)\n",
+        "\n",
+        "    backend = service.backend(\"ibmq_qasm_simulator\")\n",
+        "    \n",
+        "    # Use estimator to get the expected values corresponding to each ansatz\n",
+        "    estimator = Estimator(session=backend)\n",
+        "    job = estimator.run(ansatz, observable, theta)\n",
+        "    values = job.result().values\n",
+        "\n",
+        "    return values"
+      ]
+    },
+    {
+      "attachments": {
+      },
+      "cell_type": "markdown",
+      "metadata": {
+      },
+      "source": [
+        "このコスト関数とオプティマイザーを使って、最適なパラメーターを算出することができます"
+      ]
+    },
+    {
+      "cell_type": "code",
+      "execution_count": null,
+      "metadata": {
+      },
+      "outputs": [
+
+      ],
+      "source": [
+        "from qiskit.algorithms.optimizers import SPSA\n",
+        "\n",
+        "initial_theta = np.ones(8)\n",
+        "optimizer = SPSA()\n",
+        "\n",
+        "optimizer_result = optimizer.minimize(fun=cost_function_vqe, x0=initial_theta)\n",
+        "\n",
+        "optimal_parameters = optimizer_result.x\n",
+        "print(optimal_parameters)"
+      ]
+    },
+    {
+      "attachments": {
+      },
+      "cell_type": "markdown",
+      "metadata": {
+      },
+      "source": [
+        "このタイプの最適化の主な欠点は、収束速度が非常に遅くなることと、最適解を達成する保証がないことです。\n",
+        "\n",
+        "![theta に対する f(theta) のグラフ。複数の点は、曲線の最小値を見つける勾配降下法アルゴリズムのさまざまな状態を示します。](images/optimization_gradient_descent.png)\n",
+        "\n",
+        "### 勾配なし\n",
+        "\n",
+        "勾配を必要としない最適化アルゴリズムは、勾配情報を必要としないため、勾配の計算が困難であったり、コストがかかったり、ノイズが多すぎたりする場合に有効です。また、勾配を利用した手法が局所最適値に収束する傾向があるのに対して、大域最適値を見つける上でより頑健である傾向があります。ここでは、勾配なしオプティマイザーが不毛な台地を回避するのに役立ついくつかの例を紹介します。しかし、勾配なし手法は、特に高次元の探索空間を持つ問題に対してより高い計算リソースを必要とします。\n",
+        "\n",
+        "ここでは、代わりに [`COBYLA`](https://qiskit.org/documentation/stubs/qiskit.algorithms.optimizers.COBYLA.html#qiskit.algorithms.optimizers.COBYLA) オプティマイザーを使用する例を紹介します："
+      ]
+    },
+    {
+      "cell_type": "code",
+      "execution_count": null,
+      "metadata": {
+      },
+      "outputs": [
+
+      ],
+      "source": [
+        "from qiskit.algorithms.optimizers import COBYLA\n",
+        "\n",
+        "initial_theta = np.ones(8)\n",
+        "optimizer = COBYLA()\n",
+        "\n",
+        "optimizer_result = optimizer.minimize(fun=cost_function_vqe, x0=initial_theta)\n",
+        "\n",
+        "optimal_parameters = optimizer_result.x\n",
+        "print(optimal_parameters)"
+      ]
+    },
+    {
+      "attachments": {
+      },
+      "cell_type": "markdown",
+      "metadata": {
+        "gloss": {
+          "barren-plateaus": {
+            "text": "パラメータ化された量子回路の勾配が量子ビット数に対して指数関数的に小さくなると、最適化が難しくなり、最適化計算が不能になる可能性があります。",
+            "title": "不毛な台地"
+          }
+        }
+      },
+      "source": [
+        "## 不毛な台地\n",
+        "\n",
+        "実際、コストの起伏は、下の例のように丘と谷のように非常に複雑になることがあります。最適化手法は、黒い点と線で示されるように、コストの起伏をナビゲートし、最小値を探します。3つの探索のうち2つは、グローバルな最小値ではなく、ローカルな最小値で終わっていることがわかります。\n",
+        "\n",
+        "![コストの起伏](images/optimization_loss_landscape.png)\n",
+        "\n",
+        "最適化手法の種類にかかわらず、コストの起伏が比較的平坦である場合、その手法が適切な探索方向を決定することは困難な場合があります。このようなシナリオは[不毛な台地](gloss:barren-plateaus)と呼ばれ、コストの起伏が徐々に平坦にります（したがって、最小値への方向性を決定するのがより困難になります）。パラメータ化された幅広い量子回路において、合理的な方向に沿った勾配が一定の精度でゼロでない確率は、量子ビットの数が増えるにつれて指数関数的に減少することが分かっています。\n",
+        "\n",
+        "![Barren Plateaus](images/optimization_barren_plateaus.png)\n",
+        "\n",
+        "この分野はまだ活発な研究が行われていますが、最適化性能を向上させるためのいくつかの推奨事項があります：\n",
+        "\n",
+        "- **ブートストラップ** は、最適化ループが勾配が小さいパラメータ空間で立ち往生するのを避けるのに役立ちます。\n",
+        "- **ハードウェア効率の良いansatzの実験**: ブラックボックスオラクルとしてノイズの多い量子系を使用しているため、その評価の質はオプティマイザーの性能に影響を与えることがあります。[`EfficientSU2`](https://qiskit.org/documentation/stubs/qiskit.circuit.library.EfficientSU2.html) のようなハードウェア効率の良いansatzを使用することで、指数関数的に小さな勾配が発生することを回避できる可能性があります。\n",
+        "- **エラー抑制とエラー緩和の実験**: Qiskit Runtime Primitiveは、様々な `optimization_level`と `resilience_setting` をそれぞれ実験するためのシンプルなインターフェースを提供します。これにより、ノイズの影響を軽減し、最適化プロセスをより効率的に行うことができます。\n",
+        "- **勾配なしオプティマイザーを使った実験**: `COBYLA`のようなオプティマイザーは、勾配ベースの最適化アルゴリズムとは異なり、勾配情報に頼らずにパラメータを最適化するため、不毛な台地の影響を受けにくいです。"
+      ]
+    },
+    {
+      "attachments": {
+      },
+      "cell_type": "markdown",
+      "metadata": {
+      },
+      "source": [
+        "このレッスンで、あなたは最適化ループを定義する方法を学びました：\n",
+        "\n",
+        "- 最適化ループをブートストラップする。\n",
+        "- ローカルオプティマイザーとグローバルオプティマイザーを使い分けながら、トレードオフを理解する。\n",
+        "- 不毛の台地とそれを避ける方法を探る\n",
+        "\n",
+        "ハイレベルの変分ワークロードは完了です：\n",
+        "\n",
+        "![最適化回路](images/optimization_circuit.png)\n",
+        "\n",
+        "次に、このフレームワークを意識して、具体的な変分アルゴリズムを探っていきます。"
+      ]
+    }
+  ],
+  "metadata": {
+    "kernelspec": {
+      "display_name": "Python 3",
+      "language": "python",
+      "name": "python3"
+    },
+    "language_info": {
+      "codemirror_mode": {
+        "name": "ipython",
+        "version": 3
+      },
+      "file_extension": ".py",
+      "mimetype": "text/x-python",
+      "name": "python",
+      "nbconvert_exporter": "python",
+      "pygments_lexer": "ipython3",
+      "version": "3.9"
+    }
+  },
+  "nbformat": 4,
+  "nbformat_minor": 2
+}
diff --git a/translations/ja/algorithm-design/reference/reference.ipynb b/translations/ja/algorithm-design/reference/reference.ipynb
new file mode 100644
index 000000000..593838003
--- /dev/null
+++ b/translations/ja/algorithm-design/reference/reference.ipynb
@@ -0,0 +1,301 @@
+{
+  "cells": [
+    {
+      "attachments": {
+      },
+      "cell_type": "markdown",
+      "metadata": {
+        "gloss": {
+          "reference": {
+            "text": "問題の初期状態、決まった開始状態。通常 $|0\\rangle$ を指しますが、問題を適切に表現する様々な状態に設定することができます。",
+            "title": "参照状態"
+          }
+        }
+      },
+      "source": [
+        "## 参照状態\n",
+        "\n",
+        "このレッスンでは、変分アルゴリズムをより速く収束させるために、参照状態で系を初期化する方法について学びます。まず、 [*参照状態*](gloss:reference) を手動で構築する方法を学び、次に変分アルゴリズムで使用可能ないくつかの標準的な選択肢を探ります。\n",
+        "\n",
+        "![Reference Workflow](images/reference_workflow.png)"
+      ]
+    },
+    {
+      "attachments": {
+      },
+      "cell_type": "markdown",
+      "metadata": {
+      },
+      "source": [
+        "## デフォルト状態\n",
+        "\n",
+        "*参照状態* とは、問題に対する最初の固定開始状態を意味します。参照状態を準備するために、量子回路の開始時に適切なパラメータ化されていないユニタリー回路 $U_R$ を適用し、 $|\\rho\\rangle = U_R |0\\rangle$ となるようにする必要があります。既存の最適解からの推測やデータ点がある場合、それを出発点として使用すると変分アルゴリズムが早く収束する可能性があります。\n",
+        "\n",
+        "最も単純な参照状態はデフォルト状態であり、 $n$ 量子ビット回路の開始状態 $|0\\rangle^{\\otimes n}$ を使用します。 デフォルト状態の場合、ユニタリー演算子 $U_R \\equiv I$ が使われます。その単純さゆえに、デフォルト状態は多くのシナリオで使用される有効な参照状態です。"
+      ]
+    },
+    {
+      "attachments": {
+      },
+      "cell_type": "markdown",
+      "metadata": {
+      },
+      "source": [
+        "## 古典的な参照状態\n",
+        "\n",
+        "$3$ 量子ビット系で、デフォルトの状態 $|000\\rangle$ の代わりに状態 $|001\\rangle$ で始めたいとします。これは純粋に古典的な参照状態の例で、これを構築するには、(Qiskitの量子ビット順序に従って)量子ビット $0$ に [X ゲート](https://qiskit.org/documentation/stubs/qiskit.circuit.library.XGate.html)  を適用するだけで、 $|001\\rangle = X_0 |000\\rangle$ となります。\n",
+        "\n",
+        "この場合、ユニタリー演算子は $U_R \\equiv X_0$ となり、参照状態 $|\\rho\\rangle \\equiv |001\\rangle$ を作ります。"
+      ]
+    },
+    {
+      "cell_type": "code",
+      "execution_count": null,
+      "metadata": {
+      },
+      "outputs": [
+
+      ],
+      "source": [
+        "from qiskit import QuantumCircuit\n",
+        "\n",
+        "qc = QuantumCircuit(3)\n",
+        "qc.x(0)\n",
+        "\n",
+        "qc.draw(\"mpl\")"
+      ]
+    },
+    {
+      "attachments": {
+      },
+      "cell_type": "markdown",
+      "metadata": {
+      },
+      "source": [
+        "## 量子的な参照状態\n",
+        "\n",
+        "例えば、 $\\frac{1}{\\sqrt{2}}(|100\\rangle+|111\\rangle)$ のような重ね合わせやエンタングルメントを含むより複雑な状態から始めたいとします。\n",
+        "\n",
+        "この状態を $|000\\rangle$ から得る方法の一つに、量子ビット $0$ に対して [アダマールゲート](https://qiskit.org/documentation/stubs/qiskit.circuit.library.HGate.html)（ $H_0$ ）、量子ビット $0$ を制御量子ビット、量子ビット $1$ をターゲット量子ビットとする [CNOT（CX）](https://qiskit.org/documentation/stubs/qiskit.circuit.library.CXGate.html)ゲート（ $CNOT_{01}$ ）、最後に量子ビット $2$ に対する $X$ ゲート（ $X_2$ ）を適用する方法があります。\n",
+        "\n",
+        "このとき、ユニタリー演算子は $U_{R} \\equiv X_2CNOT_{01}H_0|000\\rangle$ 、参照状態は $|\\rho\\rangle \\equiv \\frac{1}{\\sqrt{2}}(|100\\rangle+|111\\rangle)$ です。"
+      ]
+    },
+    {
+      "cell_type": "code",
+      "execution_count": null,
+      "metadata": {
+      },
+      "outputs": [
+
+      ],
+      "source": [
+        "qc = QuantumCircuit(3)\n",
+        "qc.h(0)\n",
+        "qc.cx(0, 1)\n",
+        "qc.x(2)\n",
+        "\n",
+        "qc.draw(\"mpl\")"
+      ]
+    },
+    {
+      "attachments": {
+      },
+      "cell_type": "markdown",
+      "metadata": {
+      },
+      "source": [
+        "## テンプレート回路を用いた参照状態の構築\n",
+        "\n",
+        "また、複数の調整可能なパラメータやエンタングルメントを簡単に表現できる [`TwoLocal`](https://qiskit.org/documentation/stubs/qiskit.circuit.library.TwoLocal.html) など、様々なテンプレート回路を利用することができます。これらのテンプレート回路については、次回のレッスンで詳しく説明しますが、パラメータをバインドした *場合* 、参照状態に利用することができます。"
+      ]
+    },
+    {
+      "cell_type": "code",
+      "execution_count": null,
+      "metadata": {
+      },
+      "outputs": [
+
+      ],
+      "source": [
+        "from qiskit.circuit.library import TwoLocal\n",
+        "from math import pi\n",
+        "\n",
+        "reference_circuit = TwoLocal(2, \"rx\", \"cz\", entanglement=\"linear\", reps=1)\n",
+        "theta_list = [pi / 2, pi / 3, pi / 3, pi / 2]\n",
+        "\n",
+        "reference_circuit = reference_circuit.bind_parameters(theta_list)\n",
+        "\n",
+        "reference_circuit.decompose().draw(\"mpl\")"
+      ]
+    },
+    {
+      "attachments": {
+      },
+      "cell_type": "markdown",
+      "metadata": {
+      },
+      "source": [
+        "## アプリケーション固有の参照状態"
+      ]
+    },
+    {
+      "attachments": {
+      },
+      "cell_type": "markdown",
+      "metadata": {
+      },
+      "source": [
+        "### 量子化学\n",
+        "\n",
+        "量子化学では、 *Hartree-Fock* 状態は原子や分子の基底状態の近似値です。正確な基底状態を求める変分アルゴリズムを作成することが目的であれば、この既知の古典的近似を参照状態として使用することで、アルゴリズムの収束を早くすることができます。\n",
+        "\n",
+        "この例では、2つの空間軌道と2つの電子（1つのアルファ・スピン、1つのベータ・スピン）を持つ問題に対して、次のように電子構造の問題を生成することができます。"
+      ]
+    },
+    {
+      "cell_type": "code",
+      "execution_count": null,
+      "metadata": {
+      },
+      "outputs": [
+
+      ],
+      "source": [
+        "from qiskit_nature.second_q.circuit.library import HartreeFock\n",
+        "from qiskit_nature.second_q.mappers import JordanWignerMapper\n",
+        "\n",
+        "num_spatial_orbitals = 2\n",
+        "num_particles = (1, 1)\n",
+        "\n",
+        "mapper = JordanWignerMapper()\n",
+        "\n",
+        "h2_reference_state = HartreeFock(\n",
+        "    num_spatial_orbitals=num_spatial_orbitals,\n",
+        "    num_particles=num_particles,\n",
+        "    qubit_mapper=mapper,\n",
+        ")\n",
+        "\n",
+        "h2_reference_state.decompose().draw(\"mpl\")"
+      ]
+    },
+    {
+      "attachments": {
+      },
+      "cell_type": "markdown",
+      "metadata": {
+      },
+      "source": [
+        "また、水素化リチウム（LiH）のような特定の分子を構成することも可能です。"
+      ]
+    },
+    {
+      "cell_type": "code",
+      "execution_count": null,
+      "metadata": {
+      },
+      "outputs": [
+
+      ],
+      "source": [
+        "from qiskit_nature.second_q.drivers import PySCFDriver\n",
+        "from qiskit_nature.second_q.circuit.library import HartreeFock\n",
+        "from qiskit_nature.second_q.mappers import JordanWignerMapper\n",
+        "from qiskit_nature.second_q.formats.molecule_info import MoleculeInfo\n",
+        "\n",
+        "molecule = MoleculeInfo(\n",
+        "    # Coordinates in Angstrom\n",
+        "    symbols=[\"Li\", \"H\"],\n",
+        "    coords=([0.0, 0.0, 0.0], [0.2, 0.0, 0.0]),\n",
+        "    multiplicity=1,  # = 2*spin + 1\n",
+        "    charge=0,\n",
+        ")\n",
+        "\n",
+        "driver = PySCFDriver.from_molecule(molecule)\n",
+        "LiH_problem = driver.run()\n",
+        "\n",
+        "mapper = JordanWignerMapper()\n",
+        "LiH_reference_state = HartreeFock(\n",
+        "    num_spatial_orbitals=LiH_problem.num_spatial_orbitals,\n",
+        "    num_particles=LiH_problem.num_particles,\n",
+        "    qubit_mapper=mapper,\n",
+        ")\n",
+        "\n",
+        "LiH_reference_state.decompose().draw(\"mpl\")"
+      ]
+    },
+    {
+      "attachments": {
+      },
+      "cell_type": "markdown",
+      "metadata": {
+      },
+      "source": [
+        "### 量子機械学習\n",
+        "\n",
+        "[変分量子分類器（VQC）](https://learn.qiskit.org/course/machine-learning/variational-classification) の文脈では、訓練データは、 *特徴量マップ* として知られるパラメータ化された回路で量子状態に符号化され、各パラメータの値は訓練データセットのデータ点を表します。 [ZZFeatureMap](https://qiskit.org/documentation/stubs/qiskit.circuit.library.ZZFeatureMap.html) は、この特徴量マップにデータ点($x$) を渡すパラメータ化回路の一種です。"
+      ]
+    },
+    {
+      "cell_type": "code",
+      "execution_count": null,
+      "metadata": {
+      },
+      "outputs": [
+
+      ],
+      "source": [
+        "from qiskit.circuit.library import ZZFeatureMap\n",
+        "\n",
+        "data = [0.1, 0.2]\n",
+        "\n",
+        "zz_feature_map_reference = ZZFeatureMap(feature_dimension=2, reps=2)\n",
+        "zz_feature_map_reference = zz_feature_map_reference.bind_parameters(data)\n",
+        "zz_feature_map_reference.decompose().draw(\"mpl\")"
+      ]
+    },
+    {
+      "attachments": {
+      },
+      "cell_type": "markdown",
+      "metadata": {
+      },
+      "source": [
+        "このレッスンで、あなたは、以下を使ってシステムを初期化する方法を学びました：\n",
+        "\n",
+        "- デフォルトの参照状態\n",
+        "- 古典的な参照状態\n",
+        "- 量子的な参照状態\n",
+        "- アプリケーション固有の参照状態\n",
+        "\n",
+        "このハイレベルな変分ワークロードは次のようになります。\n",
+        "\n",
+        "![Reference Circuit](images/reference_circuit.png)\n",
+        "\n",
+        "参照状態は固定された最初の出発点であるのに対し、 *変分形式* を使って、変分アルゴリズムが探索するためのパラメータ化された状態の集合を表す *ansatz* を定義することができます。"
+      ]
+    }
+  ],
+  "metadata": {
+    "kernelspec": {
+      "display_name": "Python 3",
+      "language": "python",
+      "name": "python3"
+    },
+    "language_info": {
+      "codemirror_mode": {
+        "name": "ipython",
+        "version": 3
+      },
+      "file_extension": ".py",
+      "mimetype": "text/x-python",
+      "name": "python",
+      "nbconvert_exporter": "python",
+      "pygments_lexer": "ipython3",
+      "version": "3.9"
+    }
+  },
+  "nbformat": 4,
+  "nbformat_minor": 2
+}
diff --git a/translations/ja/toc.yaml b/translations/ja/toc.yaml
index b3eb33dda..1eac8b5b4 100644
--- a/translations/ja/toc.yaml
+++ b/translations/ja/toc.yaml
@@ -113,10 +113,10 @@
 #---- Below was added by machine ---
 
 - sections:
-  - id: environment-setup-guide-to-work-with-qiskit-textbook
+  - id: setting-the-environment
     url: v1/ch-prerequisites/setting-the-environment
     uuid: a5a86ddc-b519-11ec-b909-0242ac120002
-  - id: introduction-to-python-and-jupyter-notebooks
+  - id: python-and-jupyter-notebooks
     url: v1/ch-prerequisites/python-and-jupyter-notebooks
     uuid: a9f5ea68-b519-11ec-b909-0242ac120002
   title: "\u524D\u63D0\u6761\u4EF6"
@@ -134,7 +134,7 @@
   - id: single-qubit-gates
     url: v1/ch-states/single-qubit-gates
     uuid: bd4334a4-b519-11ec-b909-0242ac120002
-  - id: the-case-for-quantum-computers
+  - id: case-for-quantum
     url: v1/ch-states/case-for-quantum
     uuid: c0bbff62-b519-11ec-b909-0242ac120002
   title: "\u91CF\u5B50\u72B6\u614B\u3068\u91CF\u5B50\u30D3\u30C3\u30C8"
@@ -149,29 +149,29 @@
   - id: phase-kickback
     url: v1/ch-gates/phase-kickback
     uuid: cb7353ec-b519-11ec-b909-0242ac120002
-  - id: basic-circuit-identities
+  - id: more-circuit-identities
     url: v1/ch-gates/more-circuit-identities
     uuid: ce8ce75a-b519-11ec-b909-0242ac120002
   - id: proving-universality
     url: v1/ch-gates/proving-universality
     uuid: d15d3548-b519-11ec-b909-0242ac120002
-  - id: classical-computation-on-a-quantum-computer
+  - id: oracles
     url: v1/ch-gates/oracles
     uuid: d4f6e424-b519-11ec-b909-0242ac120002
   title: "\u8907\u6570\u91CF\u5B50\u30D3\u30C3\u30C8\u3068\u3082\u3064\u308C\u72B6\
     \u614B"
   url: /v1_ch-gates
 - sections:
-  - id: quantum-circuits
+  - id: defining-quantum-circuits
     url: v1/ch-algorithms/defining-quantum-circuits
     uuid: d90cbfb6-b519-11ec-b909-0242ac120002
-  - id: deutsch-jozsa-algorithm
+  - id: deutsch-jozsa
     url: v1/ch-algorithms/deutsch-jozsa
     uuid: dc9d3aa2-b519-11ec-b909-0242ac120002
-  - id: bernstein-vazirani-algorithm
+  - id: bernstein-vazirani
     url: v1/ch-algorithms/bernstein-vazirani
     uuid: df6e5a40-b519-11ec-b909-0242ac120002
-  - id: simons-algorithm
+  - id: simon
     url: v1/ch-algorithms/simon
     uuid: e2714126-b519-11ec-b909-0242ac120002
   - id: quantum-fourier-transform
@@ -180,16 +180,19 @@
   - id: quantum-phase-estimation
     url: v1/ch-algorithms/quantum-phase-estimation
     uuid: e8ac9ec8-b519-11ec-b909-0242ac120002
-  - id: shors-algorithm
+  - id: shor
     url: v1/ch-algorithms/shor
     uuid: ec157cd8-b519-11ec-b909-0242ac120002
-  - id: grovers-algorithm
+  - id: grover
     url: v1/ch-algorithms/grover
     uuid: eebb7488-b519-11ec-b909-0242ac120002
   - id: quantum-counting
     url: v1/ch-algorithms/quantum-counting
     uuid: f1a9942c-b519-11ec-b909-0242ac120002
-  - id: quantum-teleportation
+  - id: quantum-walk-search-algorithm
+    url: v1/ch-algorithms/quantum-walk-search-algorithm
+    uuid: 9a061c81-d490-4fff-bb71-727408234082
+  - id: teleportation
     url: v1/ch-algorithms/teleportation
     uuid: f4f4a090-b519-11ec-b909-0242ac120002
   - id: superdense-coding
@@ -202,32 +205,32 @@
     \u30B4\u30EA\u30BA\u30E0"
   url: /v1_ch-algorithms
 - sections:
-  - id: solving-linear-systems-of-equations-using-hhl-and-its-qiskit-implementation
+  - id: hhl_tutorial
     url: v1/ch-applications/hhl_tutorial
     uuid: 01484810-b51a-11ec-b909-0242ac120002
-  - id: simulating-molecules-using-vqe
+  - id: vqe-molecules
     url: v1/ch-applications/vqe-molecules
     uuid: 041df288-b51a-11ec-b909-0242ac120002
-  - id: solving-combinatorial-optimization-problems-using-qaoa
+  - id: qaoa
     url: v1/ch-applications/qaoa
     uuid: 06ff7ada-b51a-11ec-b909-0242ac120002
-  - id: solving-satisfiability-problems-using-grovers-algorithm
+  - id: satisfiability-grover
     url: v1/ch-applications/satisfiability-grover
     uuid: 09d1529c-b51a-11ec-b909-0242ac120002
-  - id: hybrid-quantum-classical-neural-networks-with-pytorch-and-qiskit
+  - id: machine-learning-qiskit-pytorch
     url: v1/ch-machine-learning/machine-learning-qiskit-pytorch
     uuid: 0cac4d14-b51a-11ec-b909-0242ac120002
-  - id: the-variational-quantum-linear-solver
+  - id: vqls
     url: v1/ch-paper-implementations/vqls
     uuid: 1163f456-b51a-11ec-b909-0242ac120002
-  - id: flexible-representation-of-quantum-images-frqi
+  - id: image-processing-frqi-neqr
     url: v1/ch-applications/image-processing-frqi-neqr
     uuid: 14acf6a8-b51a-11ec-b909-0242ac120002
   title: "\u91CF\u5B50\u30A2\u30D7\u30EA\u30B1\u30FC\u30B7\u30E7\u30F3\u306E\u30A2\
     \u30EB\u30B4\u30EA\u30BA\u30E0"
   url: /v1_ch-applications
 - sections:
-  - id: introduction-to-quantum-error-correction-via-the-repetition-code
+  - id: error-correction-repetition-code
     url: v1/ch-quantum-hardware/error-correction-repetition-code
     uuid: 197f7f66-b51a-11ec-b909-0242ac120002
   - id: measurement-error-mitigation
@@ -243,45 +246,45 @@
     \u30A6\u30A7\u30A2\u306E\u63A2\u7A76"
   url: /v1_quantum-hardware
 - sections:
-  - id: calibrating-qubits-using-qiskit-pulse
+  - id: calibrating-qubits-pulse
     url: v1/ch-quantum-hardware/calibrating-qubits-pulse
     uuid: 292100ca-b51a-11ec-b909-0242ac120002
-  - id: accessing-higher-energy-states-with-qiskit-pulse
+  - id: accessing_higher_energy_states
     url: v1/ch-quantum-hardware/accessing_higher_energy_states
     uuid: 2bf0d316-b51a-11ec-b909-0242ac120002
-  - id: introduction-to-transmon-physics
+  - id: transmon-physics
     url: v1/ch-quantum-hardware/transmon-physics
     uuid: 2e944e0e-b51a-11ec-b909-0242ac120002
-  - id: circuit-quantum-electrodynamics
+  - id: cQED-JC-SW
     url: v1/ch-quantum-hardware/cQED-JC-SW
     uuid: 31b2ccdc-b51a-11ec-b909-0242ac120002
   title: "\u30DE\u30A4\u30AF\u30ED\u6CE2\u30D1\u30EB\u30B9\u306B\u3088\u308B\u91CF\
     \u5B50\u30CF\u30FC\u30C9\u30A6\u30A7\u30A2\u306E\u63A2\u7A76"
   url: /v1_quantum-hardware-pulses
 - sections:
-  - id: lab-1-quantum-circuits
+  - id: Lab01_QuantumCircuits
     url: v1/ch-labs/Lab01_QuantumCircuits
     uuid: 3ff5a7ec-b51a-11ec-b909-0242ac120002
-  - id: lab-2-quantum-measurement
+  - id: Lab02_QuantumMeasurement
     url: v1/ch-labs/Lab02_QuantumMeasurement
     uuid: 82eb9557-1420-487b-a3a5-72fdf38878e8
   title: "\u91CF\u5B50\u30B3\u30F3\u30D4\u30E5\u30FC\u30C6\u30A3\u30F3\u30B0\u30FB\
     \u30E9\u30DC"
   url: /v1_ch-labs
 - sections:
-  - id: an-introduction-to-linear-algebra-for-quantum-computing
+  - id: linear_algebra
     url: v1/ch-appendix/linear_algebra
     uuid: 5d2bb2ca-b51a-11ec-b909-0242ac120002
-  - id: basic-qiskit-syntax
+  - id: qiskit
     url: v1/ch-appendix/qiskit
     uuid: 5fd6b79a-b51a-11ec-b909-0242ac120002
   title: "Appendix(\u4ED8\u9332)"
   url: /v1_ch-appendix
 - sections:
-  - id: bonus-level-sandbox
+  - id: hello-qiskit
     url: v1/ch-ex/hello-qiskit
     uuid: 63a85f0e-b51a-11ec-b909-0242ac120002
-  - id: estimating-pi-using-quantum-phase-estimation-algorithm
+  - id: piday-code
     url: v1/ch-demos/piday-code
     uuid: 69b9707c-b51a-11ec-b909-0242ac120002
   title: "\u30B2\u30FC\u30E0\u3068\u30C7\u30E2"
@@ -382,331 +385,18 @@
   - id: quantum-counting
     url: v2/ch-algorithms/quantum-counting
     uuid: f1a9942c-b519-11ec-b909-0242ac120002
-  - id: quantum-teleportation
-    url: v2/ch-algorithms/teleportation
-    uuid: f4f4a090-b519-11ec-b909-0242ac120002
-  - id: superdense-coding
-    url: v2/ch-algorithms/superdense-coding
-    uuid: f87cbd42-b519-11ec-b909-0242ac120002
+  - id: quantum-walk-search-algorithm
+    url: v2/ch-algorithms/quantum-walk-search-algorithm
+    uuid: b2ef52cf-0ce4-494a-8142-fd00e0e8330d
   - id: quantum-key-distribution
     url: v2/ch-algorithms/quantum-key-distribution
     uuid: fbafa8c6-b519-11ec-b909-0242ac120002
-  title: "\u91CF\u5B50\u30D7\u30ED\u30C8\u30B3\u30EB\u3068\u91CF\u5B50\u30A2\u30EB\
-    \u30B4\u30EA\u30BA\u30E0"
-  url: /v2_ch-algorithms
-- sections:
-  - id: solving-linear-systems-of-equations-using-hhl-and-its-qiskit-implementation
-    url: v2/ch-applications/hhl_tutorial
-    uuid: 01484810-b51a-11ec-b909-0242ac120002
-  - id: simulating-molecules-using-vqe
-    url: v2/ch-applications/vqe-molecules
-    uuid: 041df288-b51a-11ec-b909-0242ac120002
-  - id: solving-combinatorial-optimization-problems-using-qaoa
-    url: v2/ch-applications/qaoa
-    uuid: 06ff7ada-b51a-11ec-b909-0242ac120002
-  - id: solving-satisfiability-problems-using-grovers-algorithm
-    url: v2/ch-applications/satisfiability-grover
-    uuid: 09d1529c-b51a-11ec-b909-0242ac120002
-  - id: hybrid-quantum-classical-neural-networks-with-pytorch-and-qiskit
-    url: v2/ch-machine-learning/machine-learning-qiskit-pytorch
-    uuid: 0cac4d14-b51a-11ec-b909-0242ac120002
-  - id: the-variational-quantum-linear-solver
-    url: v2/ch-paper-implementations/vqls
-    uuid: 1163f456-b51a-11ec-b909-0242ac120002
-  - id: flexible-representation-of-quantum-images-frqi
-    url: v2/ch-applications/image-processing-frqi-neqr
-    uuid: 14acf6a8-b51a-11ec-b909-0242ac120002
-  - id: facial-expression-recognition
-    url: v2/ch-applications/facial-expression-recognition
-    uuid: 22cb8bf2-cae9-11ec-9d64-0242ac120002
-  title: "\u91CF\u5B50\u30A2\u30D7\u30EA\u30B1\u30FC\u30B7\u30E7\u30F3\u306E\u30A2\
-    \u30EB\u30B4\u30EA\u30BA\u30E0"
-  url: /v2_ch-applications
-- sections:
-  - id: introduction-to-quantum-error-correction-via-the-repetition-code
-    url: v2/quantum-hardware/error-correction-repetition-code
-    uuid: 197f7f66-b51a-11ec-b909-0242ac120002
-  - id: measurement-error-mitigation
-    url: v2/quantum-hardware/measurement-error-mitigation
-    uuid: 1c5bd31a-b51a-11ec-b909-0242ac120002
-  - id: randomized-benchmarking
-    url: v2/quantum-hardware/randomized-benchmarking
-    uuid: 1fa1d22c-b51a-11ec-b909-0242ac120002
-  - id: measuring-quantum-volume
-    url: v2/quantum-hardware/measuring-quantum-volume
-    uuid: 22ab0bd2-b51a-11ec-b909-0242ac120002
-  - id: density-matrix
-    url: v2/quantum-hardware/density-matrix
-    uuid: 498dce44-cae9-11ec-9d64-0242ac120002
-  title: "\u91CF\u5B50\u56DE\u8DEF\u306B\u3088\u308B\u91CF\u5B50\u30CF\u30FC\u30C9\
-    \u30A6\u30A7\u30A2\u306E\u63A2\u7A76"
-  url: /v2_quantum-hardware
-- sections:
-  - id: calibrating-qubits-using-qiskit-pulse
-    url: v2/ch-quantum-hardware/calibrating-qubits-pulse
-    uuid: 292100ca-b51a-11ec-b909-0242ac120002
-  - id: accessing-higher-energy-states-with-qiskit-pulse
-    url: v2/ch-quantum-hardware/accessing_higher_energy_states
-    uuid: 2bf0d316-b51a-11ec-b909-0242ac120002
-  - id: introduction-to-transmon-physics
-    url: v2/ch-quantum-hardware/transmon-physics
-    uuid: 2e944e0e-b51a-11ec-b909-0242ac120002
-  - id: circuit-quantum-electrodynamics
-    url: v2/ch-quantum-hardware/cQED-JC-SW
-    uuid: 31b2ccdc-b51a-11ec-b909-0242ac120002
-  - id: exploring-the-jaynes-cummings-hamiltonian-with-qiskit-pulse
-    url: v2/ch-quantum-hardware/Jaynes-Cummings-model
-    uuid: 346d7eea-b51a-11ec-b909-0242ac120002
-  title: "\u30DE\u30A4\u30AF\u30ED\u6CE2\u30D1\u30EB\u30B9\u306B\u3088\u308B\u91CF\
-    \u5B50\u30CF\u30FC\u30C9\u30A6\u30A7\u30A2\u306E\u63A2\u7A76"
-  url: /v2_quantum-hardware-pulses
-- sections:
-  - id: lab-1-quantum-circuits
-    url: v2/ch-labs/Lab01_QuantumCircuits
-    uuid: 3ff5a7ec-b51a-11ec-b909-0242ac120002
-  - id: lab-2-single-qubit-gates
-    url: v2/ch-labs/Lab02_Single_Qubit_Gates
-    uuid: 43418a88-b51a-11ec-b909-0242ac120002
-  - id: lab-3-quantum-measurements
-    url: v2/ch-labs/Lab03_QuantumMeasurement
-    uuid: 45aa5412-b51a-11ec-b909-0242ac120002
-  - id: lab-4-bell-ghz-circuit
-    url: v2/ch-labs/Lab04_Bell_GHZ_Circuit
-    uuid: 4a04a0f8-b51a-11ec-b909-0242ac120002
-  - id: lab-9-quantum-simulation-as-a-search-algorithm
-    url: v2/ch-labs/Lab09_QuantumSimulationSearchAlgorithm
-    uuid: 59a076f4-b51a-11ec-b909-0242ac120002
-  title: "\u91CF\u5B50\u30B3\u30F3\u30D4\u30E5\u30FC\u30C6\u30A3\u30F3\u30B0\u30FB\
-    \u30E9\u30DC"
-  url: /v2_ch-labs
-- sections:
-  - id: an-introduction-to-linear-algebra-for-quantum-computing
-    url: v2/ch-appendix/linear_algebra
-    uuid: 5d2bb2ca-b51a-11ec-b909-0242ac120002
-  - id: basic-qiskit-syntax
-    url: v2/ch-appendix/qiskit
-    uuid: 5fd6b79a-b51a-11ec-b909-0242ac120002
-  title: "Appendix(\u4ED8\u9332)"
-  url: /v2_ch-appendix
-- sections:
-  - id: bonus-level-sandbox
-    url: v2/ch-ex/hello-qiskit
-    uuid: 63a85f0e-b51a-11ec-b909-0242ac120002
-  - id: estimating-pi-using-quantum-phase-estimation-algorithm
-    url: v2/ch-demos/piday-code
-    uuid: 69b9707c-b51a-11ec-b909-0242ac120002
-  - id: local-reality-and-the-chsh-inequality
-    url: v2/ch-demos/chsh
-    uuid: 6c507826-b51a-11ec-b909-0242ac120002
-  - id: widgets-demonstration
-    url: v2/ch-demos/widgets-index
-    uuid: 6f1f26ce-b51a-11ec-b909-0242ac120002
-  - id: first-quantum-game
-    url: v2/ch-demos/first-quantum-game
-    uuid: 53a49e62-cae9-11ec-9d64-0242ac120002
-  title: "\u30B2\u30FC\u30E0\u3068\u30C7\u30E2"
-  url: /v2_ch-demos
-- sections:
-  - id: annotations
-    url: v2/examples/annotations
-    uuid: 745440a2-b51a-11ec-b909-0242ac120002
-  - id: blocks
-    url: v2/examples/blocks
-    uuid: 771a4f16-b51a-11ec-b909-0242ac120002
-  - id: widgets
-    url: v2/examples/widgets
-    uuid: 79da872a-b51a-11ec-b909-0242ac120002
-  - id: locale
-    url: v2/examples/locale
-    uuid: 7cbf7978-b51a-11ec-b909-0242ac120002
-  title: "\u4F8B"
-  type: docs
-  url: /v2_examples
-- sections:
-  - id: phase-kickback
-    url: v2/algorithms/phase-kickback
-    uuid: 976e9da6-6c33-4e2f-b7eb-f9588f700f07
-  title: "\u91CF\u5B50\u30A2\u30EB\u30B4\u30EA\u30BA\u30E0"
-  type: learning-path
-  url: /v2_algorithms
-- sections:
-  - id: why-quantum-computing
-    url: v2/intro/why-quantum-computing
-    uuid: bdb1d662-e0f6-428a-8830-befe6b47f320
-  - id: the-atoms-of-computation
-    url: v2/intro/atoms-of-computation
-    uuid: 37ef477e-ab8d-11ec-b909-0242ac120002
-  - id: what-is-quantum
-    url: v2/intro/what-is-quantum
-    uuid: 7c765036-aebc-11ec-b909-0242ac120002
-  - id: describing-quantum-computers
-    url: v2/intro/describing-quantum-computers
-    uuid: 9260a554-aebc-11ec-b909-0242ac120002
-  - id: entangled-states
-    url: v2/intro/entangled-states
-    uuid: 98aa9032-aebc-11ec-b909-0242ac120002
-  - id: visualizing-entanglement
-    url: v2/intro/visualizing-entanglement
-    uuid: 9e8ceb9e-aebc-11ec-b909-0242ac120002
-  - id: grovers-search-algorithm
-    url: v2/intro/grover-intro
-    uuid: a2e6f7a2-aebc-11ec-b909-0242ac120002
-  - id: project
-    url: v2/intro/project
-    uuid: e25c3f90-aebd-11ec-b909-0242ac120002
-  title: "\u91CF\u5B50\u30B3\u30F3\u30D4\u30E5\u30FC\u30BF\u30FC\u5165\u9580"
-  type: learning-path
-  url: /v2_introduction
-- sections:
-  - id: introduction
-    url: v2/quantum-machine-learning/introduction
-    uuid: 16e935fc-b51b-11ec-b909-0242ac120002
-  - id: parameterized-quantum-circuits
-    url: v2/quantum-machine-learning/pqc
-    uuid: 19a9f4de-b51b-11ec-b909-0242ac120002
-  - id: data-encoding
-    url: v2/quantum-machine-learning/encoding
-    uuid: 1d09af66-b51b-11ec-b909-0242ac120002
-  - id: training-quantum-circuits
-    url: v2/quantum-machine-learning/training
-    uuid: 1fb38444-b51b-11ec-b909-0242ac120002
-  - id: supervised-learning
-    url: v2/quantum-machine-learning/supervised
-    uuid: 22cf51bc-b51b-11ec-b909-0242ac120002
-  - id: variational-classification
-    url: v2/quantum-machine-learning/vqc
-    uuid: 26951034-b51b-11ec-b909-0242ac120002
-  - id: quantum-feature-maps-kernels
-    url: v2/quantum-machine-learning/kernel
-    uuid: 29b0fda0-b51b-11ec-b909-0242ac120002
-  - id: unsupervised-learning
-    url: v2/quantum-machine-learning/unsupervised
-    uuid: 2cc6b386-b51b-11ec-b909-0242ac120002
-  - id: quantum-generative-adversarial-networks
-    url: v2/quantum-machine-learning/qgan
-    uuid: 2ff2a3e4-b51b-11ec-b909-0242ac120002
-  - id: project
-    url: v2/quantum-machine-learning/project
-    uuid: 345f0c2e-b51b-11ec-b909-0242ac120002
-  title: "\u91CF\u5B50\u6A5F\u68B0\u5B66\u7FD2"
-  type: learning-path
-  url: /v2_machine-learning
-
-- title: Variational Algorithm Design
-  url: /algorithm-design
-  type: learning-path
-  sections:
-    - id: variational
-      uuid: 4fbef8a1-f8c0-4f08-af4f-5b8d723bd274
-      url: /algorithm-design/variational/variational
-    - id: badge
-      uuid: c6038861-dcf9-4089-bf6f-03843ed52cb2
-      url: /algorithm-design/badge/badge
-
-
-#---- Below was added by machine ---
-
-- sections:
-  - id: single-systems
-    url: v2/basics/single-systems
-    uuid: dd5680be-7490-401e-9e68-c1ec7d0f5d61
-  - id: multiple-systems
-    url: v2/basics/multiple-systems
-    uuid: 327b128f-4fec-410a-a3f9-079939002bf2
-  - id: quantum-circuits
-    url: v2/basics/quantum-circuits
-    uuid: 57d5452e-85cd-40f8-a290-11466213c1a6
-  - id: entanglement-in-action
-    url: v2/basics/entanglement-in-action
-    uuid: 366f7150-4c04-4763-8269-ae6180812b73
-  title: "\u91CF\u5B50\u60C5\u5831\u306E\u57FA\u790E"
-  url: /v2_basics
-- sections:
-  - id: environment-setup-guide-to-work-with-qiskit-textbook
-    url: v2/ch-prerequisites/setting-the-environment
-    uuid: a5a86ddc-b519-11ec-b909-0242ac120002
-  - id: introduction-to-python-and-jupyter-notebooks
-    url: v2/ch-prerequisites/python-and-jupyter-notebooks
-    uuid: a9f5ea68-b519-11ec-b909-0242ac120002
-  title: "\u524D\u63D0\u6761\u4EF6"
-  url: /v2_ch-prerequisites
-- sections:
-  - id: introduction
-    url: v2/ch-states/introduction
-    uuid: afcc559e-b519-11ec-b909-0242ac120002
-  - id: the-atoms-of-computation
-    url: v2/ch-states/atoms-computation
-    uuid: b31c20ee-b519-11ec-b909-0242ac120002
-  - id: representing-qubit-states
-    url: v2/ch-states/representing-qubit-states
-    uuid: b81717ca-b519-11ec-b909-0242ac120002
-  - id: single-qubit-gates
-    url: v2/ch-states/single-qubit-gates
-    uuid: bd4334a4-b519-11ec-b909-0242ac120002
-  - id: the-case-for-quantum-computers
-    url: v2/ch-states/case-for-quantum
-    uuid: c0bbff62-b519-11ec-b909-0242ac120002
-  title: "\u91CF\u5B50\u72B6\u614B\u3068\u91CF\u5B50\u30D3\u30C3\u30C8"
-  url: /v2_ch-states
-- sections:
-  - id: introduction
-    url: v2/ch-gates/introduction
-    uuid: c50f4c90-b519-11ec-b909-0242ac120002
-  - id: multiple-qubits-and-entangled-states
-    url: v2/ch-gates/multiple-qubits-entangled-states
-    uuid: c7cfe502-b519-11ec-b909-0242ac120002
-  - id: phase-kickback
-    url: v2/algorithms/phase-kickback
-    uuid: cb7353ec-b519-11ec-b909-0242ac120002
-  - id: basic-circuit-identities
-    url: v2/ch-gates/more-circuit-identities
-    uuid: ce8ce75a-b519-11ec-b909-0242ac120002
-  - id: proving-universality
-    url: v2/ch-gates/proving-universality
-    uuid: d15d3548-b519-11ec-b909-0242ac120002
-  - id: classical-computation-on-a-quantum-computer
-    url: v2/ch-gates/oracles
-    uuid: d4f6e424-b519-11ec-b909-0242ac120002
-  title: "\u8907\u6570\u91CF\u5B50\u30D3\u30C3\u30C8\u3068\u3082\u3064\u308C\u72B6\
-    \u614B"
-  url: /v2_ch-gates
-- sections:
-  - id: quantum-circuits
-    url: v2/ch-algorithms/defining-quantum-circuits
-    uuid: d90cbfb6-b519-11ec-b909-0242ac120002
-  - id: deutsch-jozsa-algorithm
-    url: v2/ch-algorithms/deutsch-jozsa
-    uuid: dc9d3aa2-b519-11ec-b909-0242ac120002
-  - id: bernstein-vazirani-algorithm
-    url: v2/ch-algorithms/bernstein-vazirani
-    uuid: df6e5a40-b519-11ec-b909-0242ac120002
-  - id: simons-algorithm
-    url: v2/ch-algorithms/simon
-    uuid: e2714126-b519-11ec-b909-0242ac120002
-  - id: quantum-fourier-transform
-    url: v2/ch-algorithms/quantum-fourier-transform
-    uuid: e59ba472-b519-11ec-b909-0242ac120002
-  - id: quantum-phase-estimation
-    url: v2/ch-algorithms/quantum-phase-estimation
-    uuid: e8ac9ec8-b519-11ec-b909-0242ac120002
-  - id: shors-algorithm
-    url: v2/ch-algorithms/shor
-    uuid: ec157cd8-b519-11ec-b909-0242ac120002
-  - id: grovers-algorithm
-    url: v2/ch-algorithms/grover
-    uuid: eebb7488-b519-11ec-b909-0242ac120002
-  - id: quantum-counting
-    url: v2/ch-algorithms/quantum-counting
-    uuid: f1a9942c-b519-11ec-b909-0242ac120002
-  - id: quantum-teleportation
-    url: v2/ch-algorithms/teleportation
-    uuid: f4f4a090-b519-11ec-b909-0242ac120002
   - id: superdense-coding
     url: v2/ch-algorithms/superdense-coding
     uuid: f87cbd42-b519-11ec-b909-0242ac120002
-  - id: quantum-key-distribution
-    url: v2/ch-algorithms/quantum-key-distribution
-    uuid: fbafa8c6-b519-11ec-b909-0242ac120002
+  - id: quantum-teleportation
+    url: v2/ch-algorithms/teleportation
+    uuid: f4f4a090-b519-11ec-b909-0242ac120002
   title: "\u91CF\u5B50\u30D7\u30ED\u30C8\u30B3\u30EB\u3068\u91CF\u5B50\u30A2\u30EB\
     \u30B4\u30EA\u30BA\u30E0"
   url: /v2_ch-algorithms
@@ -796,16 +486,16 @@
     \u30E9\u30DC"
   url: /v2_ch-labs
 - sections:
-  - id: an-introduction-to-linear-algebra-for-quantum-computing
+  - id: linear_algebra
     url: v2/ch-appendix/linear_algebra
     uuid: 5d2bb2ca-b51a-11ec-b909-0242ac120002
-  - id: basic-qiskit-syntax
+  - id: qiskit
     url: v2/ch-appendix/qiskit
     uuid: 5fd6b79a-b51a-11ec-b909-0242ac120002
   title: "Appendix(\u4ED8\u9332)"
   url: /v2_ch-appendix
 - sections:
-  - id: bonus-level-sandbox
+  - id: hello-qiskit
     url: v2/ch-ex/hello-qiskit
     uuid: 63a85f0e-b51a-11ec-b909-0242ac120002
   - id: estimating-pi-using-quantum-phase-estimation-algorithm
diff --git a/translations/ja/v1/ch-algorithms/quantum-walk-search-algorithm.ipynb b/translations/ja/v1/ch-algorithms/quantum-walk-search-algorithm.ipynb
index 41f3e802e..c7a437c26 100644
--- a/translations/ja/v1/ch-algorithms/quantum-walk-search-algorithm.ipynb
+++ b/translations/ja/v1/ch-algorithms/quantum-walk-search-algorithm.ipynb
@@ -2,7 +2,7 @@
  "cells": [
   {
    "cell_type": "markdown",
-   "id": "ba9d6e23",
+   "id": "a97639dd",
    "metadata": {
     "tags": [
      "remove_cell"
@@ -14,153 +14,142 @@
   },
   {
    "cell_type": "markdown",
-   "id": "fb9cec4d",
+   "id": "17f20a60",
    "metadata": {},
    "source": [
     "量子ウォークは古典のマルコフ連鎖の量子版であり、多くの量子アルゴリズムにとって重要なものです。この章ではグラフ上にマークした要素を見つける量子ウォークによる探索アルゴリズムを実装します。このアルゴリズムは古典に比べて二次加速を持ちます。\n",
-    "## 目次\n",
-    "1. [古典マルコフ連鎖](#markov)\n",
-    "2. [量子ウォーク](#qwalks)\n",
-    "    1. [コイン量子ウォーク](#coined)\n",
-    "    2. [セゲディ量子ウォーク](#szegedy)\n",
-    "    3. [コインとセゲディ量子ウォークの等価性](#equivalence)\n",
-    "3. [例: 超立方体上の量子ウォーク](#example1)\n",
-    "4. [量子ウォークによる探索アルゴリズム](#qwalkalgo)\n",
-    "5. [例: 4次元超立方体上の量子ウォークによる探索](#qiskiti)\n",
-    "6. [リファレンス](#references)\n",
+    "\n",
     "\n",
     "## 1. 古典マルコフ連鎖 <a name=\"markov\"></a>\n",
     "マルコフ連鎖は現実のプロセスをモデル化するためによく使われる確率過程です。状態と遷移確率からなり、各時間ステップでの状態間の確率の移動を記述します。ここで紹介する離散時間マルコフ連鎖では、時間ステップは離散的です。マルコフ連鎖はマルコフ性を満たします、つまり次のステップでの確率過程は現在のステップのみに依存して、それ以前には依存しません。マルコフ連鎖は状態間の確率の移動を表現する遷移行列 P を持ちます。ここで、マルコフ連鎖の例と、その遷移行列 $P$ を示します。\n",
-    "\\begin{equation}\n",
-    "P=\n",
-    "\\begin{pmatrix}\n",
-    "0.1 & 0.3 & 0.3\\\\\n",
-    "0.1 & 0.1 & 0.2 \\\\\n",
-    "0.8 & 0.6 & 0.5\n",
-    "\\end{pmatrix}\n",
-    "\\label{eq:matrix_example}\n",
-    "\\end{equation}\n",
+    "\n",
+    "\n",
+    "   $$\n",
+    "     P\n",
+    "     = \\begin{pmatrix}\n",
+    "    0.1 & 0.3 & 0.3\\\\\n",
+    "    0.1 & 0.1 & 0.2 \\\\\n",
+    "    0.8 & 0.6 & 0.5\n",
+    "     \\end{pmatrix}.\n",
+    "   $$\n",
+    "   \n",
     "<img src=\"images/markov_example.png\" width='480'>\n",
     "\n",
-    "遷移行列 $P$ が与えられると、$t$ 時間ステップ後の確率分布を $P^t$ により計算できます。\n",
+    "遷移行列 $P$ が与えられると、 $t$ 時間ステップ後の確率分布を $P^t$ により計算できます。\n",
     "\n",
     "## 2. 量子ウォーク <a name=\"qwalks\"></a>\n",
     "量子ウォークは古典のマルコフ連鎖の量子版です。測定するまでは、量子ウォークは全ての可能な経路の重ね合わせを取ります。また、量子干渉によりある状態は打ち消しあいます。この現象を、間違えた答えを消去するような設計に利用することで、量子ウォークアルゴリズムをランダムウォークよりも速くします。コイン量子ウォークとセゲディ量子ウォークがよく用いられ、これらはある条件下では等価になります。コイン量子ウォークはグラフ上の頂点を、セゲディ量子ウォークは辺を動きます。量子ウォークの実装の前にこの二つのモデルを紹介します。\n",
     "\n",
     " $\\newcommand{\\ket}[1]{\\left|{#1}\\right\\rangle}$\n",
     " $\\newcommand{\\bra}[1]{\\left\\langle{#1}\\right|}$\n",
-    "### 2.1 コイン量子ウォーク <a name=\"coined\"></a>\n",
+    "### 2.A コイン量子ウォーク <a name=\"coined\"></a>\n",
     "　コイン量子ウォークの簡単な例は、無限長の整数上の運動です。この場合、整数 $\\mathbb{Z}$ 上での位置を $\\{\\ket{j} : j \\in  \\mathbb{Z} \\}$ とします。そして、コインの計算基底は $\\{\\ket{0}, \\ket{1}\\}$ とし、コインが $\\ket{0}$ の場合はある方向に動き、コインが $\\ket{1}$ の場合はもう一方に動きます。\n",
     "\n",
     "コイン量子ウォークはグラフの頂点上を動くので、頂点を状態とします。辺で繋がっている状態間のみ動くことができます。コインモデルでは、二つの量子状態空間と二つの演算子を使います。一つ目の状態空間は位置を表す位置状態です。上記の運動では、整数上のどこでも移動できるので、整数となります。もう一方は、コイン状態です。コイン状態は次のステップでどのように動くかを決定します。このコイン状態と位置状態とを合わせて、ヒルベルト空間上のベクトルとして表現できます。コイン状態を $\\mathcal{H}_C$ のベクトルとして表し、位置状態を $\\mathcal{H}_P$ のベクトルとして表すと、運動のすべての量子状態空間は $\\mathcal{H} = \\mathcal{H}_C \\otimes \\mathcal{H}_P$ となります。\n",
     "\n",
     "すでに述べたように、モデルには二つの演算子がありコイン演算子 $C$ とシフト演算子 $S$ です。コイン演算子は時間ステップ毎に $\\mathcal{H}_C$ に作用して重ね合わせ状態を作るため、全ての可能な経路を同時に動きます。整数上の運動では、時間ステップ毎に左と右の両方に動くことになります。いろいろなコイン演算子がありますが、よく用いられるのはアダマールコインとグローバーコインです。アダマールコインはアダマールゲートにより等確率での重ね合わせ状態を作ります: \n",
     "\n",
-    "\\begin{equation}\n",
-    "    H = \\frac{1}{\\sqrt{2}}\n",
-    "\\begin{bmatrix}\n",
-    "1 & 1 \\\\\n",
-    "1 & -1 \n",
-    "\\end{bmatrix}\n",
-    "\\end{equation}\n",
+    "   $$\n",
+    "     H\n",
+    "     = \\frac{1}{\\sqrt{2}}\n",
+    "     \\begin{bmatrix}\n",
+    "    1 & 1 \\\\\n",
+    "    1 & -1 \n",
+    "     \\end{bmatrix}.\n",
+    "   $$\n",
     "\n",
     "グローバーコインはグローバーのアルゴリズムにあるグローバーの拡散演算子を用い、以下のように定義します\n",
     "\n",
-    "\\begin{equation}\n",
-    "    G = \n",
-    "    \\begin{bmatrix}\n",
-    "\\frac{2}{n} -1 & \\frac{2}{n} & \\ldots & \\frac{2}{n}\\\\\n",
-    "\\frac{2}{n} & \\frac{2}{n} - 1  & \\ldots & \\frac{2}{n} \\\\\n",
-    "\\vdots & \\vdots & \\ddots & \\vdots \\\\\n",
-    "\\frac{2}{n} & \\frac{2}{n} & \\ldots & \\frac{2}{n} -1\n",
-    "\\end{bmatrix}\n",
-    "\\end{equation}\n",
+    "   $$\n",
+    "     G\n",
+    "     = \\begin{bmatrix}\n",
+    "    \\frac{2}{n} -1 & \\frac{2}{n} & \\ldots & \\frac{2}{n}\\\\\n",
+    "    \\frac{2}{n} & \\frac{2}{n} - 1  & \\ldots & \\frac{2}{n} \\\\\n",
+    "    \\vdots & \\vdots & \\ddots & \\vdots \\\\\n",
+    "    \\frac{2}{n} & \\frac{2}{n} & \\ldots & \\frac{2}{n} -1\n",
+    "     \\end{bmatrix}.\n",
+    "   $$\n",
     "\n",
-    "アダマールコインのようにグローバーコインも重ね合わせ状態を作ります。しかし、振る舞いは少し異なります。位置 $\\ket{000}$ にグローバーコインを作用させると、下図のような状態ベクトルの確率分布を得ます。このように、アダマールコインのような等確率の重ね合わせ状態を作りません。他の状態に比べて、$\\ket{000}$ により大きな確率を持ちます。\n",
+    "アダマールコインのようにグローバーコインも重ね合わせ状態を作ります。しかし、振る舞いは少し異なります。位置 $\\ket{000}$ にグローバーコインを作用させると、下図のような状態ベクトルの確率分布を得ます。このように、アダマールコインのような等確率の重ね合わせ状態を作りません。他の状態に比べて、 $\\ket{000}$ により大きな確率を持ちます。\n",
     "\n",
     "<img src=\"images/grover_prob.svg\" width='480'>\n",
     "\n",
     "モデルのもう一つの演算子のシフト演算子は $\\mathcal{H}_P$ に作用して、次の位置に移動させます。整数上の運動では、シフト演算子により、コインが $\\ket{0}$ の時は左に動き、コインが $\\ket{1}$ の時は右に動きます: \n",
     "\n",
-    "\\begin{equation}\n",
-    "    S \\ket{0}\\ket{j} = \\ket{0}\\ket{j+1}\n",
-    "\\end{equation}\n",
-    "\n",
-    "\\begin{equation}\n",
-    "    S \\ket{1}\\ket{j} = \\ket{1}\\ket{j-1}\n",
-    "\\end{equation}\n",
-    "\n",
+    "$$\n",
+    "\\begin{aligned}\n",
+    "  S \\vert 0 \\rangle \\vert j \\rangle & = \\vert 0 \\rangle \\vert j+1 \\rangle \\\\[1mm]\n",
+    "   S \\vert 1 \\rangle \\vert j \\rangle & = \\vert 1 \\rangle \\vert j-1 \\rangle\n",
+    "\\end{aligned}\n",
+    "$$\n",
     "\n",
     "上記で定義されたシフト演算子で、一ステップ分のコイン量子ウォークを以下のユニタリ演算子として表現できます\n",
-    "\\begin{equation}\n",
+    "$$\n",
     "    U = SC,\n",
-    "\\end{equation}\n",
+    "$$\n",
     "\n",
     "ここで C はコイン演算子です。整数上の量子ウォークでアダマールコインを用いましたが、C はアダマールコインでも、グローバーコインでも、その他のコイン演算子でも用いることができます。\n",
     "\n",
     "\n",
     "先に進みましょう。$t$ 時間ステップ後の量子状態 $\\ket{\\psi}$ は以下で表すことができます\n",
-    "\\begin{equation}\n",
+    "$$\n",
     "    \\ket{\\psi (t)} = U^t \\ket{\\psi(0)},\n",
-    "\\end{equation}\n",
-    "ここで、$\\ket{\\psi(0)}$ は初期状態で、U は一ステップ分の演算子です[1]。\n",
+    "$$\n",
+    "ここで、 $\\ket{\\psi(0)}$ は初期状態で、U は一ステップ分の演算子です[1]。\n",
     "\n",
     "コイン量子ウォークは多くの正則グラフ(全ての頂点で隣接する頂点の数が同じグラフ)で安定しています[2]。 もう一つのモデルであるセゲディ量子ウォークは非正則グラフに適しており、次の章で解説します。\n",
     "\n",
-    "### 2.2 セゲディ量子ウォーク <a name=\"szegedy\"></a>\n",
-    "コイン量子ウォークはグラフの頂点を動きましたが、セゲディ量子ウォークは元のグラフの二部二重被覆グラフ上の辺を動きます。二重被覆グラフとは元のグラフの二倍の頂点を持つグラフです。二部二重被覆グラフの二つの頂点は、元のグラフでその頂点が繋がっている場合にのみ、辺で繋がります。このモデルを作るには、古典のランダムウォークの遷移確率行列 P から始めます。1 章で示したように、古典の離散時間ランダムウォークは遷移行列 P で記述されます。任意の $N \\times N$ の遷移行列 $P$ を持つ $N$-頂点グラフに対して、離散時間量子ウォークをヒルベルト空間 $\\mathcal{H}^N \\otimes \\mathcal{H}^N$ 上のユニタリ演算子として定義できます。$P_{jk}$ を $j$ から $k$ へ遷移する確率と定義しましょう。運動を定義する前に、正規化された状態を定義します\n",
+    "### 2.B セゲディ量子ウォーク <a name=\"szegedy\"></a>\n",
+    "コイン量子ウォークはグラフの頂点を動きましたが、セゲディ量子ウォークは元のグラフの二部二重被覆グラフ上の辺を動きます。二重被覆グラフとは元のグラフの二倍の頂点を持つグラフです。二部二重被覆グラフの二つの頂点は、元のグラフでその頂点が繋がっている場合にのみ、辺で繋がります。このモデルを作るには、古典のランダムウォークの遷移確率行列 P から始めます。1 章で示したように、古典の離散時間ランダムウォークは遷移行列 P で記述されます。任意の $N \\times N$ の遷移行列 $P$ を持つ $N$-頂点グラフに対して、離散時間量子ウォークをヒルベルト空間 $\\mathcal{H}^N \\otimes \\mathcal{H}^N$ 上のユニタリ演算子として定義できます。 $P_{jk}$ を $j$ から $k$ へ遷移する確率と定義しましょう。運動を定義する前に、正規化された状態を定義します\n",
     "\n",
-    "\\begin{equation}\n",
+    "$$\n",
     "    \\ket{\\psi_j} := \\sum_{k=1}^N \\sqrt{P_{kj}} \\ket{j,k}, \\; j=1,...,N\n",
-    "\\end{equation}\n",
+    "$$\n",
     "\n",
-    "そして、${\\ket{\\psi_j}}:j=1,...,N$ への射影は\n",
+    "そして、 $\\vert \\psi_j \\rangle :j=1,...,N$ への射影は\n",
     "\n",
-    "\\begin{equation}\n",
+    "$$\n",
     "    \\Pi := \\sum_{j=1}^N \\ket{\\psi_j} \\bra{\\psi_j}\n",
-    "\\label{eq:sz_pi}\n",
-    "\\end{equation}\n",
+    "$$\n",
     "\n",
     "シフト演算子を導入します:\n",
     "\n",
-    "\\begin{equation}\n",
+    "$$\n",
     "    S := \\sum_{j,k=1}^N \\ket{j,k} \\bra{k,j}\n",
-    "\\label{eq:sz_s}\n",
-    "\\end{equation}\n",
+    "$$\n",
     "\n",
     "上で定義した $S$ と $\\Pi$ を用いて一ステップ分の離散時間量子ウォークを定義します:\n",
     "\n",
-    "\\begin{equation}\n",
+    "$$\n",
     "    U := S(2 \\Pi - 1),\n",
-    "\\label{eq:sz_op}\n",
-    "\\end{equation}\n",
+    "$$\n",
     "\n",
-    "ここで、$(2 \\Pi - 1)$ は反射演算子です。また、$t$ ステップ後の状態は $U^t$ により定義できます[2]。\n",
+    "ここで、 $(2 \\Pi - 1)$ は反射演算子です。また、 $t$ ステップ後の状態は $U^t$ により定義できます[2]。\n",
     "\n",
-    "### 2.3 コインとセゲディ量子ウォークの等価性 <a name=\"equivalence\"></a>\n",
+    "### 2.C コインとセゲディ量子ウォークの等価性 <a name=\"equivalence\"></a>\n",
     "グローバーコインによるコイン量子ウォークはセゲディ量子ウォークと等価であることが知られています。詳細は Thomas G. Wong による文献[3]を参照ください、文献では演算子の等価性についても示されています。\n",
     "\n",
     "\n",
     "## 3. 例: 超立方体上の量子ウォーク <a name=\"example1\"></a>\n",
-    "超立方体は $3$次元の立方体を $n$次元化したものです。全ての頂点の次数は $n$ で、超立方体は $N=2^n$個の頂点を持ちます。二値の $n$タプルを用いて超立方体の頂点を表現できます。隣接する頂点の二値表現は一つの二値のみが異なります。例えば4次元超立方体の場合、頂点 $0000$ の隣接頂点は $0001$, $0010$, $0100$, $1000$ です。したがって、ハミング距離が1の全ての頂点と接続しています。辺も同時にラベルできます。二つの隣接する頂点のa番目のビットが異なる場合、$a$とラベルされた辺で繋がります。\n",
+    "超立方体は $3$ 次元の立方体を $n$ 次元化したものです。全ての頂点の次数は $n$ で、超立方体は $N=2^n$ 個の頂点を持ちます。二値の $n$ タプルを用いて超立方体の頂点を表現できます。隣接する頂点の二値表現は一つの二値のみが異なります。例えば4次元超立方体の場合、頂点 $0000$ の隣接頂点は $0001$, $0010$, $0100$, $1000$ です。したがって、ハミング距離が1の全ての頂点と接続しています。辺も同時にラベルできます。二つの隣接する頂点のa番目のビットが異なる場合、 $a$ とラベルされた辺で繋がります。\n",
     "\n",
-    "超立方体のコイン量子ウォークを表現するヒルベルト空間は $\\mathcal{H} = \\mathcal{H}^n \\otimes \\mathcal{H}^{2^n}$ です、ここで、$\\mathcal{H}^n$ はコイン空間で、$\\mathcal{H}^{2^n}$ は位置です。計算基底では\n",
+    "超立方体のコイン量子ウォークを表現するヒルベルト空間は $\\mathcal{H} = \\mathcal{H}^n \\otimes \\mathcal{H}^{2^n}$ です。ここで、　$\\mathcal{H}^n$ はコイン空間で、　$\\mathcal{H}^{2^n}$ は位置です。計算基底では\n",
     "\n",
-    "\\begin{equation}\n",
+    "$$\n",
     "    \\big\\{ \\ket{a,\\vec{v}}, 0 \\leq a \\leq n-1, \\vec{v} \\in  \\{(00...00), (00...01), ....., (11...11 )\\} \\big\\}. \n",
-    "\\end{equation}\n",
+    "$$\n",
     "\n",
-    "辺 $a$ を表すコイン計算基底 $a$ は、移動の方向を表します。$a=0$ の場合、現在の辺とは一番目の二値が異なる辺へ移動します。また、$a=1$ の場合、現在の辺とは二番目の二値が異なる辺へ移動します。$\\vec{e}_a$ を $a$ 以外は全て $0$ をとる二値の n タプルとします。そして、シフト演算子 $S$ は 状態 $\\ket{a} \\ket{\\vec{v}}$ を $\\ket{a} \\ket{\\vec{v} \\oplus \\vec{e}_a}$ へ移動します。\n",
+    "辺 $a$ を表すコイン計算基底 $a$ は、移動の方向を表します。　$a=0$ の場合、現在の辺とは一番目の二値が異なる辺へ移動します。また、　$a=1$ の場合、現在の辺とは二番目の二値が異なる辺へ移動します。　$\\vec{e}_a$ を $a$ 以外は全て $0$ をとる二値の n タプルとします。そして、シフト演算子 $S$ は 状態 $\\ket{a} \\ket{\\vec{v}}$ を $\\ket{a} \\ket{\\vec{v} \\oplus \\vec{e}_a}$ へ移動します。\n",
     "\n",
-    "\\begin{equation}\n",
+    "$$\n",
     "    S \\ket{a} \\ket{\\vec{v}} = \\ket{a} \\ket{\\vec{v} \\oplus \\vec{e}_a}.\n",
-    "\\end{equation}\n",
+    "$$\n",
     "\n",
     "ここでは、グローバーコイン $G$ を用います。そして発展演算子は\n",
     "\n",
-    "\\begin{equation}\n",
+    "$$\n",
     "    U = SG.\n",
-    "\\end{equation}\n",
+    "$$\n",
     "\n",
     "次に、4次元超立方体の量子ウォークをどのように実装するのかを紹介します。コイン演算子とシフト演算子の実装が必要となります。Qiskitから必要なライブラリをインポートするところから始めます。"
    ]
@@ -168,7 +157,7 @@
   {
    "cell_type": "code",
    "execution_count": 2,
-   "id": "de5e4475",
+   "id": "6592de64",
    "metadata": {},
    "outputs": [],
    "source": [
@@ -188,7 +177,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "5f13edaa",
+   "id": "fa16553e",
    "metadata": {},
    "source": [
     "回路は位置に4個、コインに2個用いて、6量子ビットからなります。前に述べたようにコインはグローバーコインで、グローバーのアルゴリズムの拡散演算子です。この実装からはじめます。"
@@ -197,7 +186,7 @@
   {
    "cell_type": "code",
    "execution_count": 3,
-   "id": "427c1f4a",
+   "id": "872212b3",
    "metadata": {},
    "outputs": [
     {
@@ -226,7 +215,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "ef61d782",
+   "id": "06243b9e",
    "metadata": {},
    "source": [
     "次にシフト演算子を実装します。隣接する頂点にだけ移動します、また、全ての隣接する頂点は一ビットだけ異なります。コインに従って移動するため、頂点の量子ビットの一つに対してNOTゲートを作用します。コインが $\\ket{11}$ 状態の時は、一番目の頂点の量子ビットが異なる状態に移動します。コインが $\\ket{10}$ または $\\ket{01}$ 状態の時は、それぞれ二番目、三番目の量子ビットが異なる状態に移動します。最後にグローバーコインが $\\ket{00}$ の時は、4番目の量子ビットを反転します。これは、グローバーコインの後に、CCNOTゲートとNOTゲートにより実装します。これら全てで4次元超立方体の量子ウォークの一ステップ分になります。"
@@ -235,7 +224,7 @@
   {
    "cell_type": "code",
    "execution_count": 4,
-   "id": "d3ab5492",
+   "id": "34cd6edb",
    "metadata": {},
    "outputs": [
     {
@@ -269,18 +258,18 @@
   },
   {
    "cell_type": "markdown",
-   "id": "159bca03",
+   "id": "44c7294b",
    "metadata": {},
    "source": [
     "## 4. 量子ウォークによる探索アルゴリズム <a name=\"qwalkalgo\"></a>\n",
-    "次にグラフ上にマークした要素を見つける量子ウォークによる探索アルゴリズムを実装します。はじめにアルゴリズムを紹介し、その後で実装します。量子ウォークによる探索アルゴリズムは、量子ウォークを用いて、グラフ上にマークした要素を見つける問題を解きます。つまり、ある頂点集合 $|M|$ をマークして、グラフ上の任意の頂点から始め、マークした複数の頂点を見つけるまで移動します。量子ウォークによる探索では基底として、現在の頂点と直前の頂点に対応する二つのレジスタを必要とします。つまりグラフの辺に対応します。遷移行列 $P$ による古典マルコフ連鎖を基にして、$\\mathcal{H}$ 上のユニタリ演算子 $W(P)$ により量子ウォークを記述します。また、頂点 $x$ に隣接する頂点の一様な重ね合わせ状態として $\\ket{p_x} = \\sum_y \\sqrt{P_{xy}}\\ket{y}$ を定義します。$\\ket{x}\\ket{y}$ を一つの基底とします。もし $x$ がマークした頂点の場合、基底 $\\ket{x}\\ket{y}$ を ''good'' と定義します。それ以外を ''bad'' とします。そして、''good'' と ''bad'' 状態は: \n",
+    "次にグラフ上にマークした要素を見つける量子ウォークによる探索アルゴリズムを実装します。はじめにアルゴリズムを紹介し、その後で実装します。量子ウォークによる探索アルゴリズムは、量子ウォークを用いて、グラフ上にマークした要素を見つける問題を解きます。つまり、ある頂点集合 $|M|$ をマークして、グラフ上の任意の頂点から始め、マークした複数の頂点を見つけるまで移動します。量子ウォークによる探索では基底として、現在の頂点と直前の頂点に対応する二つのレジスタを必要とします。つまりグラフの辺に対応します。遷移行列 $P$ による古典マルコフ連鎖を基にして、　$\\mathcal{H}$ 上のユニタリ演算子 $W(P)$ により量子ウォークを記述します。また、頂点 $x$ に隣接する頂点の一様な重ね合わせ状態として $\\ket{p_x} = \\sum_y \\sqrt{P_{xy}}\\ket{y}$ を定義します。 $\\ket{x}\\ket{y}$ を一つの基底とします。もし $x$ がマークした頂点の場合、基底 $\\ket{x}\\ket{y}$ を ''good'' と定義します。それ以外を ''bad'' とします。そして、''good'' と ''bad'' 状態は: \n",
     "\n",
-    "\\begin{equation}\n",
+    "$$\n",
     "    \\ket{G} = \\frac{1}{\\sqrt{|M|}} \\sum_{x \\in M} \\ket{x} \\ket{p_x}, \\;\n",
     "    \\ket{B} = \\frac{1}{\\sqrt{N-|M|}} \\sum_{x \\notin M} \\ket{x} \\ket{p_x},\n",
-    "\\end{equation}\n",
+    "$$\n",
     "\n",
-    "これらは good および bad の基底の重ね合わせ状態です。次に、$\\epsilon = |M|/N$ と $\\theta = \\arcsin(\\sqrt{\\epsilon})$ を定義します。\n",
+    "これらは good および bad の基底の重ね合わせ状態です。次に、 $\\epsilon = |M|/N$ と $\\theta = \\arcsin(\\sqrt{\\epsilon})$ を定義します。\n",
     "\n",
     "\n",
     "まとめると、アルゴリズムは以下の三つのステップからなります:\n",
@@ -296,37 +285,37 @@
     "3. 計算基底での測定を実施\n",
     "\n",
     "\n",
-    "ステップ $1$ は、アダマールゲートと、$x$ が一つ目のレジスタと等しい時にだけ位相をシフトする位相オラクルを用いて反射を行うことで簡単に実装できます。\n",
+    "ステップ $1$ は、アダマールゲートと、 $x$ が一つ目のレジスタと等しい時にだけ位相をシフトする位相オラクルを用いて反射を行うことで簡単に実装できます。\n",
     "\n",
     "\n",
     "ステップ 2(b) は以下の写像を行うユニタリ変換 $R(P)$ を探すことと等価です:\n",
-    "\\begin{align}\n",
+    "$$\n",
     "\\label{eq:mapping_1}\n",
     "    \\ket{U} &\\mapsto \\ket{U}, \\: \\text{と} \\\\\n",
     "    \\ket{\\psi} &\\mapsto -\\ket{\\psi}, \\: \\forall \\ket{\\psi} \\text{$\\ket{U}$ に直交する $W(P)$ の固有ベクトルに含まれる場合}\n",
     "\\label{eq:mapping_2}\n",
-    "\\end{align}\n",
+    "$$\n",
     "\n",
-    "この演算子を探すために、$W(P)$ の位相推定を行います。上記で $W(P)$ をランダムウォークの発展演算子として定義しました。2.1 章で見たように、これはユニタリ演算子です。従って、$W(P)$ の固有値のノルムは $1$ となります。そこで、$W(P)$ の固有値を $e^{\\pm 2i\\theta_j}$ と書き直します。ユニタリ演算子 $W(P)$ は固有値が $1$ の固有ベクトル $\\ket{U}$ を持ちます。これは、$\\theta_1=0$ により得られます。$R(P)$ は補助量子ビットを加えて $O(1/\\sqrt{\\delta})$ の精度で位相推定を行うことで $\\ket{U}$ を見つけます、ここで、$\\delta$ は $P$ のスペクトルギャップです。これを行うには、$W(P)$ を $O(1/\\sqrt{\\delta})$ 回作用する必要があります。$\\ket{w}$ を $W(P)$ の固有ベクトルで、固有値 $e^{\\pm 2i\\theta_j}$ を持つとしましょう。 $\\tilde{\\theta_j}$ を位相推定による $\\theta_j$ のもっとも良い推定値と仮定します。ステップ 2(b) の $\\ket{w}$ の写像を与える演算子 $R(P)$ は文献[4]で与えられています。\n",
+    "この演算子を探すために、 $W(P)$ の位相推定を行います。上記で $W(P)$ をランダムウォークの発展演算子として定義しました。2.1 章で見たように、これはユニタリ演算子です。従って、 $W(P)$ の固有値のノルムは $1$ となります。そこで、 $W(P)$ の固有値を $e^{\\pm 2i\\theta_j}$ と書き直します。ユニタリ演算子 $W(P)$ は固有値が $1$ の固有ベクトル $\\ket{U}$ を持ちます。これは、 $\\theta_1=0$ により得られます。　$R(P)$ は補助量子ビットを加えて $O(1/\\sqrt{\\delta})$ の精度で位相推定を行うことで $\\ket{U}$ を見つけます、ここで、 $\\delta$ は $P$ のスペクトルギャップです。これを行うには、 $W(P)$ を $O(1/\\sqrt{\\delta})$ 回作用する必要があります。 $\\ket{w}$ を $W(P)$ の固有ベクトルで、固有値 $e^{\\pm 2i\\theta_j}$ を持つとしましょう。 $\\tilde{\\theta_j}$ を位相推定による $\\theta_j$ のもっとも良い推定値と仮定します。ステップ 2(b) の $\\ket{w}$ の写像を与える演算子 $R(P)$ は文献[4]で与えられています。\n",
     "\n",
-    "\\begin{equation}\n",
+    "$$\n",
     "    \\ket{w} \\ket{0} \\mapsto \\ket{w} \\ket{\\tilde{\\theta_j}} \\mapsto (-1)^{|\\tilde{\\theta_j} \\neq 0|} \\ket{w} \\ket{\\tilde{\\theta_j}} \\mapsto (-1)^{|\\tilde{\\theta_j} \\neq 0|} \\ket{w} \\ket{0}\n",
-    "\\end{equation}\n",
+    "$$\n",
     "\n",
     "\n",
     "## 5.例: 4次元超立方体上の量子ウォークによる探索 <a name=\"qiskiti\"></a>\n",
     "\n",
     "\n",
-    "量子ウォークによる探索アルゴリズムは $O(1/\\sqrt{\\epsilon})$ ステップでマークされた頂点の集合を探すことができます、ここで $\\epsilon =  |M|/N$、$M$ はマークした頂点数、$N$ は全ての頂点数です。このアルゴリズムは元はセゲディ量子ウォークを使い、量子状態を表現する二つの頂点レジスタを用いました。しかしグローバーコインを用いるコイン量子ウォークはセゲディ量子ウォークと等価で、コイン量子ウォークは一般的に簡素に実装できるため、コイン量子ウォークによる実装を選びます。3 章で紹介した4次元超立方体を用います。\n",
+    "量子ウォークによる探索アルゴリズムは $O(1/\\sqrt{\\epsilon})$ ステップでマークされた頂点の集合を探すことができます、ここで $\\epsilon =  |M|/N$、 $M$ はマークした頂点数、 $N$ は全ての頂点数です。このアルゴリズムは元はセゲディ量子ウォークを使い、量子状態を表現する二つの頂点レジスタを用いました。しかしグローバーコインを用いるコイン量子ウォークはセゲディ量子ウォークと等価で、コイン量子ウォークは一般的に簡素に実装できるため、コイン量子ウォークによる実装を選びます。3 章で紹介した4次元超立方体を用います。\n",
     "\n",
-    "まとめると、以下のようにアルゴリズムを実装します。ステップ 1 を行うには、頂点レジスタの全ての量子ビットにアダマールゲートを作用させて全ての辺の一様な重ね合わせ状態を作ります。ステップ 2(a) では、位相オラクルを実装します。ステップ 2(b) は、超立方体の量子ウォークの一ステップ分の位相推定と、$\\theta \\neq 0$ となるすべての量子状態のマークを行います。これを補助量子ビットの回転により行います。最後のステップは、位相推定の逆演算を行います。位相推定に必要な量子ビットは $\\theta$ の精度に依存します。\n",
+    "まとめると、以下のようにアルゴリズムを実装します。ステップ 1 を行うには、頂点レジスタの全ての量子ビットにアダマールゲートを作用させて全ての辺の一様な重ね合わせ状態を作ります。ステップ 2(a) では、位相オラクルを実装します。ステップ 2(b) は、超立方体の量子ウォークの一ステップ分の位相推定と、 $\\theta \\neq 0$ となるすべての量子状態のマークを行います。これを補助量子ビットの回転により行います。最後のステップは、位相推定の逆演算を行います。位相推定に必要な量子ビットは $\\theta$ の精度に依存します。\n",
     "\n",
     "以下、4次元超立方体の量子ウォークによる探索アルゴリズムを実装します。"
    ]
   },
   {
    "cell_type": "markdown",
-   "id": "5e1296db",
+   "id": "b7e7a842",
    "metadata": {},
    "source": [
     "このアルゴリズムでは、一ステップ分のゲートに対して以前実装した反転を必要とします。これはビルトイン関数 inverse() で行えます。"
@@ -335,7 +324,7 @@
   {
    "cell_type": "code",
    "execution_count": 5,
-   "id": "60fb5044",
+   "id": "4bb02dd5",
    "metadata": {},
    "outputs": [
     {
@@ -358,7 +347,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "a30255e6",
+   "id": "64aef93a",
    "metadata": {},
    "source": [
     "一ステップ分のゲートの反転は後で位相推定の逆計算に使用します。3 章で実装した一ステップ分のゲートとその反転のゲートは、制御ゲートを必要とします。これらを後に制御量子ビットに応じて使用します。"
@@ -367,7 +356,7 @@
   {
    "cell_type": "code",
    "execution_count": 6,
-   "id": "cb819da1",
+   "id": "c0a202d6",
    "metadata": {},
    "outputs": [],
    "source": [
@@ -380,7 +369,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "ed43f489",
+   "id": "c15fdfdf",
    "metadata": {},
    "source": [
     "制御された一ステップ分のゲートと、制御された一ステップ分の反転ゲートは位相推定に用いられます。位相推定には量子フーリエ変換も用います。Qiskit では量子フーリエ変換を実装した QFT 関数があります。位相推定には逆量子フーリエ変換を用いますが、位相推定の逆計算にも通常の QFT を使用します。"
@@ -389,7 +378,7 @@
   {
    "cell_type": "code",
    "execution_count": 7,
-   "id": "e5cb4a98",
+   "id": "78c4f2bc",
    "metadata": {},
    "outputs": [
     {
@@ -415,7 +404,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "253629e1",
+   "id": "2c545266",
    "metadata": {},
    "source": [
     "位相推定を実装する前に、1011 と 1111 をマークする位相オラクルを実装します。その後、回路を作成します。これはアルゴリズムのステップ 2(a) です。"
@@ -424,7 +413,7 @@
   {
    "cell_type": "code",
    "execution_count": 8,
-   "id": "29d92511",
+   "id": "0e97be9e",
    "metadata": {
     "scrolled": true
    },
@@ -464,16 +453,16 @@
   },
   {
    "cell_type": "markdown",
-   "id": "d940c64e",
+   "id": "1051c836",
    "metadata": {},
    "source": [
-    "他の量子ビットがゼロでない場合に補助量子ビットを回転するゲートを実装します。このゲートは位相推定で用いられ、$\\theta \\neq 0$ の時に補助量子ビットを回転します。"
+    "他の量子ビットがゼロでない場合に補助量子ビットを回転するゲートを実装します。このゲートは位相推定で用いられ、 $\\theta \\neq 0$ の時に補助量子ビットを回転します。"
    ]
   },
   {
    "cell_type": "code",
    "execution_count": 9,
-   "id": "e180e3cb",
+   "id": "67c9ba50",
    "metadata": {},
    "outputs": [
     {
@@ -505,16 +494,16 @@
   },
   {
    "cell_type": "markdown",
-   "id": "abfa2ff1",
+   "id": "50f2c353",
    "metadata": {},
    "source": [
-    "次にアルゴリズムのステップ 2(b) を実装します。このステップは、量子ウォーク一ステップ分の位相推定と、$\\theta \\neq 0$ の時の補助量子ビットの回転操作を含みます。このために、直前に作成した mark_auxiliary_gate を用います。その後、位相推定の逆演算を行います。"
+    "次にアルゴリズムのステップ 2(b) を実装します。このステップは、量子ウォーク一ステップ分の位相推定と、 $\\theta \\neq 0$ の時の補助量子ビットの回転操作を含みます。このために、直前に作成した mark_auxiliary_gate を用います。その後、位相推定の逆演算を行います。"
    ]
   },
   {
    "cell_type": "code",
    "execution_count": 11,
-   "id": "0b18ab93",
+   "id": "b074b0fa",
    "metadata": {
     "scrolled": false
    },
@@ -565,16 +554,16 @@
   },
   {
    "cell_type": "markdown",
-   "id": "101cce65",
+   "id": "3520d6e1",
    "metadata": {},
    "source": [
-    "ここで、これまで作成したゲートを用いて、量子ウォークによる探索アルゴリズムの全体を実装します。最初は、ステップ 1 の頂点とコインの量子ビットにアダマールゲートを作用させます。その後、ステップ 2(a) と 2(b) の位相オラクルゲートと位相推定ゲートを繰り返し作用させます。4 章でアルゴリズムを説明したように、$O(1/\\sqrt{\\epsilon})$ 回の繰り返しが必要です。最後に頂点量子ビットを測定します。"
+    "ここで、これまで作成したゲートを用いて、量子ウォークによる探索アルゴリズムの全体を実装します。最初は、ステップ 1 の頂点とコインの量子ビットにアダマールゲートを作用させます。その後、ステップ 2(a) と 2(b) の位相オラクルゲートと位相推定ゲートを繰り返し作用させます。4 章でアルゴリズムを説明したように、 $O(1/\\sqrt{\\epsilon})$  回の繰り返しが必要です。最後に頂点量子ビットを測定します。"
    ]
   },
   {
    "cell_type": "code",
    "execution_count": 12,
-   "id": "5573b106",
+   "id": "8acd737e",
    "metadata": {},
    "outputs": [
     {
@@ -616,7 +605,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "2b4e8cdb",
+   "id": "d7aee277",
    "metadata": {},
    "source": [
     "最後に、この実装を qasm シミュレータで実行します。マークした状態へとはっきりと集まっていることが分かります。"
@@ -625,7 +614,7 @@
   {
    "cell_type": "code",
    "execution_count": 13,
-   "id": "682d2f40",
+   "id": "4ad3171f",
    "metadata": {},
    "outputs": [
     {
@@ -651,7 +640,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "be8303f3",
+   "id": "2c421e09",
    "metadata": {},
    "source": [
     "## 6. 参考文献 <a id='references'></a>\n",
@@ -665,7 +654,7 @@
   {
    "cell_type": "code",
    "execution_count": 14,
-   "id": "7c11b1c1",
+   "id": "a6831f83",
    "metadata": {},
    "outputs": [
     {
diff --git a/translations/ja/v1/ch-demos/piday-code.ipynb b/translations/ja/v1/ch-demos/piday-code.ipynb
index e522a1490..b75a83232 100644
--- a/translations/ja/v1/ch-demos/piday-code.ipynb
+++ b/translations/ja/v1/ch-demos/piday-code.ipynb
@@ -8,21 +8,21 @@
     ]
    },
    "source": [
-    "# [量子位相推定](../ch-algorithms/quantum-phase-estimation.html)を使った円周率($\\pi$)の推定"
+    "# 量子位相推定を使った円周率($\\pi$)の推定"
    ]
   },
   {
    "cell_type": "markdown",
    "metadata": {},
    "source": [
-    "### 1.[量子位相推定アルゴリズム](../ch-algorithms/quantum-phase-estimation.html)の概要\n",
+    "## 1.[量子位相推定アルゴリズム](../ch-algorithms/quantum-phase-estimation.html)の概要\n",
     "\n",
     "量子推定アルゴリズム（QPE）は多くの非常に複雑な量子アルゴリズムの一角を担う量子アルゴリズムの１つです。その中心にあるのは、QPEはかなり簡単な問題を解きます。$U\\vert\\psi\\rangle = \\exp\\left(2 \\pi i  \\theta\\right)\\vert\\psi\\rangle$）となっている演算子$U$の量子状態$\\vert\\psi\\rangle$が与えられているとき、$\\theta$の推定値を得ることができますか？\n",
     "\n",
     "この答えは”yes\"です。QPEアルゴリズムから$2^n\\theta$を得られます。ここで$n$とは$\\theta$を推定するために使っている量子ビット数です。\n",
     "\n",
     "\n",
-    "### 2.円周率$\\pi$の推定\n",
+    "## 2.円周率$\\pi$の推定\n",
     "\n",
     "このデモでは、演算子と状態を次のように選びます。\n",
     "\n",
diff --git a/translations/ja/v1/ch-ex/hello-qiskit.ipynb b/translations/ja/v1/ch-ex/hello-qiskit.ipynb
index 9ca4fd802..7e80997d2 100755
--- a/translations/ja/v1/ch-ex/hello-qiskit.ipynb
+++ b/translations/ja/v1/ch-ex/hello-qiskit.ipynb
@@ -8,7 +8,7 @@
     ]
    },
    "source": [
-    "# こんにちはQiskitゲーム"
+    "## こんにちはQiskitゲーム"
    ]
   },
   {
@@ -17,7 +17,7 @@
    "source": [
     "量子ビットと量子ゲートを使い始めるための簡単なパズル。\n",
     "\n",
-    "# レベル 1: ビットから始める\n",
+    "## レベル 1: ビットから始める\n",
     "\n",
     "このテキストブックの他の章と同様に、このドキュメントはJupyter notebookです。ただし、他の多くの章とは異なり、実際に実行して機能させる必要があります。\n",
     "\n",
@@ -64,9 +64,9 @@
    "cell_type": "markdown",
    "metadata": {},
    "source": [
-    "## パズル 1\n",
+    "### パズル 1\n",
     "\n",
-    "### 導入\n",
+    "#### 導入\n",
     "量子コンピューターは*量子ビット*に基づいています：ビットの量子力学のルールに従います。しかし、正確にはビットとは何ですか？そして、それらはコンピューターでどのように使用されていますか？\n",
     "\n",
     "ビットの特徴は、2つの可能な出力値があることです。これらは、`1`と`0`、または「オン」と「オフ」、または「真」と「偽」と呼ぶことができます。私たちが使用する名前は重要ではありません。重要な点は、2つあるということです。\n",
@@ -75,7 +75,7 @@
     "\n",
     "以下で試してみてください。\n",
     "\n",
-    "### 練習問題\n",
+    "#### 練習問題\n",
     "*`NOT`ゲートを3回使用してみましょう。"
    ]
   },
@@ -99,7 +99,7 @@
    "cell_type": "markdown",
    "metadata": {},
    "source": [
-    "### 終結\n",
+    "#### 終結\n",
     "ここでは、オン(白)またはオフ(黒)の円を使用してビットを視覚化しました。`NOT`ゲートの効果は、ビットの2つの状態を切り替えて、オンとオフを切り替えることでした。"
    ]
   },
@@ -107,12 +107,12 @@
    "cell_type": "markdown",
    "metadata": {},
    "source": [
-    "## パズル 2\n",
+    "### パズル 2\n",
     "\n",
-    "### 導入\n",
+    "#### 導入\n",
     "1つだけではなく2つのビットで遊ぶ方が面白いです。ここに遊ぶための別のものがあります。以前と同じように見えます。しかし、それは別のビットなので、別の場所にあります。\n",
     "\n",
-    "### 練習問題\n",
+    "#### 練習問題\n",
     "* もう一方のビットをオンにします。"
    ]
   },
@@ -134,7 +134,7 @@
    "cell_type": "markdown",
    "metadata": {},
    "source": [
-    "### 終結\n",
+    "#### 終結\n",
     "これで、コンピューティングの最も基本的な構成要素である`NOT`ゲートをマスターしました。"
    ]
   },
@@ -142,9 +142,9 @@
    "cell_type": "markdown",
    "metadata": {},
    "source": [
-    "## パズル 3\n",
+    "### パズル 3\n",
     "\n",
-    "### 導入\n",
+    "#### 導入\n",
     "「ゲート」という言葉を使用して、量子ビットを操作するために使用できる簡単なツールを説明します。`NOT`ゲートは最も単純な例です。しかし、ビットを使って面白いことをするためには、単にオンとオフを切り替える以上のことをする必要があります。\n",
     "\n",
     "もう少し洗練されたゲートは`CNOT`と呼ばれ、「controled-NOT」の略です。これはビットのペアで使用されます。\n",
@@ -153,7 +153,7 @@
     "\n",
     "これを理解する最良の方法は、試してみることです。\n",
     "\n",
-    "### 練習問題\n",
+    "#### 練習問題\n",
     "\n",
     "*CNOTを使用して、右側のビットをオンにします。\n",
     "\n",
@@ -178,12 +178,12 @@
    "cell_type": "markdown",
    "metadata": {},
    "source": [
-    "## パズル 4\n",
+    "### パズル 4\n",
     "\n",
-    "### 導入・導入\n",
+    "#### 導入・導入\n",
     "私たちが実行するプログラムはすべて、多くの量子ビット上の多くのゲートで構成されています。これに向けたステップとして、`CNOT`をいくつか使用する必要があるものを試してみましょう。\n",
     "\n",
-    "### 練習問題\n",
+    "#### 練習問題\n",
     "\n",
     "* 複数のCNOTを使用して、左ビットをオフにし、右ビットをオンにします。"
    ]
@@ -206,7 +206,7 @@
    "cell_type": "markdown",
    "metadata": {},
    "source": [
-    "### 終結\n",
+    "#### 終結\n",
     "素晴らしい！\n",
     "\n",
     "これらの種類の操作は、あらゆるコンピューティングがコンパイルするものです。より多くのビットと制御制御`NOT`ゲートがあれば、テトリスから自動運転車まですべてを構築できます。"
@@ -216,16 +216,16 @@
    "cell_type": "markdown",
    "metadata": {},
    "source": [
-    "## パズル 5\n",
+    "### パズル 5\n",
     "\n",
-    "### 導入\n",
+    "#### 導入\n",
     "お気づきかもしれませんが、パズルボードにはたくさんの空きスペースがあります。これは、ビットに関するより多くの情報を視覚化するために使用できるためです。たとえば、ビット値がランダムなプロセスによって生成された場合はどうなるでしょうか？\n",
     "\n",
     "0または1のビット値を表すために黒と白の円を使用しました。したがって、等確率で0または1が得られるランダムビットの場合、灰色の円を使用します。\n",
     "\n",
     "これで、ゲートがこのランダム性をどのように操作するかを見ることができます。\n",
     "\n",
-    "### 練習問題\n",
+    "#### 練習問題\n",
     "* CNOTを使用して、右側のビットをランダムにします。"
    ]
   },
@@ -247,7 +247,7 @@
    "cell_type": "markdown",
    "metadata": {},
    "source": [
-    "### 終結\n",
+    "#### 終結\n",
     "\n",
     "\n",
     "素晴らしいでしょう！\n",
@@ -264,16 +264,16 @@
    "cell_type": "markdown",
    "metadata": {},
    "source": [
-    "## パズル 6\n",
+    "### パズル 6\n",
     "\n",
-    "### 導入\n",
+    "#### 導入\n",
     "下のパズルでは、新しい円が表示されます。\n",
     "\n",
     "これまでに見た円とは異なり、新しいビットを表すものではありません。代わりに、2ビットが一致するかどうかを示します。\n",
     "\n",
     "それらが一致することが確実であるとき、それはオフになります。それらが一致しないことが確実な場合、それはオンになります。それらの一致と不一致がランダムである場合、それは灰色になります。\n",
     "\n",
-    "### 練習問題\n",
+    "#### 練習問題\n",
     "\n",
     "* 2つのビットを常に一致させないようにします(つまり、真ん中の円を白にします)。"
    ]
@@ -298,12 +298,12 @@
    "cell_type": "markdown",
    "metadata": {},
    "source": [
-    "## パズル 7\n",
+    "### パズル 7\n",
     "\n",
-    "### 導入\n",
+    "#### 導入\n",
     "これで、ビットについて知る必要があるすべてがほぼ必要であることがわかりました。先に進む前に、もう1つ演習を行いましょう。\n",
     "\n",
-    "### 練習問題\n",
+    "#### 練習問題\n",
     "* 右側のビットをオンにします。"
    ]
   },
@@ -325,7 +325,7 @@
    "cell_type": "markdown",
    "metadata": {},
    "source": [
-    "### 終結\n",
+    "#### 終結\n",
     "次に、ビットが量子になったときに何が起こるかを見てみましょう。量子ビットの時間です！"
    ]
   },
@@ -333,19 +333,19 @@
    "cell_type": "markdown",
    "metadata": {},
    "source": [
-    "# レベル 2: 基本的な単一量子ビットゲート"
+    "## レベル 2: 基本的な単一量子ビットゲート"
    ]
   },
   {
    "cell_type": "markdown",
    "metadata": {},
    "source": [
-    "## パズル 1\n",
+    "### パズル 1\n",
     "\n",
-    "### 導入\n",
+    "#### 導入\n",
     "量子力学の講義から始める代わりに、ここで遊ぶための量子ビットがあります。`X`と呼ばれる最も単純な量子ビットゲートを試してみてください。\n",
     "\n",
-    "### 練習問題\n",
+    "#### 練習問題\n",
     "* `X`ゲートを3回使用して、何が起こるかを確認します。"
    ]
   },
@@ -369,7 +369,7 @@
    "cell_type": "markdown",
    "metadata": {},
    "source": [
-    "### 終結\n",
+    "#### 終結\n",
     "このパズルはとてもなじみのあるものに思えたはずです。何も変化しなかった円があったこと以外は、ビットの最初のものとまったく同じでした。\n",
     "`X`ゲートは`NOT`ゲートとまったく同じ効果がありました。"
    ]
@@ -378,9 +378,9 @@
    "cell_type": "markdown",
    "metadata": {},
    "source": [
-    "## パズル 2\n",
+    "### パズル 2\n",
     "\n",
-    "### 導入\n",
+    "#### 導入\n",
     "さきほどのパズルには2つの円がありました。これらは異なるビットではなく、両方とも同じ量子ビットを表しています。\n",
     "\n",
     "量子ビットはビットの量子バージョンです。それらはビットと同じプロパティのいくつかを持っていますが、いくつかの追加機能も持っています。\n",
@@ -391,7 +391,7 @@
     "\n",
     "円の色は以前とまったく同じように使用されます。円が黒の場合、対応する出力は値`0`になります。白い円は`1`を取得することを意味します。\n",
     "\n",
-    "### 練習問題\n",
+    "#### 練習問題\n",
     "* Z出力をオフにします。"
    ]
   },
@@ -413,7 +413,7 @@
    "cell_type": "markdown",
    "metadata": {},
    "source": [
-    "### 終結\n",
+    "#### 終結\n",
     "量子ビットからビットを抽出するプロセスは、「測定」と呼ばれます。\n",
     "\n",
     "X出力とZ出力の両方を個別に抽出することはできません。1つだけ選択する必要があります。"
@@ -423,12 +423,12 @@
    "cell_type": "markdown",
    "metadata": {},
    "source": [
-    "## パズル 3\n",
+    "### パズル 3\n",
     "\n",
-    "### 導入\n",
+    "#### 導入\n",
     "このパズルでは、別の量子ビットが表示されます。これもまた、2つの円で表される内部動作を持ちます。この量子ビットの場合、これらの円はパズルボードの右側にあります。\n",
     "\n",
-    "### 練習問題\n",
+    "#### 練習問題\n",
     "* もう片方の量子ビットのZ出力をオフにします。"
    ]
   },
@@ -450,7 +450,7 @@
    "cell_type": "markdown",
    "metadata": {},
    "source": [
-    "### 終結\n",
+    "#### 終結\n",
     "これから、プログラムで使用される名前で量子ビットの呼び出しを開始します。左側の量子ビットは`q[0]`になり、右側の量子ビットは`q[1]`になります。\n",
     "\n",
     "また、Z出力およびX出力という用語は広く使用されていないことに注意してください。「Z測定の出力」などと言うのがより一般的です。しかし、それはこれらのパズルにとって少し長い表現です。"
@@ -460,16 +460,16 @@
    "cell_type": "markdown",
    "metadata": {},
    "source": [
-    "## パズル 4\n",
+    "### パズル 4\n",
     "\n",
-    "### 導入\n",
+    "#### 導入\n",
     "次に、新しいゲートであるHゲートを試してみましょう。\n",
     "\n",
     "これは、単純なビットでは不可能なことです。それは、それが適用される量子ビットの2つの円を交換する効果があります。\n",
     "\n",
     "これを素敵なアニメーション形式で見たい場合は、[Hello Quantumアプリ](https://helloquantum.mybluemix.net/)をチェックしてください。しかし、ここでは、3回繰り返すという古いトリックでこれをテストしてください。\n",
     "\n",
-    "### 練習問題\n",
+    "#### 練習問題\n",
     "* `H`ゲートを3回使用します。"
    ]
   },
@@ -491,7 +491,7 @@
    "cell_type": "markdown",
    "metadata": {},
    "source": [
-    "### 終結\n",
+    "#### 終結\n",
     "ここから先は、下のコードをパズルの下に置き、量子プログラムを表示します。\n",
     "\n",
     "これらは、実際の量子コンピューターで実行できる実際のQiskitプログラムです。それらは、いわゆる回路図で表すこともできます。量子プログラムの回路図を表示するには、以下のコードセルを実行します。"
@@ -510,12 +510,12 @@
    "cell_type": "markdown",
    "metadata": {},
    "source": [
-    "## パズル 5\n",
+    "### パズル 5\n",
     "\n",
-    "### 導入\n",
+    "#### 導入\n",
     "Z出力が一定の場合(完全にオフまたはオンの場合)、`X`ゲートは単に値を反転します。しかし、さきほどのパズルで見たように、Z出力がランダムである場合はどうなりますか？この練習問題で、その答えがわかります。\n",
     "\n",
-    "### 練習問題\n",
+    "#### 練習問題\n",
     "* Z出力を完全にオフにします。 `h`ゲートはいくつでも使用できますが、`x`は正確に3回使用します。"
    ]
   },
@@ -546,7 +546,7 @@
    "cell_type": "markdown",
    "metadata": {},
    "source": [
-    "### 終結\n",
+    "#### 終結\n",
     "たとえそれをひっくり返したとしても、ランダムな結果はただのランダムな結果であることがわかります。"
    ]
   },
@@ -554,12 +554,12 @@
    "cell_type": "markdown",
    "metadata": {},
    "source": [
-    "## パズル 6\n",
+    "### パズル 6\n",
     "\n",
-    "### 導入\n",
+    "#### 導入\n",
     "`x`ゲートがZ出力を反転することを確認しましたが、X出力には影響しません。そのためには、新しいゲート、`z`ゲートが必要です。\n",
     "\n",
-    "### 練習問題\n",
+    "#### 練習問題\n",
     "* X出力をオフにします。"
    ]
   },
@@ -590,7 +590,7 @@
    "cell_type": "markdown",
    "metadata": {},
    "source": [
-    "### 終結\n",
+    "#### 終結\n",
     "`x`ゲートはZ出力を反転し、`z`ゲートはx出力を反転します。今のところ、これは物事に名前を付ける奇妙な方法のように思えるかもしれません。しかし、量子コンピューティングについてさらに学ぶにつれて、それは意味を持つようになるでしょう。"
    ]
   },
@@ -598,12 +598,12 @@
    "cell_type": "markdown",
    "metadata": {},
    "source": [
-    "## パズル 7\n",
+    "### パズル 7\n",
     "\n",
-    "### 導入\n",
+    "#### 導入\n",
     "ゲートを組み合わせることで、新しい効果を得ることができます。簡単な例として、`z`と`h`を組み合わせることで、`x`の作用をすることができます。\n",
     "\n",
-    "### 練習問題\n",
+    "#### 練習問題\n",
     "* `x`ゲートを使用せずにZ出力をオンにします"
    ]
   },
@@ -634,14 +634,14 @@
    "cell_type": "markdown",
    "metadata": {},
    "source": [
-    "## パズル 8\n",
+    "### パズル 8\n",
     "\n",
-    "### 導入\n",
+    "#### 導入\n",
     "Z出力が完全にオンまたはオフの場合、X出力は常にランダムであることに気付くかもしれません。これは、量子ビットがそれぞれの出力について同時に確実になることは決してないためです。それらが一方について確実である場合、もう一方はランダムでなければなりません。\n",
     "\n",
     "もし上記ではない場合、Z出力とX出力を使用して2ビットを格納できてしまいます。つまり量子ビットが持っているより多くのメモリーを持つことになってしまいます。複数の方法で出力を抽出できるという事実にもかかわらず、それでも、量子ビットにビット以上を格納することは許可されていません。\n",
     "\n",
-    "### 練習問題\n",
+    "#### 練習問題\n",
     "* X出力をオフにし、Z出力をランダムにします。"
    ]
   },
@@ -674,15 +674,15 @@
    "cell_type": "markdown",
    "metadata": {},
    "source": [
-    "## パズル 9\n",
+    "### パズル 9\n",
     "\n",
-    "## 導入\n",
+    "### 導入\n",
     "量子ビットの限られた確実性は、2つの出力間で共有することもできます。たとえば、両方の出力を、それらが提供する出力についてほぼ確実(完全に確実ではない)にすることで妥協することができます。\n",
     "\n",
     "さまざまなグレーの色合いを使用して、これを視覚化します。円が暗いほど、出力は`0`になる可能性が高くなります。円が明るいほど、出力は`1`になる可能性が高くなります。\n",
     "\n",
     "\n",
-    "### 練習問題\n",
+    "#### 練習問題\n",
     "* `q[1]`の2つの円を両方とも明るい灰色にします。これは、`1`を出力する可能性が高いことを意味しますが、確実ではありません。"
    ]
   },
@@ -713,12 +713,12 @@
    "cell_type": "markdown",
    "metadata": {},
    "source": [
-    "## パズル 10\n",
+    "### パズル 10\n",
     "\n",
-    "### 導入\n",
+    "#### 導入\n",
     "これで基本的なツールがわかったので、両方の量子ビットに同時に取り組むことができます。\n",
     "\n",
-    "### 練習問題\n",
+    "#### 練習問題\n",
     "*  両方のZ出力をランダムにします。"
    ]
   },
@@ -749,7 +749,7 @@
    "cell_type": "markdown",
    "metadata": {},
    "source": [
-    "### 終結\n",
+    "#### 終結\n",
     "ここでの各Z出力は、ランダムに`0`または`1`を出力します。しかし、それらの出力は相関しますか？ 反相関？ 完全に無関係ですか？\n",
     "\n",
     "ビットで行ったように、いくつかの追加の円を使用してこの情報を追跡します。"
@@ -759,12 +759,12 @@
    "cell_type": "markdown",
    "metadata": {},
    "source": [
-    "## パズル 11\n",
+    "### パズル 11\n",
     "\n",
-    "### 導入\n",
+    "#### 導入\n",
     "このパズルでは、4つの新しい円が表示されます。そのうちの1つは、レベル1ですでに見たものです。そこでは、2つのビット値が確実に一致するか(黒)、一致しないか(白)を追跡していました。ここでは、両方の量子ビットのZ出力に対して同じ作用をします。\n",
     "\n",
-    "### 練習問題\n",
+    "#### 練習問題\n",
     "* Z出力が確実に一致しないようにします。"
    ]
   },
@@ -795,12 +795,12 @@
    "cell_type": "markdown",
    "metadata": {},
    "source": [
-    "## パズル 12\n",
+    "### パズル 12\n",
     "\n",
-    "### 導入\n",
+    "#### 導入\n",
     "一番上の新しい円にも同様の作用があります。この場合、両方の量子ビットのX出力について一致する確率と一致しない確率を追跡します。\n",
     "\n",
-    "### 練習問題\n",
+    "#### 練習問題\n",
     "* X出力が確実に一致するようにします。"
    ]
   },
@@ -831,7 +831,7 @@
    "cell_type": "markdown",
    "metadata": {},
    "source": [
-    "### 終結\n",
+    "#### 終結\n",
     "この情報を表すのがこの新しい円である理由がわからない場合は、一方の量子ビットのX出力から伸びる行と、もう一方の量子ビットのX出力から伸びる別の行について考えてみてください。最上部の円は、これら2つの行が交わる場所にあります。これが、2つのX出力間の一致を表す理由です。"
    ]
   },
@@ -839,12 +839,12 @@
    "cell_type": "markdown",
    "metadata": {},
    "source": [
-    "## パズル 13\n",
+    "### パズル 13\n",
     "\n",
-    "### 導入\n",
+    "#### 導入\n",
     "説明する必要のある新しい円がまだ2つあります。1つは、`q[0]`からのZ出力行が`q[1]`からのX出力行と交わる場所です。これは、`q[0]`のZ出力が`q[1]`からのX出力と一致するかどうかを示します。もう1つも同じですが、`q[0]`がX出力と`q[1]`がZ出力です。\n",
     "\n",
-    "### 練習問題\n",
+    "#### 練習問題\n",
     "* `q[0]`のX出力が`q[1]`のZ出力と一致しないことを確認します。"
    ]
   },
@@ -875,12 +875,12 @@
    "cell_type": "markdown",
    "metadata": {},
    "source": [
-    "## パズル 14\n",
+    "### パズル 14\n",
     "\n",
-    "### 導入\n",
+    "#### 導入\n",
     "`x`、`z`、`h`ゲートが新しい円にどのように影響するかに注目してください。具体的には、xゲートは、単一のZ出力だけでなく、それらの行全体に影響を与えます。つまり、各円を黒から白(または暗い色から明るい色)に、またはその逆に反転します。\n",
     "\n",
-    "### 練習問題\n",
+    "#### 練習問題\n",
     "*  2つのZ出力をオンにします。"
    ]
   },
@@ -911,7 +911,7 @@
    "cell_type": "markdown",
    "metadata": {},
    "source": [
-    "### 終結\n",
+    "#### 終結\n",
     "次のパズルで分かるように、Zゲートは同じような方法でX出力の列に影響します。"
    ]
   },
@@ -919,12 +919,12 @@
    "cell_type": "markdown",
    "metadata": {},
    "source": [
-    "## パズル 15\n",
+    "### パズル 15\n",
     "\n",
-    "### 導入\n",
+    "#### 導入\n",
     "以前、`h`ゲートが円のペアを交換する方法を見てきました。これと同じように行のペアに影響を与えます。行全体が反転されるまで、円の各ペアを反転します。再度、[Hello Quantumアプリ](https://helloquantum.mybluemix.net/)で素敵なアニメーションをチェックすることをお勧めします。\n",
     "\n",
-    "### 練習問題\n",
+    "#### 練習問題\n",
     "* X出力をオフにします。"
    ]
   },
@@ -955,7 +955,7 @@
    "cell_type": "markdown",
    "metadata": {},
    "source": [
-    "### 結果\n",
+    "#### 結果\n",
     "これで、2つの量子ビットがどのように見えるか、およびそれらを個別に操作する方法の基本を理解しました。しかし、2つの量子ビットゲートを使い始めると、本当の楽しみが出てきます。"
    ]
   },
@@ -963,23 +963,23 @@
    "cell_type": "markdown",
    "metadata": {},
    "source": [
-    "# レベル 3: ２量子ビットのゲート"
+    "## レベル 3: ２量子ビットのゲート"
    ]
   },
   {
    "cell_type": "markdown",
    "metadata": {},
    "source": [
-    "### パズル 1\n",
+    "#### パズル 1\n",
     "\n",
-    "### 導入\n",
+    "#### 導入\n",
     "ビットの演習では、`CNOT`ゲートを使用しました。量子ビットの場合、NOTの量子バージョンとしてxを使用して、同様のゲートがあります。このため、Qiskitプログラムでは`cx`と呼びます。\n",
     "\n",
     "従来の`CNOT`と同様に、`cx`ゲートには「コントロール」と「ターゲット」があります。コントロールのZ出力を調べ、その結果で、xがターゲット量子ビットに適用されるかどうかを決定します。\n",
     "\n",
     "このゲートを適用すると、選択した量子ビットがターゲットとして機能します。その場合、他の量子ビットがコントロールになります。\n",
     "\n",
-    "### 練習問題\n",
+    "#### 練習問題\n",
     "* `cx`を1回か2回使用して、q[1]のZ出力をオンにし、q[0]のZ出力をオフにします。"
    ]
   },
@@ -1012,12 +1012,12 @@
    "cell_type": "markdown",
    "metadata": {},
    "source": [
-    "## パズル 2\n",
+    "### パズル 2\n",
     "\n",
-    "### 導入\n",
+    "#### 導入\n",
     "`cx`ゲートだけでなく、`cz`もあります。これは同じことを行いますが、`x`ではなく`z`をターゲットに適用する可能性がある点が異なります。\n",
     "\n",
-    "### 練習問題\n",
+    "#### 練習問題\n",
     "* q [0]のX出力をオンにします。"
    ]
   },
@@ -1048,14 +1048,14 @@
    "cell_type": "markdown",
    "metadata": {},
    "source": [
-    "## パズル 2b\n",
+    "### パズル 2b\n",
     "\n",
-    "### 導入\n",
+    "#### 導入\n",
     "量子ゲートの興味深い点は、量子ゲートが何をしているのかを説明する方法が複数あることが多いということです。これらの説明は完全に互換性がないように見える場合がありますが、同じ操作です。\n",
     "\n",
     "たとえば、`cz`は、*ターゲット*の潜在的なZ出力に応じて、*コントロール*量子ビットにzを適用するゲートとして説明することもできます。以前とまったく同じ説明ですが、量子ビットの役割が逆になっています。それにもかかわらず、それは等しく真実です。\n",
     "\n",
-    "###  練習問題\n",
+    "####  練習問題\n",
     "* 前回の練習問題と同じですが、量子ビットが逆になっています。ただし、czは以前と同じ方法です。"
    ]
   },
@@ -1086,16 +1086,16 @@
    "cell_type": "markdown",
    "metadata": {},
    "source": [
-    "## パズル 3\n",
+    "### パズル 3\n",
     "\n",
-    "### 導入\n",
+    "#### 導入\n",
     "ここで、czについて別の説明をします。`h`ゲートと同様に、円が入れ替わっていると考えることができます。czの作用は以下のようなものです。\n",
     "\n",
     "- q[0]のX出力を右上の隣接する円と交換します。\n",
     "- q[1]のX出力(左上の隣の場合)でも同じことを行います。czはまた、盤面の上部にある円で何か奇妙なことをしますが、それは後で解決される謎です！\n",
     "ここでも、[Hello Quantumアプリ](https://helloquantum.mybluemix.net/)で素晴らしいアニメーションをみることができます。\n",
     "\n",
-    "### 練習問題\n",
+    "#### 練習問題\n",
     "* コントロールとして各量子ビットで2回`cz`を実行し、何が起こるかを確認します。"
    ]
   },
@@ -1128,7 +1128,7 @@
    "cell_type": "markdown",
    "metadata": {},
    "source": [
-    "###  終結\n",
+    "####  終結\n",
     "これで、`cz`について非常に役立つことがわかりました。コントロールとしてどの量子ビットを選択しても、`cz`はどちらの場合も同じことを行います。このため、これからは`cz`のコントロール量子ビットを選択する必要はありません。"
    ]
   },
@@ -1136,16 +1136,16 @@
    "cell_type": "markdown",
    "metadata": {},
    "source": [
-    "## パズル 3b\n",
+    "### パズル 3b\n",
     "\n",
-    "### 導入\n",
+    "#### 導入\n",
     "前述のように、XおよびZ出力は、量子ビットから出力を取得する2つの方法に対応します。XおよびZ測定です。これらの名前が示すように、Y測定として知られる3番目の方法もあります。\n",
     "\n",
     "これらのパズルでは、物事を少し簡単にするために、Y測定をほとんど無視します。ただし、量子ビットの完全な説明については、Y出力がどのように見えるかを追跡する必要があります。これは、各量子ビットのY出力に対して、さらに2、3行の円を追加することを意味します。\n",
     "\n",
     "次の演習は、Y出力行が表示されることを除いて、前回とまったく同じです。これらを使用すると、最後のパズルで見られた奇妙な効果はまったく奇妙になりません。ご自分の目で確かめてみてください!\n",
     "\n",
-    "### 練習問題\n",
+    "#### 練習問題\n",
     "* `cz`を2回実行して、何が起こるかを確認します。"
    ]
   },
@@ -1176,7 +1176,7 @@
    "cell_type": "markdown",
    "metadata": {},
    "source": [
-    "### 終結\n",
+    "#### 終結\n",
     "ここでは、`cz`によって行われた3番目の円の交換が表示されます。上部の円(両方の量子ビットのX出力の相関を表す)が中央の円(両方の量子ビットのY出力の相関を表す)と交換されます。\n",
     "\n",
     "量子ビットの説明が完全ではなかったという理由だけで、中央の行が欠落しているとき、これは奇妙に見えました。それでも、Y出力のない単純な盤面を引き続き使用します。`hello_quantum.run_game()`で`mode = 'y'引数`を使用すると自分でY出力を追加できます。"
@@ -1186,12 +1186,12 @@
    "cell_type": "markdown",
    "metadata": {},
    "source": [
-    "## パズル 4\n",
+    "### パズル 4\n",
     "\n",
-    "### 導入\n",
+    "#### 導入\n",
     "前の練習問題では、`z`ゲートといくつかの`h`ゲートから`x`を作成しました。同様に、`cz`といくつかの`h`から`cx`を構築することも可能です。\n",
     "\n",
-    "### 練習問題\n",
+    "#### 練習問題\n",
     "* q[1]のZ出力をオンにします。"
    ]
   },
@@ -1222,7 +1222,7 @@
    "cell_type": "markdown",
    "metadata": {},
    "source": [
-    "### 終結\n",
+    "#### 終結\n",
     "`cz`とは異なり、`cx`は対称ではありません。代わりに、ターゲットがq[0]であるcxを作成したい場合は、代わりにq[0]で`h`を実行する必要があります。"
    ]
   },
@@ -1230,16 +1230,16 @@
    "cell_type": "markdown",
    "metadata": {},
    "source": [
-    "## パズル 5\n",
+    "### パズル 5\n",
     "\n",
-    "### 導入\n",
+    "#### 導入\n",
     "`cz`の「逆方向性」を解釈することができました：つまり、ターゲット量子ビットがコントロール量子ビットの役割を果たし、また、その逆の役割も果たすという解釈です。`cx`でも同じことをします。ただし、このゲートには対称効果がないため、これはもう少し注意が必要です。\n",
     "\n",
     "具体的には、コントロールのZ出力の実行内容に応じてターゲットに対してxを実行すると考えるのではなく、ターゲットのX出力の実行内容に応じてコントロールに対してzを実行すると考えることができます。\n",
     "\n",
     "この演習では、ターゲットがコントロールを行っているように見え、コントロールがターゲットであることがわかります。\n",
     "\n",
-    "###  練習問題\n",
+    "####  練習問題\n",
     "* `q[0]`のX出力をオンにします。"
    ]
   },
@@ -1270,7 +1270,7 @@
    "cell_type": "markdown",
    "metadata": {},
    "source": [
-    "### 終結\n",
+    "#### 終結\n",
     "`cx`がどのように機能するかについてのこれらの2つの異なる話は矛盾しているように見えるかもしれませんが、それらは等しく有効な説明です。量子ゲートの奇妙で素晴らしい性質の素晴らしい例です。"
    ]
   },
@@ -1278,14 +1278,14 @@
    "cell_type": "markdown",
    "metadata": {},
    "source": [
-    "## パズル 6\n",
+    "### パズル 6\n",
     "\n",
-    "### 導入\n",
+    "#### 導入\n",
     "これで、`cx`のこれら2つの解釈がわかったので、非常に便利なことを行うことができます。\n",
     "\n",
     "このパズルでは、`q[1]`をターゲットとして`cx`を取得しますが、`q[0]`をターゲットとして`cx`が必要になります。いくつかの`h`ゲートのヘルプを使用して同じ効果を得る方法を理解できるかどうかを確認してください。\n",
     "\n",
-    "### 練習問題\n",
+    "#### 練習問題\n",
     "* `q[1]`のZ出力を維持しますが、`q[0]`のZ出力をオフにします。"
    ]
   },
@@ -1316,7 +1316,7 @@
    "cell_type": "markdown",
    "metadata": {},
    "source": [
-    "### 終結\n",
+    "#### 終結\n",
     "これらの練習問題から何かを覚えているなら、それはおそらくこれであるはずです。実際の量子ビットデバイスでは、`cx`を実行できる方法を制限するのが一般的であるため、それらを方向転換する機能は非常に便利です。"
    ]
   },
@@ -1324,12 +1324,12 @@
    "cell_type": "markdown",
    "metadata": {},
    "source": [
-    "## パズル  7\n",
+    "### パズル  7\n",
     "\n",
-    "### 導入\n",
+    "#### 導入\n",
     "もう1つの有用な量子ゲートは`swap`です。これは、名前が示すとおり、2つの量子ビットの状態を交換します。Qiskitでは単に`swap`コマンドを呼び出すことができますが、このゲートを`cz`または`cx`ゲートから作成する方が興味深いです。\n",
     "\n",
-    "### 練習問題\n",
+    "#### 練習問題\n",
     "* 2つの量子ビットを交換します。\n",
     " - q[0]のZ出力を白、X出力を灰色にします。\n",
     " - q[1]のZ出力を濃い灰色にし、X出力を薄い灰色にします。"
@@ -1362,7 +1362,7 @@
    "cell_type": "markdown",
    "metadata": {},
    "source": [
-    "### 終結\n",
+    "#### 終結\n",
     "このパズルの解決策は、汎用のswapではない可能性があることに注意してください。今回の解決策を、次のパズルの解決策と比較してください。"
    ]
   },
@@ -1370,12 +1370,12 @@
    "cell_type": "markdown",
    "metadata": {},
    "source": [
-    "## パズル 8\n",
+    "### パズル 8\n",
     "\n",
-    "#### 導入\n",
+    "##### 導入\n",
     "これは、`swap`を作成するというアイデアに基づいた別のパズルです。\n",
     "\n",
-    "#### 練習問題\n",
+    "##### 練習問題\n",
     "* 2つの量子ビットを交換します。\n",
     "  * q[0]のX出力を黒にします。\n",
     "  * q[1]のZ出力を白にします。\n",
@@ -1409,12 +1409,12 @@
    "cell_type": "markdown",
    "metadata": {},
    "source": [
-    "## パズル 9\n",
+    "### パズル 9\n",
     "\n",
-    "#### 導入\n",
+    "##### 導入\n",
     "また別の`swap`ベースのパズル。\n",
     "\n",
-    "#### 練習問題\n",
+    "##### 練習問題\n",
     "* 2つの量子ビットを交換します。\n",
     " * q[0]のZ出力をオンにします。\n",
     " * Z出力q[1]をオフにします。"
@@ -1449,21 +1449,21 @@
    "cell_type": "markdown",
    "metadata": {},
    "source": [
-    "# レベル 4:クリフォードゲートを越えて"
+    "## レベル 4:クリフォードゲートを越えて"
    ]
   },
   {
    "cell_type": "markdown",
    "metadata": {},
    "source": [
-    "## パズル 1a\n",
+    "### パズル 1a\n",
     "\n",
-    "### 導入\n",
+    "#### 導入\n",
     "これまでに見たゲートは「クリフォードゲート」と呼ばれています。それらは、量子コンピューターで情報を移動および操作するために非常に重要です。ただし、クリフォードゲートのみを使用して標準的なコンピューターよりも優れたパフォーマンスを発揮できるアルゴリズムを作成することはできません。新しいゲートが必要です。\n",
     "\n",
     "このパズルで試すことができます。数回実行して、何ができるかを確認してください。\n",
     "\n",
-    "### 練習問題\n",
+    "#### 練習問題\n",
     "* `ry(pi/4)`をq[0]に4回適用します。"
    ]
   },
@@ -1496,7 +1496,7 @@
    "cell_type": "markdown",
    "metadata": {},
    "source": [
-    "###  終結\n",
+    "####  終結\n",
     "よくできました！残りの問題で、これを理解するために新しいことを試してみましょう。"
    ]
   },
@@ -1504,14 +1504,14 @@
    "cell_type": "markdown",
    "metadata": {},
    "source": [
-    "## パズル 1b\n",
+    "### パズル 1b\n",
     "\n",
-    "### 導入\n",
+    "#### 導入\n",
     "今見たゲートをよりよく理解するために、量子ビットを視覚化するために少し異なる方法を使用します。この場合、確実に`0`になる出力は、白い円ではなく白い線で表されます。`1`を与えることが確実な出力は、黒い円ではなく黒い線になります。ランダムな出力の場合、灰色の円ではなく、一部が白で一部が黒の線が表示されます。\n",
     "\n",
     "これは、この新しい視覚化に慣れるのに役立つ古い練習問題です。\n",
     "\n",
-    "### 練習問題\n",
+    "#### 練習問題\n",
     "* X出力が確実に一致するようにします。"
    ]
   },
@@ -1542,12 +1542,12 @@
    "cell_type": "markdown",
    "metadata": {},
    "source": [
-    "## パズル 1c\n",
+    "### パズル 1c\n",
     "\n",
-    "### 導入\n",
+    "#### 導入\n",
     "このパズルでは、`ブロッホ`で新しいことができることがわかります。これは実際にはゲートではなく、量子プログラムには表示されません。代わりに、各量子ビットの2本の線を互いに重ねて描画することにより、見た目を変更するだけです。また、それらのレベルが交差するポイントを配置します。ブロッホを使用すると、`ry(pi / 4)`がどのように機能するかを理解できるはずです。\n",
     "\n",
-    "### 練習問題\n",
+    "#### 練習問題\n",
     "* `q[0]`の一番下の行を完全にオンにして、`ブロッホ`ゲートを使用します。"
    ]
   },
@@ -1580,7 +1580,7 @@
    "cell_type": "markdown",
    "metadata": {},
    "source": [
-    "### 終結\n",
+    "#### 終結\n",
     "ポイントをたどった場合、`ry(pi/4)`の効果はポイントを$\\pi/4$ラジアン(45度)回転させることであることに気付くはずです。\n",
     "\n",
     "線のレベルもそれに合わせて変化します。`ry(-pi/4)`ゲートの効果は、回転が反対方向であることを除いて同じです。\n",
@@ -1592,12 +1592,12 @@
    "cell_type": "markdown",
    "metadata": {},
    "source": [
-    "## パズル 2\n",
+    "### パズル 2\n",
     "\n",
-    "### 導入\n",
+    "#### 導入\n",
     "では、これらのゲートを他の量子ビットでも使用しましょう。\n",
     "\n",
-    "### 練習問題\n",
+    "#### 練習問題\n",
     "*  ボトムラインを完全にオンにします。"
    ]
   },
@@ -1630,12 +1630,12 @@
    "cell_type": "markdown",
    "metadata": {},
    "source": [
-    "## パズル 3\n",
+    "### パズル 3\n",
     "\n",
-    "### 導入\n",
+    "#### 導入\n",
     "これは`cx`、`cz`、`h`があれば解くことができるパズルですが、残念ながら、今回は、cxとhがありません。そのため、`cz`と`ry`がどのように機能するかを理解する必要があります。\n",
     "\n",
-    "### 練習問題\n",
+    "#### 練習問題\n",
     "* Z出力を一致させます。"
    ]
   },
@@ -1666,12 +1666,12 @@
    "cell_type": "markdown",
    "metadata": {},
    "source": [
-    "## パズル 4\n",
+    "### パズル 4\n",
     "\n",
-    "### 導入\n",
+    "#### 導入\n",
     "`x`または`z`を使用すると、`ry`を*反転*させて、反対方向に向けることができます。\n",
     "\n",
-    "### 練習問題\n",
+    "#### 練習問題\n",
     "* それぞれに1つの`ry(pi/4)`を付けて、Z出力を完全にオフにします。"
    ]
   },
@@ -1702,12 +1702,12 @@
    "cell_type": "markdown",
    "metadata": {},
    "source": [
-    "## パズル 5\n",
+    "### パズル 5\n",
     "\n",
-    "### 導入\n",
+    "#### 導入\n",
     "`ry`を使用すると、`cz`や`cx`よりも興味深い条件付きゲートを作成できます。たとえば、制御`h`を作成できます。\n",
     "\n",
-    "### 練習問題\n",
+    "#### 練習問題\n",
     "* q[1]の量子ビットで1回の`ry(pi/4)`と`ry(-pi/4)`を使用して、q[1]のZ出力をオフにします。"
    ]
   },
@@ -1738,7 +1738,7 @@
    "cell_type": "markdown",
    "metadata": {},
    "source": [
-    "# ボーナスレベル: Sandbox\n",
+    "## ボーナスレベル: Sandbox\n",
     "\n",
     "これで、完全に強力な量子プログラムを構築するのに十分な基本的な量子ゲートを理解できました。最終レベルでこれを味わうことができます。しかしその前に、ここにいくつかのボーナスレベルがあります。\n",
     "\n",
@@ -1804,7 +1804,7 @@
    "cell_type": "markdown",
    "metadata": {},
    "source": [
-    "# ボーナスレベル: 自分でパズルをつくってみましょう。\n",
+    "## ボーナスレベル: 自分でパズルをつくってみましょう。\n",
     "\n",
     "こちらからパズルを提供するだけでなく、みなさんがご自分でパズルを作成できるようにしました。Quantum Experienceを使用すると、自分のパズルのnotebookを簡単に作成できます。\n",
     "\n",
@@ -1921,14 +1921,14 @@
    "cell_type": "markdown",
    "metadata": {},
    "source": [
-    "# レベル 5: 量子変数の一意性の証明"
+    "## レベル 5: 量子変数の一意性の証明"
    ]
   },
   {
    "cell_type": "markdown",
    "metadata": {},
    "source": [
-    "## 古典変数を用いたベルのテスト\n",
+    "### 古典変数を用いたベルのテスト\n",
     "ここでは、量子変数(量子ビットに基づく)が標準のもの(ビットに基づく)とどのように異なるかを調査します。\n",
     "\n",
     "これを行うには、`A`と`B`と呼ぶ変数のペアを作成します。これらが何であるか、またはどのように初期化されるかについては、条件を設定しません。したがって、多くの可能性があります。\n",
@@ -2076,27 +2076,37 @@
     "\n",
     "この理由を理解するために、`hash2bit(A, H)`と`hash2bit(B,V)`がどの実行でも違う値にならないことを`P['HV'] = 0.0`が示していることから始めましょう。つまり、これは常にそれらが等しいことを期待できることを意味します。\n",
     "\n",
+    "```\n",
     "    hash2bit(A, H) = hash2bit(B, V)        (1)\n",
-    " \n",
+    "```\n",
+    "\n",
     "`P['VV'] = 0.0`および`P['VH'] = 0.0`から、同様に以下が取得できます。\n",
     "\n",
+    "```\n",
     "    hash2bit(A, V) = hash2bit(B, V)        (2)\n",
     "    \n",
     "    hash2bit(A, V) = hash2bit(B, H)        (3)\n",
-    "    \n",
+    "```\n",
+    "\n",
     "(1)と(2)を組み合わせると、\n",
     "\n",
+    "```\n",
     "    hash2bit(A, H) = hash2bit(A, V)        (4)\n",
-    "    \n",
+    "```\n",
+    "\n",
     "これを(3)と組み合わせると、\n",
     "\n",
+    "```\n",
     "    hash2bit(A, H) = hash2bit(B, H)        (5)\n",
+    "```\n",
     "\n",
     "そして、これらの値が常に等しい場合、異なる実行はありえません。これはまさに私たちが証明しようとしたものです： `P['HH'] = 0.0`\n",
     "\n",
     "より一般的には、`P['HV']`、 `P['VH']`、および`P['VV']`の値を使用して、`P['HH']`の上限を設定できます。[CHSH不等式](https://en.wikipedia.org/wiki/CHSH_inequality)を適応させることにより、次のことがわかります。\n",
     "\n",
-    "$\\,\\,\\,\\,\\,\\,\\,$ `P['HH']` $\\, \\leq \\,$ `P['HV'] + P['VH'] + P['VV']`\n",
+    "```\n",
+    "P['HH'] ≤ P['HV'] + P['VH'] + P['VV']\n",
+    "```\n",
     "\n",
     "これは、`P['HH’]`だけの特別な特徴ではありません。他のすべての確率にも当てはまります。これらの確率のそれぞれは、他の確率の合計より大きくすることはできません。\n",
     "\n",
@@ -2149,7 +2159,7 @@
    "cell_type": "markdown",
    "metadata": {},
    "source": [
-    "## 量子変数を用いたベルのテスト\n",
+    "### 量子変数を用いたベルのテスト\n",
     "ここで、変数AとBが量子変数になることを除いて、同じことをもう一度やり直します。具体的には、それらは最も単純な種類の量子変数、つまり量子ビットになります。"
    ]
   },
@@ -2220,7 +2230,7 @@
     "\n",
     "選択するということは、確率`P['HH']`、`P['HV']`、`P['VH']`、および`P['VV']`が盤面上の円に明示的に対応することを意味することに注意してください。たとえば、一番上の円は、2つのX出力が一致しない可能性がどの程度あるかを示しています。これが白の場合、`P['HH'] = 1`、黒の場合、`P['HH'] = 0`です。\n",
     "\n",
-    "### 練習問題\n",
+    "#### 練習問題\n",
     "* 両方の量子ビットのX出力が一致しない可能性が高く、他のすべての出力の組み合わせが一致する可能性が高くなるようにします。"
    ]
   },
@@ -2266,12 +2276,15 @@
    "source": [
     "`P`の値は、以下の関数で計算されます。これでは、ここでは、このnotebookの他の部分のパズルと同様に、可能な出力を与えたサンプルの数がわかる結果を得ています。出力はビット文字列、`文字列`として提供され、Qiskitは右から左に番号を付けます。 これは、`bit[0]`に対応するaの値が右から1番目であることを意味します。\n",
     "\n",
+    "```\n",
     "    a = string[-1]\n",
+    "```\n",
     "\n",
     "`b`の値は、右から2番目のすぐ隣にあります。\n",
     "\n",
+    "```\n",
     "    b = string[-2]\n",
-    "\n",
+    "```\n",
     "\n",
     "このビット文字列のサンプル数は、結果の辞書型`stats`によって`stats [string]`として提供されます。"
    ]
@@ -2363,8 +2376,10 @@
     "\n",
     "答えは、その論理には隠された仮定があるということです。その理由を理解するために、式(4)に戻りましょう。\n",
     "\n",
+    "```\n",
     "    hash2bit ( A, H ) = hash2bit ( A, V )        (4)\n",
-    "    \n",
+    "```\n",
+    "\n",
     "ここでは、変数`A`の`H`タイプのハッシュから取得する値を`V`タイプのハッシュの値と比較します。\n",
     "\n",
     "古典的な変数の場合、これは完全に理にかなっています。両方のハッシュを計算して結果を比較することを妨げるものは何もありません。変数のハッシュを計算すると変数が変更されても、それは問題ではありません。事前にコピーするだけで、両方のハッシュを問題なく実行できます。\n",
diff --git a/translations/ja/v1/toc.yaml b/translations/ja/v1/toc.yaml
index 2961b6390..e5cd9d650 100755
--- a/translations/ja/v1/toc.yaml
+++ b/translations/ja/v1/toc.yaml
@@ -1,10 +1,10 @@
 - title: 前提条件
   url: /ch-prerequisites
   sections:
-    - id: environment-setup-guide-to-work-with-qiskit-textbook
+    - id: setting-the-environment
       uuid: a5a86ddc-b519-11ec-b909-0242ac120002
       url: /ch-prerequisites/setting-the-environment
-    - id: introduction-to-python-and-jupyter-notebooks
+    - id: python-and-jupyter-notebooks
       uuid: a9f5ea68-b519-11ec-b909-0242ac120002
       url: /ch-prerequisites/python-and-jupyter-notebooks
 
@@ -24,7 +24,7 @@
     - id: single-qubit-gates
       uuid: bd4334a4-b519-11ec-b909-0242ac120002
       url: /ch-states/single-qubit-gates
-    - id: the-case-for-quantum-computers
+    - id: case-for-quantum
       uuid: c0bbff62-b519-11ec-b909-0242ac120002
       url: /ch-states/case-for-quantum
 
@@ -41,13 +41,13 @@
     - id: phase-kickback
       uuid: cb7353ec-b519-11ec-b909-0242ac120002
       url: /ch-gates/phase-kickback
-    - id: basic-circuit-identities
+    - id: more-circuit-identities
       uuid: ce8ce75a-b519-11ec-b909-0242ac120002
       url: /ch-gates/more-circuit-identities
     - id: proving-universality
       uuid: d15d3548-b519-11ec-b909-0242ac120002
       url: /ch-gates/proving-universality
-    - id: classical-computation-on-a-quantum-computer
+    - id: oracles
       uuid: d4f6e424-b519-11ec-b909-0242ac120002
       url: /ch-gates/oracles
 
@@ -55,16 +55,16 @@
 - title: 量子プロトコルと量子アルゴリズム
   url: /ch-algorithms
   sections:
-    - id: quantum-circuits
+    - id: defining-quantum-circuits
       uuid: d90cbfb6-b519-11ec-b909-0242ac120002
       url: /ch-algorithms/defining-quantum-circuits
-    - id: deutsch-jozsa-algorithm
+    - id: deutsch-jozsa
       uuid: dc9d3aa2-b519-11ec-b909-0242ac120002
       url: /ch-algorithms/deutsch-jozsa
-    - id: bernstein-vazirani-algorithm
+    - id: bernstein-vazirani
       uuid: df6e5a40-b519-11ec-b909-0242ac120002
       url: /ch-algorithms/bernstein-vazirani
-    - id: simons-algorithm
+    - id: simon
       uuid: e2714126-b519-11ec-b909-0242ac120002
       url: /ch-algorithms/simon
     - id: quantum-fourier-transform
@@ -73,16 +73,19 @@
     - id: quantum-phase-estimation
       uuid: e8ac9ec8-b519-11ec-b909-0242ac120002
       url: /ch-algorithms/quantum-phase-estimation
-    - id: shors-algorithm
+    - id: shor
       uuid: ec157cd8-b519-11ec-b909-0242ac120002
       url: /ch-algorithms/shor
-    - id: grovers-algorithm
+    - id: grover
       uuid: eebb7488-b519-11ec-b909-0242ac120002
       url: /ch-algorithms/grover
     - id: quantum-counting
       uuid: f1a9942c-b519-11ec-b909-0242ac120002
       url: /ch-algorithms/quantum-counting
-    - id: quantum-teleportation
+    - id: quantum-walk-search-algorithm
+      uuid: 9a061c81-d490-4fff-bb71-727408234082
+      url: /ch-algorithms/quantum-walk-search-algorithm
+    - id: teleportation
       uuid: f4f4a090-b519-11ec-b909-0242ac120002
       url: /ch-algorithms/teleportation
     - id: superdense-coding
@@ -96,32 +99,32 @@
 - title: 量子アプリケーションのアルゴリズム
   url: /ch-applications
   sections:
-    - id: solving-linear-systems-of-equations-using-hhl-and-its-qiskit-implementation
+    - id: hhl_tutorial
       uuid: 01484810-b51a-11ec-b909-0242ac120002
       url: /ch-applications/hhl_tutorial
-    - id: simulating-molecules-using-vqe
+    - id: vqe-molecules
       uuid: 041df288-b51a-11ec-b909-0242ac120002
       url: /ch-applications/vqe-molecules
-    - id: solving-combinatorial-optimization-problems-using-qaoa
+    - id: qaoa
       uuid: 06ff7ada-b51a-11ec-b909-0242ac120002
       url: /ch-applications/qaoa
-    - id: solving-satisfiability-problems-using-grovers-algorithm
+    - id: satisfiability-grover
       uuid: 09d1529c-b51a-11ec-b909-0242ac120002
       url: /ch-applications/satisfiability-grover
-    - id: hybrid-quantum-classical-neural-networks-with-pytorch-and-qiskit
+    - id: machine-learning-qiskit-pytorch
       uuid: 0cac4d14-b51a-11ec-b909-0242ac120002
       url: /ch-machine-learning/machine-learning-qiskit-pytorch
-    - id: the-variational-quantum-linear-solver
+    - id: vqls
       uuid: 1163f456-b51a-11ec-b909-0242ac120002
       url: /ch-paper-implementations/vqls
-    - id: flexible-representation-of-quantum-images-frqi
+    - id: image-processing-frqi-neqr
       uuid: 14acf6a8-b51a-11ec-b909-0242ac120002
       url: /ch-applications/image-processing-frqi-neqr
 
 - title: 量子回路による量子ハードウェアの探究
   url: /quantum-hardware
   sections:
-    - id: introduction-to-quantum-error-correction-via-the-repetition-code
+    - id: error-correction-repetition-code
       uuid: 197f7f66-b51a-11ec-b909-0242ac120002
       url: /ch-quantum-hardware/error-correction-repetition-code
     - id: measurement-error-mitigation
@@ -138,16 +141,16 @@
 - title: マイクロ波パルスによる量子ハードウェアの探究
   url: /quantum-hardware-pulses
   sections:
-    - id: calibrating-qubits-using-qiskit-pulse
+    - id: calibrating-qubits-pulse
       uuid: 292100ca-b51a-11ec-b909-0242ac120002
       url: /ch-quantum-hardware/calibrating-qubits-pulse
-    - id: accessing-higher-energy-states-with-qiskit-pulse
+    - id: accessing_higher_energy_states
       uuid: 2bf0d316-b51a-11ec-b909-0242ac120002
       url: /ch-quantum-hardware/accessing_higher_energy_states
-    - id: introduction-to-transmon-physics
+    - id: transmon-physics
       uuid: 2e944e0e-b51a-11ec-b909-0242ac120002
       url: /ch-quantum-hardware/transmon-physics
-    - id: circuit-quantum-electrodynamics
+    - id: cQED-JC-SW
       uuid: 31b2ccdc-b51a-11ec-b909-0242ac120002
       url: /ch-quantum-hardware/cQED-JC-SW
 
@@ -155,10 +158,10 @@
 - title: 量子コンピューティング・ラボ
   url: /ch-labs
   sections:
-    - id: lab-1-quantum-circuits
+    - id: Lab01_QuantumCircuits
       uuid: 3ff5a7ec-b51a-11ec-b909-0242ac120002
       url: /ch-labs/Lab01_QuantumCircuits
-    - id: lab-2-quantum-measurement
+    - id: Lab02_QuantumMeasurement
       uuid: 82eb9557-1420-487b-a3a5-72fdf38878e8
       url: /ch-labs/Lab02_QuantumMeasurement
     
@@ -166,10 +169,10 @@
 - title: Appendix(付録)
   url: /ch-appendix
   sections:
-    - id: an-introduction-to-linear-algebra-for-quantum-computing
+    - id: linear_algebra
       uuid: 5d2bb2ca-b51a-11ec-b909-0242ac120002
       url: /ch-appendix/linear_algebra
-    - id: basic-qiskit-syntax
+    - id: qiskit
       uuid: 5fd6b79a-b51a-11ec-b909-0242ac120002
       url: /ch-appendix/qiskit
 
@@ -177,9 +180,9 @@
 - title: ゲームとデモ
   url: /ch-demos
   sections:
-    - id: bonus-level-sandbox
+    - id: hello-qiskit
       uuid: 63a85f0e-b51a-11ec-b909-0242ac120002
       url: /ch-ex/hello-qiskit
-    - id: estimating-pi-using-quantum-phase-estimation-algorithm
+    - id: piday-code
       uuid: 69b9707c-b51a-11ec-b909-0242ac120002
       url: /ch-demos/piday-code
diff --git a/translations/ja/v2/basics/multiple-systems.ipynb b/translations/ja/v2/basics/multiple-systems.ipynb
index 4b8e8a09e..f1acc5537 100755
--- a/translations/ja/v2/basics/multiple-systems.ipynb
+++ b/translations/ja/v2/basics/multiple-systems.ipynb
@@ -1158,7 +1158,7 @@
     "4 つのベルの状態のすべての集まり\n",
     "\n",
     "$$\n",
-    "  \\bigl\\{\\vert \\phi^+ \\rangle, \\vert \\phi^- \\rangle, \\vert \\psi^+ \\rangle, \\vert \\psi^+ \\rangle\\bigr\\}\n",
+    "  \\bigl\\{\\vert \\phi^+ \\rangle, \\vert \\phi^- \\rangle, \\vert \\psi^+ \\rangle, \\vert \\psi^- \\rangle\\bigr\\}\n",
     "$$\n",
     "\n",
     "は、 *ベル基底*として知られています。 2量子ビットの任意の量子状態ベクトル、または実際には 2 ビットの 4 つの古典的な状態に対応する要素を持つ任意の複素ベクトルは、4 つのベル状態の線形結合として表現できます。例えば、次のようになります。\n",
@@ -1704,7 +1704,7 @@
     "\n",
     "2つのシステムに注目すると、 $\\mathsf{X}$ が古典状態の集合 $\\Sigma$ を持つシステム、 $\\mathsf{Y}$ が古典状態集合 $\\Gamma$ を持つシステムであれば、合成システムの古典状態の集合 $(\\mathsf{X},\\mathsf{Y})$ は $\\Sigma\\times\\Gamma$ です。 したがって、この合成システムに対して実行可能な演算の集合は、行と列が集合 $\\Sigma\\times\\Gamma$ に対応するように配置されたユニタリー行列で表現されます。この行列の行と列の並びは、システム $(\\mathsf{X},\\mathsf{Y})$ の量子状態ベクトルに用いられる並びと同じです。\n",
     "\n",
-    "例えば、 $\\Sigma = \\{1,2,3\\}$ と $\\Gamma = \\{0,1\\}$ とすると、デカルト積 $\\{1,2,3\\}\\times\\{0,1\\}$ の要素の順番は、 $(1,0)$, $(1,1)$, $(2,0)$, $(2,1)$, $(3,0)$, $(3,1)$ が標準であす。<br>以下は、 $(\\mathsf{X},\\mathsf{Y})$ 上の演算を表すユニタリー行列の例です。\n",
+    "例えば、 $\\Sigma = \\{1,2,3\\}$ と $\\Gamma = \\{0,1\\}$ とすると、デカルト積 $\\{1,2,3\\}\\times\\{0,1\\}$ の要素の順番は、 $(1,0)$, $(1,1)$, $(2,0)$, $(2,1)$, $(3,0)$, $(3,1)$ が標準です。<br>以下は、 $(\\mathsf{X},\\mathsf{Y})$ 上の演算を表すユニタリー行列の例です。\n",
     "\n",
     "$$\n",
     "U = \n",
@@ -1874,7 +1874,7 @@
     "\n",
     "授業を締めくくるために、複数システムでのユニタリー演算の例を 2 つ見てみましょう。*スワップ演算*から始めます。\n",
     "\n",
-    "$\\mathsf{X}$ と $\\mathsf{Y}$ が、同じ古典的な状態セット $\\Sigma$ を共有するシステムであるとします。ペア $(\\mathsf{X},\\mathsf{Y})$ の*スワップ*操作は、2 つのシステムの内容を交換する操作ですが、それ以外の場合はシステムをそのままにしておきます (したがって、$\\mathsf{X}$ は残ります)。左側に、$\\mathsf{Y}$ が右側に残ります)。\n",
+    "$\\mathsf{X}$ と $\\mathsf{Y}$ が、同じ古典的な状態セット $\\Sigma$ を共有するシステムであるとします。ペア $(\\mathsf{X},\\mathsf{Y})$ の*スワップ*操作は、2 つのシステムの内容を交換する操作ですが、それ以外の場合はシステムをそのままにしておきます (したがって、 $\\mathsf{X}$ は左側に、 $\\mathsf{Y}$ が右側に残ります)。\n",
     "\n",
     "この演算を $\\operatorname{SWAP}$ と書き、古典的な状態 $a,b\\in\\Sigma$ のすべての選択に対して、次のように操作します。\n",
     "\n",
@@ -2053,7 +2053,7 @@
    "cell_type": "markdown",
    "metadata": {},
    "source": [
-    "以下の別の例では、 $|{-}\\rangle$ および $\\tfrac{1}{\\sqrt{2}}(|0\\rangle + i|1\\rangle)$ 状態を表す状態ベクトルを作成し、合成して、新しい状態ベクトルを作成します。この新しいベクトルを変数`psi`に割り当てます。"
+    "以下の別の例では、 $|{+}\\rangle$ および $\\tfrac{1}{\\sqrt{2}}(|0\\rangle + i|1\\rangle)$ 状態を表す状態ベクトルを作成し、合成して、新しい状態ベクトルを作成します。この新しいベクトルを変数`psi`に割り当てます。"
    ]
   },
   {
diff --git a/translations/ja/v2/ch-algorithms/quantum-walk-search-algorithm.ipynb b/translations/ja/v2/ch-algorithms/quantum-walk-search-algorithm.ipynb
new file mode 100644
index 000000000..c72f4b83f
--- /dev/null
+++ b/translations/ja/v2/ch-algorithms/quantum-walk-search-algorithm.ipynb
@@ -0,0 +1,709 @@
+{
+ "cells": [
+  {
+   "cell_type": "markdown",
+   "id": "8c521bc3",
+   "metadata": {
+    "tags": [
+     "remove_cell"
+    ]
+   },
+   "source": [
+    "# 量子ウォークによる探索アルゴリズム"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "55932bd9",
+   "metadata": {},
+   "source": [
+    "量子ウォークは古典のマルコフ連鎖の量子版であり、多くの量子アルゴリズムにとって重要なものです。この章ではグラフ上にマークした要素を見つける量子ウォークによる探索アルゴリズムを実装します。このアルゴリズムは古典に比べて二次加速を持ちます。\n",
+    "\n",
+    "\n",
+    "## 1. 古典マルコフ連鎖 <a name=\"markov\"></a>\n",
+    "マルコフ連鎖は現実のプロセスをモデル化するためによく使われる確率過程です。状態と遷移確率からなり、各時間ステップでの状態間の確率の移動を記述します。ここで紹介する離散時間マルコフ連鎖では、時間ステップは離散的です。マルコフ連鎖はマルコフ性を満たします、つまり次のステップでの確率過程は現在のステップのみに依存して、それ以前には依存しません。マルコフ連鎖は状態間の確率の移動を表現する遷移行列 P を持ちます。ここで、マルコフ連鎖の例と、その遷移行列 $P$ を示します。\n",
+    "\n",
+    "\n",
+    "   $$\n",
+    "     P\n",
+    "     = \\begin{pmatrix}\n",
+    "    0.1 & 0.3 & 0.3\\\\\n",
+    "    0.1 & 0.1 & 0.2 \\\\\n",
+    "    0.8 & 0.6 & 0.5\n",
+    "     \\end{pmatrix}.\n",
+    "   $$\n",
+    "   \n",
+    "<img src=\"images/markov_example.png\" width='480'>\n",
+    "\n",
+    "遷移行列 $P$ が与えられると、 $t$ 時間ステップ後の確率分布を $P^t$ により計算できます。\n",
+    "\n",
+    "## 2. 量子ウォーク <a name=\"qwalks\"></a>\n",
+    "量子ウォークは古典のマルコフ連鎖の量子版です。測定するまでは、量子ウォークは全ての可能な経路の重ね合わせを取ります。また、量子干渉によりある状態は打ち消しあいます。この現象を、間違えた答えを消去するような設計に利用することで、量子ウォークアルゴリズムをランダムウォークよりも速くします。コイン量子ウォークとセゲディ量子ウォークがよく用いられ、これらはある条件下では等価になります。コイン量子ウォークはグラフ上の頂点を、セゲディ量子ウォークは辺を動きます。量子ウォークの実装の前にこの二つのモデルを紹介します。\n",
+    "\n",
+    " $\\newcommand{\\ket}[1]{\\left|{#1}\\right\\rangle}$\n",
+    " $\\newcommand{\\bra}[1]{\\left\\langle{#1}\\right|}$\n",
+    "### 2.A コイン量子ウォーク <a name=\"coined\"></a>\n",
+    "　コイン量子ウォークの簡単な例は、無限長の整数上の運動です。この場合、整数 $\\mathbb{Z}$ 上での位置を $\\{\\ket{j} : j \\in  \\mathbb{Z} \\}$ とします。そして、コインの計算基底は $\\{\\ket{0}, \\ket{1}\\}$ とし、コインが $\\ket{0}$ の場合はある方向に動き、コインが $\\ket{1}$ の場合はもう一方に動きます。\n",
+    "\n",
+    "コイン量子ウォークはグラフの頂点上を動くので、頂点を状態とします。辺で繋がっている状態間のみ動くことができます。コインモデルでは、二つの量子状態空間と二つの演算子を使います。一つ目の状態空間は位置を表す位置状態です。上記の運動では、整数上のどこでも移動できるので、整数となります。もう一方は、コイン状態です。コイン状態は次のステップでどのように動くかを決定します。このコイン状態と位置状態とを合わせて、ヒルベルト空間上のベクトルとして表現できます。コイン状態を $\\mathcal{H}_C$ のベクトルとして表し、位置状態を $\\mathcal{H}_P$ のベクトルとして表すと、運動のすべての量子状態空間は $\\mathcal{H} = \\mathcal{H}_C \\otimes \\mathcal{H}_P$ となります。\n",
+    "\n",
+    "すでに述べたように、モデルには二つの演算子がありコイン演算子 $C$ とシフト演算子 $S$ です。コイン演算子は時間ステップ毎に $\\mathcal{H}_C$ に作用して重ね合わせ状態を作るため、全ての可能な経路を同時に動きます。整数上の運動では、時間ステップ毎に左と右の両方に動くことになります。いろいろなコイン演算子がありますが、よく用いられるのはアダマールコインとグローバーコインです。アダマールコインはアダマールゲートにより等確率での重ね合わせ状態を作ります: \n",
+    "\n",
+    "   $$\n",
+    "     H\n",
+    "     = \\frac{1}{\\sqrt{2}}\n",
+    "     \\begin{bmatrix}\n",
+    "    1 & 1 \\\\\n",
+    "    1 & -1 \n",
+    "     \\end{bmatrix}.\n",
+    "   $$\n",
+    "\n",
+    "グローバーコインはグローバーのアルゴリズムにあるグローバーの拡散演算子を用い、以下のように定義します\n",
+    "\n",
+    "   $$\n",
+    "     G\n",
+    "     = \\begin{bmatrix}\n",
+    "    \\frac{2}{n} -1 & \\frac{2}{n} & \\ldots & \\frac{2}{n}\\\\\n",
+    "    \\frac{2}{n} & \\frac{2}{n} - 1  & \\ldots & \\frac{2}{n} \\\\\n",
+    "    \\vdots & \\vdots & \\ddots & \\vdots \\\\\n",
+    "    \\frac{2}{n} & \\frac{2}{n} & \\ldots & \\frac{2}{n} -1\n",
+    "     \\end{bmatrix}.\n",
+    "   $$\n",
+    "\n",
+    "アダマールコインのようにグローバーコインも重ね合わせ状態を作ります。しかし、振る舞いは少し異なります。位置 $\\ket{000}$ にグローバーコインを作用させると、下図のような状態ベクトルの確率分布を得ます。このように、アダマールコインのような等確率の重ね合わせ状態を作りません。他の状態に比べて、 $\\ket{000}$ により大きな確率を持ちます。\n",
+    "\n",
+    "<img src=\"images/grover_prob.svg\" width='480'>\n",
+    "\n",
+    "モデルのもう一つの演算子のシフト演算子は $\\mathcal{H}_P$ に作用して、次の位置に移動させます。整数上の運動では、シフト演算子により、コインが $\\ket{0}$ の時は左に動き、コインが $\\ket{1}$ の時は右に動きます: \n",
+    "\n",
+    "$$\n",
+    "\\begin{aligned}\n",
+    "  S \\vert 0 \\rangle \\vert j \\rangle & = \\vert 0 \\rangle \\vert j+1 \\rangle \\\\[1mm]\n",
+    "   S \\vert 1 \\rangle \\vert j \\rangle & = \\vert 1 \\rangle \\vert j-1 \\rangle\n",
+    "\\end{aligned}\n",
+    "$$\n",
+    "\n",
+    "上記で定義されたシフト演算子で、一ステップ分のコイン量子ウォークを以下のユニタリ演算子として表現できます\n",
+    "$$\n",
+    "    U = SC,\n",
+    "$$\n",
+    "\n",
+    "ここで C はコイン演算子です。整数上の量子ウォークでアダマールコインを用いましたが、C はアダマールコインでも、グローバーコインでも、その他のコイン演算子でも用いることができます。\n",
+    "\n",
+    "\n",
+    "先に進みましょう。$t$ 時間ステップ後の量子状態 $\\ket{\\psi}$ は以下で表すことができます\n",
+    "$$\n",
+    "    \\ket{\\psi (t)} = U^t \\ket{\\psi(0)},\n",
+    "$$\n",
+    "ここで、 $\\ket{\\psi(0)}$ は初期状態で、U は一ステップ分の演算子です[1]。\n",
+    "\n",
+    "コイン量子ウォークは多くの正則グラフ(全ての頂点で隣接する頂点の数が同じグラフ)で安定しています[2]。 もう一つのモデルであるセゲディ量子ウォークは非正則グラフに適しており、次の章で解説します。\n",
+    "\n",
+    "### 2.B セゲディ量子ウォーク <a name=\"szegedy\"></a>\n",
+    "コイン量子ウォークはグラフの頂点を動きましたが、セゲディ量子ウォークは元のグラフの二部二重被覆グラフ上の辺を動きます。二重被覆グラフとは元のグラフの二倍の頂点を持つグラフです。二部二重被覆グラフの二つの頂点は、元のグラフでその頂点が繋がっている場合にのみ、辺で繋がります。このモデルを作るには、古典のランダムウォークの遷移確率行列 P から始めます。1 章で示したように、古典の離散時間ランダムウォークは遷移行列 P で記述されます。任意の $N \\times N$ の遷移行列 $P$ を持つ $N$-頂点グラフに対して、離散時間量子ウォークをヒルベルト空間 $\\mathcal{H}^N \\otimes \\mathcal{H}^N$ 上のユニタリ演算子として定義できます。 $P_{jk}$ を $j$ から $k$ へ遷移する確率と定義しましょう。運動を定義する前に、正規化された状態を定義します\n",
+    "\n",
+    "$$\n",
+    "    \\ket{\\psi_j} := \\sum_{k=1}^N \\sqrt{P_{kj}} \\ket{j,k}, \\; j=1,...,N\n",
+    "$$\n",
+    "\n",
+    "そして、 $\\vert \\psi_j \\rangle :j=1,...,N$ への射影は\n",
+    "\n",
+    "$$\n",
+    "    \\Pi := \\sum_{j=1}^N \\ket{\\psi_j} \\bra{\\psi_j}\n",
+    "$$\n",
+    "\n",
+    "シフト演算子を導入します:\n",
+    "\n",
+    "$$\n",
+    "    S := \\sum_{j,k=1}^N \\ket{j,k} \\bra{k,j}\n",
+    "$$\n",
+    "\n",
+    "上で定義した $S$ と $\\Pi$ を用いて一ステップ分の離散時間量子ウォークを定義します:\n",
+    "\n",
+    "$$\n",
+    "    U := S(2 \\Pi - 1),\n",
+    "$$\n",
+    "\n",
+    "ここで、 $(2 \\Pi - 1)$ は反射演算子です。また、 $t$ ステップ後の状態は $U^t$ により定義できます[2]。\n",
+    "\n",
+    "### 2.C コインとセゲディ量子ウォークの等価性 <a name=\"equivalence\"></a>\n",
+    "グローバーコインによるコイン量子ウォークはセゲディ量子ウォークと等価であることが知られています。詳細は Thomas G. Wong による文献[3]を参照ください、文献では演算子の等価性についても示されています。\n",
+    "\n",
+    "\n",
+    "## 3. 例: 超立方体上の量子ウォーク <a name=\"example1\"></a>\n",
+    "超立方体は $3$ 次元の立方体を $n$ 次元化したものです。全ての頂点の次数は $n$ で、超立方体は $N=2^n$ 個の頂点を持ちます。二値の $n$ タプルを用いて超立方体の頂点を表現できます。隣接する頂点の二値表現は一つの二値のみが異なります。例えば4次元超立方体の場合、頂点 $0000$ の隣接頂点は $0001$, $0010$, $0100$, $1000$ です。したがって、ハミング距離が1の全ての頂点と接続しています。辺も同時にラベルできます。二つの隣接する頂点のa番目のビットが異なる場合、 $a$ とラベルされた辺で繋がります。\n",
+    "\n",
+    "超立方体のコイン量子ウォークを表現するヒルベルト空間は $\\mathcal{H} = \\mathcal{H}^n \\otimes \\mathcal{H}^{2^n}$ です。ここで、　$\\mathcal{H}^n$ はコイン空間で、　$\\mathcal{H}^{2^n}$ は位置です。計算基底では\n",
+    "\n",
+    "$$\n",
+    "    \\big\\{ \\ket{a,\\vec{v}}, 0 \\leq a \\leq n-1, \\vec{v} \\in  \\{(00...00), (00...01), ....., (11...11 )\\} \\big\\}. \n",
+    "$$\n",
+    "\n",
+    "辺 $a$ を表すコイン計算基底 $a$ は、移動の方向を表します。　$a=0$ の場合、現在の辺とは一番目の二値が異なる辺へ移動します。また、　$a=1$ の場合、現在の辺とは二番目の二値が異なる辺へ移動します。　$\\vec{e}_a$ を $a$ 以外は全て $0$ をとる二値の n タプルとします。そして、シフト演算子 $S$ は 状態 $\\ket{a} \\ket{\\vec{v}}$ を $\\ket{a} \\ket{\\vec{v} \\oplus \\vec{e}_a}$ へ移動します。\n",
+    "\n",
+    "$$\n",
+    "    S \\ket{a} \\ket{\\vec{v}} = \\ket{a} \\ket{\\vec{v} \\oplus \\vec{e}_a}.\n",
+    "$$\n",
+    "\n",
+    "ここでは、グローバーコイン $G$ を用います。そして発展演算子は\n",
+    "\n",
+    "$$\n",
+    "    U = SG.\n",
+    "$$\n",
+    "\n",
+    "次に、4次元超立方体の量子ウォークをどのように実装するのかを紹介します。コイン演算子とシフト演算子の実装が必要となります。Qiskitから必要なライブラリをインポートするところから始めます。"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 2,
+   "id": "501e85ad",
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "# Importing standard Qiskit libraries\n",
+    "from qiskit import QuantumCircuit, execute, Aer, IBMQ, QuantumRegister, ClassicalRegister\n",
+    "from qiskit.compiler import transpile, assemble\n",
+    "from qiskit.tools.jupyter import *\n",
+    "from qiskit.visualization import *\n",
+    "from qiskit.circuit.library import QFT\n",
+    "from numpy import pi\n",
+    "from qiskit.quantum_info import Statevector\n",
+    "from matplotlib import pyplot as plt\n",
+    "import numpy as np\n",
+    "# Loading your IBM Q account(s)\n",
+    "provider = IBMQ.load_account()"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "c9487c17",
+   "metadata": {},
+   "source": [
+    "回路は位置に4個、コインに2個用いて、6量子ビットからなります。前に述べたようにコインはグローバーコインで、グローバーのアルゴリズムの拡散演算子です。この実装からはじめます。"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 3,
+   "id": "330b2adc",
+   "metadata": {},
+   "outputs": [
+    {
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+      ],
+      "text/plain": [
+       "<Figure size 327.397x385.28 with 1 Axes>"
+      ]
+     },
+     "execution_count": 3,
+     "metadata": {},
+     "output_type": "execute_result"
+    }
+   ],
+   "source": [
+    "one_step_circuit = QuantumCircuit(6, name=' ONE STEP') \n",
+    "# Coin operator\n",
+    "one_step_circuit.h([4,5])\n",
+    "one_step_circuit.z([4,5])\n",
+    "one_step_circuit.cz(4,5)\n",
+    "one_step_circuit.h([4,5])\n",
+    "one_step_circuit.draw() "
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "f3a875b4",
+   "metadata": {},
+   "source": [
+    "次にシフト演算子を実装します。隣接する頂点にだけ移動します、また、全ての隣接する頂点は一ビットだけ異なります。コインに従って移動するため、頂点の量子ビットの一つに対してNOTゲートを作用します。コインが $\\ket{11}$ 状態の時は、一番目の頂点の量子ビットが異なる状態に移動します。コインが $\\ket{10}$ または $\\ket{01}$ 状態の時は、それぞれ二番目、三番目の量子ビットが異なる状態に移動します。最後にグローバーコインが $\\ket{00}$ の時は、4番目の量子ビットを反転します。これは、グローバーコインの後に、CCNOTゲートとNOTゲートにより実装します。これら全てで4次元超立方体の量子ウォークの一ステップ分になります。"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 4,
+   "id": "2e907955",
+   "metadata": {},
+   "outputs": [
+    {
+     "data": {
+      "image/svg+xml": [
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+      ],
+      "text/plain": [
+       "<Figure size 808.997x385.28 with 1 Axes>"
+      ]
+     },
+     "execution_count": 4,
+     "metadata": {},
+     "output_type": "execute_result"
+    }
+   ],
+   "source": [
+    "# Shift operator function for 4d-hypercube\n",
+    "def shift_operator(circuit):\n",
+    "    for i in range(0,4):\n",
+    "        circuit.x(4)\n",
+    "        if i%2==0:\n",
+    "            circuit.x(5)\n",
+    "        circuit.ccx(4,5,i)\n",
+    "\n",
+    "shift_operator(one_step_circuit)\n",
+    "\n",
+    "one_step_gate = one_step_circuit.to_instruction() \n",
+    "one_step_circuit.draw() "
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "57b0f8a4",
+   "metadata": {},
+   "source": [
+    "## 4. 量子ウォークによる探索アルゴリズム <a name=\"qwalkalgo\"></a>\n",
+    "次にグラフ上にマークした要素を見つける量子ウォークによる探索アルゴリズムを実装します。はじめにアルゴリズムを紹介し、その後で実装します。量子ウォークによる探索アルゴリズムは、量子ウォークを用いて、グラフ上にマークした要素を見つける問題を解きます。つまり、ある頂点集合 $|M|$ をマークして、グラフ上の任意の頂点から始め、マークした複数の頂点を見つけるまで移動します。量子ウォークによる探索では基底として、現在の頂点と直前の頂点に対応する二つのレジスタを必要とします。つまりグラフの辺に対応します。遷移行列 $P$ による古典マルコフ連鎖を基にして、　$\\mathcal{H}$ 上のユニタリ演算子 $W(P)$ により量子ウォークを記述します。また、頂点 $x$ に隣接する頂点の一様な重ね合わせ状態として $\\ket{p_x} = \\sum_y \\sqrt{P_{xy}}\\ket{y}$ を定義します。 $\\ket{x}\\ket{y}$ を一つの基底とします。もし $x$ がマークした頂点の場合、基底 $\\ket{x}\\ket{y}$ を ''good'' と定義します。それ以外を ''bad'' とします。そして、''good'' と ''bad'' 状態は: \n",
+    "\n",
+    "$$\n",
+    "    \\ket{G} = \\frac{1}{\\sqrt{|M|}} \\sum_{x \\in M} \\ket{x} \\ket{p_x}, \\;\n",
+    "    \\ket{B} = \\frac{1}{\\sqrt{N-|M|}} \\sum_{x \\notin M} \\ket{x} \\ket{p_x},\n",
+    "$$\n",
+    "\n",
+    "これらは good および bad の基底の重ね合わせ状態です。次に、 $\\epsilon = |M|/N$ と $\\theta = \\arcsin(\\sqrt{\\epsilon})$ を定義します。\n",
+    "\n",
+    "\n",
+    "まとめると、アルゴリズムは以下の三つのステップからなります:\n",
+    "\n",
+    "\n",
+    "1. 初期状態として全ての辺の一様な重ね合わせ状態を準備 $\\ket{U} = \\frac{1}{\\sqrt{N}} \\sum_{x} \\ket{x} \\ket{p_x} = \\sin{\\theta} \\ket{G} + \\cos{\\theta} \\ket{B}$\n",
+    "2. $O(1/\\sqrt{\\epsilon})$ 回繰り返し:\n",
+    "    \n",
+    "    (a) $\\ket{B}$ での反射\n",
+    "    \n",
+    "    (b) $\\ket{U}$ での反射\n",
+    "\n",
+    "3. 計算基底での測定を実施\n",
+    "\n",
+    "\n",
+    "ステップ $1$ は、アダマールゲートと、 $x$ が一つ目のレジスタと等しい時にだけ位相をシフトする位相オラクルを用いて反射を行うことで簡単に実装できます。\n",
+    "\n",
+    "\n",
+    "ステップ 2(b) は以下の写像を行うユニタリ変換 $R(P)$ を探すことと等価です:\n",
+    "$$\n",
+    "\\label{eq:mapping_1}\n",
+    "    \\ket{U} &\\mapsto \\ket{U}, \\: \\text{と} \\\\\n",
+    "    \\ket{\\psi} &\\mapsto -\\ket{\\psi}, \\: \\forall \\ket{\\psi} \\text{$\\ket{U}$ に直交する $W(P)$ の固有ベクトルに含まれる場合}\n",
+    "\\label{eq:mapping_2}\n",
+    "$$\n",
+    "\n",
+    "この演算子を探すために、 $W(P)$ の位相推定を行います。上記で $W(P)$ をランダムウォークの発展演算子として定義しました。2.1 章で見たように、これはユニタリ演算子です。従って、 $W(P)$ の固有値のノルムは $1$ となります。そこで、 $W(P)$ の固有値を $e^{\\pm 2i\\theta_j}$ と書き直します。ユニタリ演算子 $W(P)$ は固有値が $1$ の固有ベクトル $\\ket{U}$ を持ちます。これは、 $\\theta_1=0$ により得られます。　$R(P)$ は補助量子ビットを加えて $O(1/\\sqrt{\\delta})$ の精度で位相推定を行うことで $\\ket{U}$ を見つけます、ここで、 $\\delta$ は $P$ のスペクトルギャップです。これを行うには、 $W(P)$ を $O(1/\\sqrt{\\delta})$ 回作用する必要があります。 $\\ket{w}$ を $W(P)$ の固有ベクトルで、固有値 $e^{\\pm 2i\\theta_j}$ を持つとしましょう。 $\\tilde{\\theta_j}$ を位相推定による $\\theta_j$ のもっとも良い推定値と仮定します。ステップ 2(b) の $\\ket{w}$ の写像を与える演算子 $R(P)$ は文献[4]で与えられています。\n",
+    "\n",
+    "$$\n",
+    "    \\ket{w} \\ket{0} \\mapsto \\ket{w} \\ket{\\tilde{\\theta_j}} \\mapsto (-1)^{|\\tilde{\\theta_j} \\neq 0|} \\ket{w} \\ket{\\tilde{\\theta_j}} \\mapsto (-1)^{|\\tilde{\\theta_j} \\neq 0|} \\ket{w} \\ket{0}\n",
+    "$$\n",
+    "\n",
+    "\n",
+    "## 5.例: 4次元超立方体上の量子ウォークによる探索 <a name=\"qiskiti\"></a>\n",
+    "\n",
+    "\n",
+    "量子ウォークによる探索アルゴリズムは $O(1/\\sqrt{\\epsilon})$ ステップでマークされた頂点の集合を探すことができます、ここで $\\epsilon =  |M|/N$、 $M$ はマークした頂点数、 $N$ は全ての頂点数です。このアルゴリズムは元はセゲディ量子ウォークを使い、量子状態を表現する二つの頂点レジスタを用いました。しかしグローバーコインを用いるコイン量子ウォークはセゲディ量子ウォークと等価で、コイン量子ウォークは一般的に簡素に実装できるため、コイン量子ウォークによる実装を選びます。3 章で紹介した4次元超立方体を用います。\n",
+    "\n",
+    "まとめると、以下のようにアルゴリズムを実装します。ステップ 1 を行うには、頂点レジスタの全ての量子ビットにアダマールゲートを作用させて全ての辺の一様な重ね合わせ状態を作ります。ステップ 2(a) では、位相オラクルを実装します。ステップ 2(b) は、超立方体の量子ウォークの一ステップ分の位相推定と、 $\\theta \\neq 0$ となるすべての量子状態のマークを行います。これを補助量子ビットの回転により行います。最後のステップは、位相推定の逆演算を行います。位相推定に必要な量子ビットは $\\theta$ の精度に依存します。\n",
+    "\n",
+    "以下、4次元超立方体の量子ウォークによる探索アルゴリズムを実装します。"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "050a316c",
+   "metadata": {},
+   "source": [
+    "このアルゴリズムでは、一ステップ分のゲートに対して以前実装した反転を必要とします。これはビルトイン関数 inverse() で行えます。"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 5,
+   "id": "4911f08e",
+   "metadata": {},
+   "outputs": [
+    {
+     "data": {
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+      ],
+      "text/plain": [
+       "<Figure size 808.997x385.28 with 1 Axes>"
+      ]
+     },
+     "execution_count": 5,
+     "metadata": {},
+     "output_type": "execute_result"
+    }
+   ],
+   "source": [
+    "one_step_circuit.inverse().draw() "
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "b3f5ade1",
+   "metadata": {},
+   "source": [
+    "一ステップ分のゲートの反転は後で位相推定の逆計算に使用します。3 章で実装した一ステップ分のゲートとその反転のゲートは、制御ゲートを必要とします。これらを後に制御量子ビットに応じて使用します。"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 6,
+   "id": "de4c592a",
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "# Make controlled gates\n",
+    "inv_cont_one_step = one_step_circuit.inverse().control()\n",
+    "inv_cont_one_step_gate = inv_cont_one_step.to_instruction()\n",
+    "cont_one_step = one_step_circuit.control()\n",
+    "cont_one_step_gate = cont_one_step.to_instruction()"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "a259ae43",
+   "metadata": {},
+   "source": [
+    "制御された一ステップ分のゲートと、制御された一ステップ分の反転ゲートは位相推定に用いられます。位相推定には量子フーリエ変換も用います。Qiskit では量子フーリエ変換を実装した QFT 関数があります。位相推定には逆量子フーリエ変換を用いますが、位相推定の逆計算にも通常の QFT を使用します。"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 7,
+   "id": "1dde4f45",
+   "metadata": {},
+   "outputs": [
+    {
+     "data": {
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+      ],
+      "text/plain": [
+       "<Figure size 1170.2x264.88 with 1 Axes>"
+      ]
+     },
+     "execution_count": 7,
+     "metadata": {},
+     "output_type": "execute_result"
+    }
+   ],
+   "source": [
+    "inv_qft_gate = QFT(4, inverse=True).to_instruction()  \n",
+    "qft_gate = QFT(4, inverse=False).to_instruction()\n",
+    "\n",
+    "QFT(4, inverse=True).decompose().draw(\"mpl\")"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "fd11ab6d",
+   "metadata": {},
+   "source": [
+    "位相推定を実装する前に、1011 と 1111 をマークする位相オラクルを実装します。その後、回路を作成します。これはアルゴリズムのステップ 2(a) です。"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 8,
+   "id": "90133f6b",
+   "metadata": {
+    "scrolled": true
+   },
+   "outputs": [
+    {
+     "data": {
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+      ],
+      "text/plain": [
+       "<Figure size 447.797x385.28 with 1 Axes>"
+      ]
+     },
+     "execution_count": 8,
+     "metadata": {},
+     "output_type": "execute_result"
+    }
+   ],
+   "source": [
+    "phase_circuit =  QuantumCircuit(6, name=' phase oracle ')\n",
+    "# Mark 1011\n",
+    "phase_circuit.x(2)\n",
+    "phase_circuit.h(3)\n",
+    "phase_circuit.mct([0,1,2], 3)\n",
+    "phase_circuit.h(3)\n",
+    "phase_circuit.x(2)\n",
+    "# Mark 1111\n",
+    "phase_circuit.h(3)\n",
+    "phase_circuit.mct([0,1,2],3)\n",
+    "phase_circuit.h(3)\n",
+    "phase_oracle_gate = phase_circuit.to_instruction()\n",
+    "# Phase oracle circuit\n",
+    "phase_oracle_circuit =  QuantumCircuit(11, name=' PHASE ORACLE CIRCUIT ')\n",
+    "phase_oracle_circuit.append(phase_oracle_gate, [4,5,6,7,8,9])\n",
+    "phase_circuit.draw() "
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "d64b5d40",
+   "metadata": {},
+   "source": [
+    "他の量子ビットがゼロでない場合に補助量子ビットを回転するゲートを実装します。このゲートは位相推定で用いられ、 $\\theta \\neq 0$ の時に補助量子ビットを回転します。"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 9,
+   "id": "2badd3a6",
+   "metadata": {},
+   "outputs": [
+    {
+     "data": {
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+      ],
+      "text/plain": [
+       "<Figure size 387.597x325.08 with 1 Axes>"
+      ]
+     },
+     "execution_count": 9,
+     "metadata": {},
+     "output_type": "execute_result"
+    }
+   ],
+   "source": [
+    "# Mark q_4 if the other qubits are non-zero \n",
+    "mark_auxiliary_circuit = QuantumCircuit(5, name=' mark auxiliary ')\n",
+    "mark_auxiliary_circuit.x([0,1,2,3,4])\n",
+    "mark_auxiliary_circuit.mct([0,1,2,3], 4)\n",
+    "mark_auxiliary_circuit.z(4)\n",
+    "mark_auxiliary_circuit.mct([0,1,2,3], 4)\n",
+    "mark_auxiliary_circuit.x([0,1,2,3,4])\n",
+    "\n",
+    "mark_auxiliary_gate = mark_auxiliary_circuit.to_instruction()\n",
+    "mark_auxiliary_circuit.draw()"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "2fe04e88",
+   "metadata": {},
+   "source": [
+    "次にアルゴリズムのステップ 2(b) を実装します。このステップは、量子ウォーク一ステップ分の位相推定と、 $\\theta \\neq 0$ の時の補助量子ビットの回転操作を含みます。このために、直前に作成した mark_auxiliary_gate を用います。その後、位相推定の逆演算を行います。"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 11,
+   "id": "a9929ba0",
+   "metadata": {
+    "scrolled": false
+   },
+   "outputs": [
+    {
+     "data": {
+      "image/svg+xml": [
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+      ],
+      "text/plain": [
+       "<Figure size 1604.43x2853.48 with 1 Axes>"
+      ]
+     },
+     "execution_count": 11,
+     "metadata": {},
+     "output_type": "execute_result"
+    }
+   ],
+   "source": [
+    "# Phase estimation\n",
+    "phase_estimation_circuit = QuantumCircuit(11, name=' phase estimation ')\n",
+    "phase_estimation_circuit.h([0,1,2,3])\n",
+    "for i in range(0,4):\n",
+    "    stop = 2**i\n",
+    "    for j in range(0,stop):\n",
+    "        phase_estimation_circuit.append(cont_one_step, [i,4,5,6,7,8,9])\n",
+    "\n",
+    "# Inverse fourier transform\n",
+    "phase_estimation_circuit.append(inv_qft_gate, [0,1,2,3])\n",
+    "\n",
+    "# Mark all angles theta that are not 0 with an auxiliary qubit\n",
+    "phase_estimation_circuit.append(mark_auxiliary_gate, [0,1,2,3,10])\n",
+    "\n",
+    "# Reverse phase estimation\n",
+    "phase_estimation_circuit.append(qft_gate, [0,1,2,3])   \n",
+    "\n",
+    "for i in range(3,-1,-1):\n",
+    "    stop = 2**i\n",
+    "    for j in range(0,stop):\n",
+    "        phase_estimation_circuit.append(inv_cont_one_step, [i,4,5,6,7,8,9])\n",
+    "phase_estimation_circuit.barrier(range(0,10))\n",
+    "phase_estimation_circuit.h([0,1,2,3])\n",
+    "\n",
+    "# Make phase estimation gate\n",
+    "phase_estimation_gate = phase_estimation_circuit.to_instruction()\n",
+    "phase_estimation_circuit.draw() "
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "8f8dcc75",
+   "metadata": {},
+   "source": [
+    "ここで、これまで作成したゲートを用いて、量子ウォークによる探索アルゴリズムの全体を実装します。最初は、ステップ 1 の頂点とコインの量子ビットにアダマールゲートを作用させます。その後、ステップ 2(a) と 2(b) の位相オラクルゲートと位相推定ゲートを繰り返し作用させます。4 章でアルゴリズムを説明したように、 $O(1/\\sqrt{\\epsilon})$  回の繰り返しが必要です。最後に頂点量子ビットを測定します。"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 12,
+   "id": "55c1c857",
+   "metadata": {},
+   "outputs": [
+    {
+     "data": {
+      "image/svg+xml": [
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+      ],
+      "text/plain": [
+       "<Figure size 1306.69x746.48 with 1 Axes>"
+      ]
+     },
+     "execution_count": 12,
+     "metadata": {},
+     "output_type": "execute_result"
+    }
+   ],
+   "source": [
+    "# Implementation of the full quantum walk search algorithm\n",
+    "theta_q = QuantumRegister(4, 'theta')\n",
+    "node_q = QuantumRegister(4, 'node')\n",
+    "coin_q = QuantumRegister(2, 'coin')\n",
+    "auxiliary_q = QuantumRegister(1, 'auxiliary')\n",
+    "creg_c2 = ClassicalRegister(4, 'c')\n",
+    "circuit = QuantumCircuit(theta_q, node_q, coin_q, auxiliary_q, creg_c2)\n",
+    "# Apply Hadamard gates to the qubits that represent the nodes and the coin\n",
+    "circuit.h([4,5,6,7,8,9])\n",
+    "iterations = 2\n",
+    "\n",
+    "for i in range(0,iterations):\n",
+    "    circuit.append(phase_oracle_gate, [4,5,6,7,8,9])\n",
+    "    circuit.append(phase_estimation_gate, [0,1,2,3,4,5,6,7,8,9,10])\n",
+    "\n",
+    "circuit.measure(node_q[0], creg_c2[0])\n",
+    "circuit.measure(node_q[1], creg_c2[1])\n",
+    "circuit.measure(node_q[2], creg_c2[2])\n",
+    "circuit.measure(node_q[3], creg_c2[3])\n",
+    "circuit.draw()"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "38ec1434",
+   "metadata": {},
+   "source": [
+    "最後に、この実装を qasm シミュレータで実行します。マークした状態へとはっきりと集まっていることが分かります。"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 13,
+   "id": "659702ef",
+   "metadata": {},
+   "outputs": [
+    {
+     "data": {
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+      ],
+      "text/plain": [
+       "<Figure size 504x360 with 1 Axes>"
+      ]
+     },
+     "execution_count": 13,
+     "metadata": {},
+     "output_type": "execute_result"
+    }
+   ],
+   "source": [
+    "backend = Aer.get_backend('qasm_simulator') \n",
+    "job = execute( circuit, backend, shots=1024 ) \n",
+    "hist = job.result().get_counts() \n",
+    "plot_histogram( hist )"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "d055c293",
+   "metadata": {},
+   "source": [
+    "## 6. 参考文献 <a id='references'></a>\n",
+    "\n",
+    "1. Renato Portugal. Quantum Walks and Search Algorithms. New York, NY: Springer New York, 2013\n",
+    "2. Markus G. Kuhn.Some Introductory Notes on Quantum Computing. Apr. 2000\n",
+    "3. Thomas G. Wong. “Equivalence of Szegedy’s and coined quantum walks”. In: Quantum InformationProcessing 16.9 (July 2017). ISSN: 1573-1332. DOI:10.1007/s11128-017-1667-y. URL:http://dx.doi.org/10.1007/s11128-017-1667-y.37\n",
+    "4. Ronald de Wolf. Quantum Computing: Lecture Notes. 2021. arXiv:1907.09415 [quant-ph]"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 14,
+   "id": "97872b22",
+   "metadata": {},
+   "outputs": [
+    {
+     "name": "stderr",
+     "output_type": "stream",
+     "text": [
+      "/usr/local/anaconda3/lib/python3.7/site-packages/qiskit/aqua/__init__.py:86: DeprecationWarning: The package qiskit.aqua is deprecated. It was moved/refactored to qiskit-terra For more information see <https://github.com/Qiskit/qiskit-aqua/blob/main/README.md#migration-guide>\n",
+      "  warn_package('aqua', 'qiskit-terra')\n"
+     ]
+    },
+    {
+     "data": {
+      "text/html": [
+       "<h3>Version Information</h3><table><tr><th>Qiskit Software</th><th>Version</th></tr><tr><td><code>qiskit-terra</code></td><td>0.18.0</td></tr><tr><td><code>qiskit-aer</code></td><td>0.8.2</td></tr><tr><td><code>qiskit-ignis</code></td><td>0.6.0</td></tr><tr><td><code>qiskit-ibmq-provider</code></td><td>0.15.0</td></tr><tr><td><code>qiskit-aqua</code></td><td>0.9.4</td></tr><tr><td><code>qiskit</code></td><td>0.28.0</td></tr><tr><th>System information</th></tr><tr><td>Python</td><td>3.7.7 (default, May  6 2020, 04:59:01) \n",
+       "[Clang 4.0.1 (tags/RELEASE_401/final)]</td></tr><tr><td>OS</td><td>Darwin</td></tr><tr><td>CPUs</td><td>8</td></tr><tr><td>Memory (Gb)</td><td>32.0</td></tr><tr><td colspan='2'>Tue Oct 05 13:51:31 2021 BST</td></tr></table>"
+      ],
+      "text/plain": [
+       "<IPython.core.display.HTML object>"
+      ]
+     },
+     "metadata": {},
+     "output_type": "display_data"
+    }
+   ],
+   "source": [
+    "import qiskit.tools.jupyter\n",
+    "%qiskit_version_table"
+   ]
+  }
+ ],
+ "metadata": {
+  "kernelspec": {
+   "display_name": "Python 3",
+   "language": "python",
+   "name": "python3"
+  },
+  "language_info": {
+   "codemirror_mode": {
+    "name": "ipython",
+    "version": 3
+   },
+   "file_extension": ".py",
+   "mimetype": "text/x-python",
+   "name": "python",
+   "nbconvert_exporter": "python",
+   "pygments_lexer": "ipython3",
+   "version": "3.9"
+  }
+ },
+ "nbformat": 4,
+ "nbformat_minor": 5
+}
diff --git a/translations/ja/v2/quantum-hardware/error-correction-repetition-code.ipynb b/translations/ja/v2/quantum-hardware/error-correction-repetition-code.ipynb
index e85349740..a066fe961 100755
--- a/translations/ja/v2/quantum-hardware/error-correction-repetition-code.ipynb
+++ b/translations/ja/v2/quantum-hardware/error-correction-repetition-code.ipynb
@@ -157,7 +157,7 @@
      "name": "stdout",
      "output_type": "stream",
      "text": [
-      "{'000': 992, '001': 6, '010': 18, '100': 8}\n"
+      "{'001': 6, '100': 10, '010': 11, '000': 997}\n"
      ]
     }
    ],
@@ -174,6 +174,20 @@
     "print(counts)"
    ]
   },
+  {
+   "cell_type": "code",
+   "execution_count": 5,
+   "metadata": {
+    "tags": [
+     "sanity-check"
+    ]
+   },
+   "outputs": [],
+   "source": [
+    "# ...almost all results still come out '000',...\n",
+    "assert counts['000'] > sum(counts.values())*0.95"
+   ]
+  },
   {
    "cell_type": "markdown",
    "metadata": {},
@@ -185,14 +199,14 @@
   },
   {
    "cell_type": "code",
-   "execution_count": 5,
+   "execution_count": 6,
    "metadata": {},
    "outputs": [
     {
      "name": "stdout",
      "output_type": "stream",
      "text": [
-      "{'011': 16, '101': 13, '110': 16, '111': 979}\n"
+      "{'001': 1, '110': 16, '101': 19, '011': 9, '111': 979}\n"
      ]
     }
    ],
@@ -208,6 +222,21 @@
     "print(counts)"
    ]
   },
+  {
+   "cell_type": "code",
+   "execution_count": 7,
+   "metadata": {
+    "tags": [
+     "sanity-check"
+    ]
+   },
+   "outputs": [],
+   "source": [
+    "# ...number of samples that come out with a majority in the wrong state\n",
+    "# (0 in this case) is again much less than 10,...\n",
+    "assert counts.get('000', 0) < 10"
+   ]
+  },
   {
    "cell_type": "markdown",
    "metadata": {},
@@ -219,14 +248,14 @@
   },
   {
    "cell_type": "code",
-   "execution_count": 6,
+   "execution_count": 8,
    "metadata": {},
    "outputs": [
     {
      "name": "stdout",
      "output_type": "stream",
      "text": [
-      "{'000': 127, '001': 123, '010': 133, '011': 152, '100': 117, '101': 122, '110': 118, '111': 132}\n"
+      "{'100': 124, '001': 126, '110': 119, '101': 133, '010': 117, '000': 127, '011': 139, '111': 139}\n"
      ]
     }
    ],
@@ -236,6 +265,21 @@
     "print(counts)"
    ]
   },
+  {
+   "cell_type": "code",
+   "execution_count": 9,
+   "metadata": {
+    "tags": [
+     "sanity-check"
+    ]
+   },
+   "outputs": [],
+   "source": [
+    "# ...all outcomes occur with equal probability, with\n",
+    "# differences in results being due only to statistical noise.\n",
+    "assert max(counts.values()) < 200"
+   ]
+  },
   {
    "cell_type": "markdown",
    "metadata": {},
@@ -280,17 +324,19 @@
   },
   {
    "cell_type": "code",
-   "execution_count": 7,
+   "execution_count": 10,
    "metadata": {},
    "outputs": [
     {
      "data": {
-      "image/png": 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+      ],
       "text/plain": [
-       "<Figure size 382.292x264.88 with 1 Axes>"
+       "<Figure size 530.961x367.889 with 1 Axes>"
       ]
      },
-     "execution_count": 7,
+     "execution_count": 10,
      "metadata": {},
      "output_type": "execute_result"
     }
@@ -298,13 +344,13 @@
    "source": [
     "from qiskit import QuantumRegister, ClassicalRegister\n",
     "\n",
-    "cq = QuantumRegister(2,'code_qubit')\n",
-    "lq = QuantumRegister(1,'ancilla_qubit')\n",
-    "sb = ClassicalRegister(1,'syndrome_bit')\n",
-    "qc = QuantumCircuit(cq,lq,sb)\n",
-    "qc.cx(cq[0],lq[0])\n",
-    "qc.cx(cq[1],lq[0])\n",
-    "qc.measure(lq,sb)\n",
+    "cq = QuantumRegister(2, 'code_qubit')\n",
+    "lq = QuantumRegister(1, 'ancilla_qubit')\n",
+    "sb = ClassicalRegister(1, 'syndrome_bit')\n",
+    "qc = QuantumCircuit(cq, lq, sb)\n",
+    "qc.cx(cq[0], lq[0])\n",
+    "qc.cx(cq[1], lq[0])\n",
+    "qc.measure(lq, sb)\n",
     "qc.draw()"
    ]
   },
@@ -319,30 +365,32 @@
   },
   {
    "cell_type": "code",
-   "execution_count": 8,
+   "execution_count": 11,
    "metadata": {},
    "outputs": [
     {
      "data": {
-      "image/png": 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+      ],
       "text/plain": [
-       "<Figure size 382.292x264.88 with 1 Axes>"
+       "<Figure size 530.961x367.889 with 1 Axes>"
       ]
      },
-     "execution_count": 8,
+     "execution_count": 11,
      "metadata": {},
      "output_type": "execute_result"
     }
    ],
    "source": [
-    "qc_init = QuantumCircuit(cq)\n",
+    "qc_init = QuantumCircuit(cq, lq, sb)\n",
     "\n",
-    "(qc_init+qc).draw()"
+    "qc_init.compose(qc).draw()"
    ]
   },
   {
    "cell_type": "code",
-   "execution_count": 9,
+   "execution_count": 12,
    "metadata": {},
    "outputs": [
     {
@@ -354,8 +402,22 @@
     }
    ],
    "source": [
-    "counts = execute( qc_init+qc, Aer.get_backend('qasm_simulator')).result().get_counts()\n",
-    "print('Results:',counts)"
+    "counts = execute(qc_init.compose(qc), Aer.get_backend('qasm_simulator')).result().get_counts()\n",
+    "print('Results:', counts)"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 13,
+   "metadata": {
+    "tags": [
+     "sanity-check"
+    ]
+   },
+   "outputs": [],
+   "source": [
+    "# The outcome, in all cases, is 0.\n",
+    "assert list(counts.keys()) == ['0']"
    ]
   },
   {
@@ -369,31 +431,33 @@
   },
   {
    "cell_type": "code",
-   "execution_count": 10,
+   "execution_count": 14,
    "metadata": {},
    "outputs": [
     {
      "data": {
-      "image/png": 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9ezY3b9406UOr1RITE4OPjw9OTk4EBgaSnp7OsWPHCAgIMDvnJ598QmBgILVq1cLd3Z2IiAjy8sr3KSitVsvy5ctp1aoVTk5OdO7cmYSEBNq0acOUKVMAKC4uxtnZmejoaJNji4qKcHJyYunSpca2pKQkAgIC2Lp1K35+fjg7O+Pn58fBgwfNzt29e3dGjRoFQMeOHZkxYwY6nY66deuiUqno0aOHVdegVlfJ/yUA0N27x72pMyhe85FJe3HsF9x7YTy6W7dsVFn1k5wF/9gFW47Av9Ph6zR49wAs3wfXbpZ5uHiAKvfbFRUVxfDhw/Hy8mL9+vXs3LmTYcOGkZeXh1arJSwsjLlz5zJy5Ej27NnDa6+9xsaNG5k4caJJP5GRkSxcuJApU6awb98+QkJCGDZsGGlpaWbTCTNnziQyMpIePXoQGxvL4sWL2b9/P2PHji1X7ZGRkbz11lu8+OKL7Nu3j1GjRjFmzBjOnTtnPGdGRga3b9+mS5cuJseeOnWKu3fvmrQnJSWRnp7OunXriI6O5tNPPwUgNDSU7Oxs437FxcWcPHnSeOyGDRto27YtwcHBHDlyhCNHjvDxxx+X61qqIpWDA/avvow2bi/a4ykA6M5noV2/Cbt5c1HVqmXbAquJtIuw+Xu4fc/8uf9eh/cOwM07ytdVHVSpKYVdu3axYMEC1q1bR2RkpLF90KBBALz33nts376d+Ph4goKCAOjbty9Xr14lKiqKNWvW4OrqypYtW9i0aROHDh3iqaeeMu534sQJzp49axK427ZtY9WqVWzevJnw8HBje82aNRkzZgxZWVk0b968zNq3bt3Kpk2biI+PN8599u3bl+TkZHbt2mU8Z3JyMoBZ4Bpubhnac3JyuHTpEkFBQXz99dfY2dkB4OPjg6+vL3FxcUybNg2A06dPm4R4ly5duHDhAmFhYVaPbB8VqubNUEeOp3jZClSrVlK0JAZ16FDUnTraurRqQaeDL5NBBegsPQ/8VgjfZcKgTgoXVw1UqRFuVFQUffv2NQnb+61YsYIhQ4YYw9agZcuWgP4OO8CiRYsYOHCgMWwN2rVrB5jeMHvzzTfx9/dnzJgxFBUVGbf27fXfJHX+/Hmral+8eDEhISFmN5p8fHxwcHCgY0d9ICQnJ6PRaGjSpInJfsnJyXh6etKgQQNAP7oFWLhwoTFsAdq3b4+9vT1XrlwxthnCumvXroB+FF1YWGh8XN2oh4eiataUohdfArUa9fjwsg8SVsm6Cr8WWA7b+/14RpFyqp0qE7jZ2dmkp6czevRoi8+fPXuWrKwshgwZYvbchQsXAGjcuDEXL14kPT3dOJ95v5ycHJo0aWK8YZadnU1GRgZJSUk4ODiYbJ07dwagbt26ZdZ+8eJF0tLSGDlypMXafH19qVGjBqAP1pKjW9CHZsnpBHd3d7MXl6tXr1JUVISXl5fJsR4eHsbrMoS1pfMoQaVSWbXFx8c/fP+dOkJ+PurgfqgcHMrdR3x8vNV1VtRW3p+PLbaBQ81/byzJv121r8NWf7dlqTJTCjk5OQB4enpafP7XX38F9KFa0oEDB/Dz80Oj0fDjjz8C0KhRI5N9bt26RVxcHD179jQ755o1ax44GjSMTEtjGFm7u7ubtN++fZuEhASefvppY1tqaqrxBprBjRs3OHbsGPPmzTO2JSYmWrzWbdu2oVarGTBggLGtZIgnJyfj7u5u8fjqQHf+PNqtn6J+fiTaT7aiDgpE9QisOnkU3C3Mt2q/32/b5nuMH3VVZoTr4eEB8MBF9oZ51F9++cWkfe/evRw6dIg5c+YAUL9+fUB/E+p+y5Yt48qVKyYrFAyjRHt7ewICAixuDlaMngznzMzMNGmPiYkhNzfXGOYFBQXk5+ebjE4BVq5cafGGWVZWFnfu/HF34vr16yxZsoSwsDCTKYmUlBSTF4z09HTjlIgt6HQ6q7Y+ffqUv+/f71G0ZBnqEcOxmxiBKvAJimPeRqfVlqufPn36WF1nRW3l/fnYYjt7/CucHUv/2amA3h1dbF5rVdqsVWVGuE2bNiU4ONg4Z+nv709eXh4HDhwgLCyMoKAghg0bxsKFC6lTpw4tW7bk22+/ZenSpbz44ovGG17e3t506tSJxYsX4+bmRsuWLYmNjWXDhg2A6fytl5cXISEhzJ07l7y8PPz9/bl79y7Z2dns3buXtWvXmqxtfRDDORctWkS9evXw9PTk888/Z+/evSbndHFxwcPDgx07djB8+HB0Oh0bN27kgw8+AExvmF2+fJkWLVowduxYpk2bxtWrV1mwYAEuLi68++67xnOfO3eO/Px8k7B2c3MjISGBPXv2oNFoaN68udnouzSff/458McHLw4cOEBGRga1atUy3sC0Fe36Dajs7VGHhwFgN30qRVNnoN0Zi93IZ21aW3VgbwdPtYe4FMvPqwC1Gp5so2RV1YdKV554rmR5eXm88cYbxMXFkZubi0ajITAwkNWrV6PRaMjPz2fevHnExcXx22+/0b59e6ZPn05ERIRJP5mZmUydOpWffvoJZ2dnBgwYQLt27YiKiuLKlSvGG1MAv/32GwsWLOCLL77g4sWLuLi44OPjQ0hICP/4xz+srj0zM5MXX3yRo0ePotFoGD9+PC4uLvz973/nxo0b1KxZE4B///vfzJgxg3PnztGsWTMiIiK4ceMGH374IdevXwf065BDQ0NJSUlh8eLFxMXFUbNmTUaMGMHChQuNI2qAnTt38txzz3H+/Hnju4CMjAwiIyM5fvw4d+7cITY2luHDh1t9LQ+ak2rWrBlZWVlW91OW4MSfyvUvPmiPp1D8jzexf/9dVE3+eJegTTtF8f++jv27K1C1aGFVX0+61eNgQPdy1/xnzN6i/+/KMEVPW25aHew6Bod/MV+tYK+GiCfB1/LMnyhDlQrc6iY8PJzU1FROnDhh61KqpPIGbkWSwC3bhWtwOBOOntM/froT9PCBOjVtW9ejrMrM4VZHiYmJj8R3NghhSVMNjH3ij8cDOkrY/lkSuFa4f32upc3Sm4SbN2+SmZlZpdbCPsx1CCEqjgRuGbKysszW6JbcNm3aZHZc7dq1KS4uZubMmTao2tzDXocQouJUmVUKVZWHh4fxKx0fpIWVN2psqbpchxCPMgncMjg6Olr8drFHTXW5DiEeZTKlIIQQCpHAFUIIhUjgCiGEQiRwhRBCIXLTTNhMZxeXx/Lc4vElgSts5u02tvtGMyFsQaYUhBBCIRK4QgihEAlcIYRQiASuEEIoRAJXCCEUIoErhBAKkcAVQgiFSOAKIYRCJHCFEEIhErhCCKEQCVwhhFCIBK4QQihEAlcIIRQi3xYmhKiyZs+eTUpKiuLn9fPzY+XKlRXerwSuEKLKSklJISEhwdZlVBiZUhBCCIVI4AohhEIkcIVQwM07f/z5wjW4W2S7WqxVrIX/Xv/jcd5N0OlsV091IHO4QlSSKzfg+1/gxAW4XvhH+4r9oFKBe13o1gL+4g21nWxX5/1+L4KUC/DjGfjPNX3oGrz5JdSqAW0bQ2AraNFAfx3CehK4QlSwO/dgdzL8cObB++h0kPsb7D4O+07AED8IagNqGwbYqRz47CfIv/3gfW7dhaQs/dbaHUZ3h3q1larw0SdTCkJUoMv5ELOn9LAt6V4xxCbB6n/rw1ppWh3sSoSP4ksP25IyL8HSPfqgFtapVoF78uRJVCoV+/btA+DcuXOoVCp27Nhh3MdSW1WVm5uLSqVi27Ztpe5X8roBCgsLmT9/PsePHy/XOfPz85k6dSoNGzbE2dmZwMBADh8+/FD1P25+vQHvHYC8Ww93fOYlfegqOb+r08GOo/Dtzw93/N0iWJcA6Y9w6NatW1exc1WrwPXw8ODIkSP069cPgKSkJAD8/f2N+1hqq6oSExMB6NatW6n7lbxu0K9fjI6OpqCgwOrz6XQ6QkNDiY2NZfny5ezevZv69evTv3//cgf346ZYCxsPw827D95nZZh+K03WVf10hFKOnoMjZYzGy6pbq4PN38NvhQ/eRwnt2rXjlVde4bPPPiMpKYkTJ07w3XffsWrVKl544QVq1apldkyXLl04c+YMERERitRYreZwNRoNGo3G+DgpKQk3Nze8vb1LbauqEhMTcXNzw8fHp9T9Sl43wPHjx1GpVPj5+Vl9vri4OBISEti7dy+DBg0C4Mknn8TX15e///3v7N27t9zX8Lg4lA4518vezxrf/wJ+zaBVo4rp70Hyb+unMirCnXv6kfLkPhXTX3kEBgayYMEC+vSxfPJevXrx0ksvkZ+fz7p165g/fz4FBQV06dKFgwcPUq9ePYYMGcKGDRsqvdZKG+HGx8czfPhwmjRpgpOTE56enkyaNIn8/HzjPv7+/owZM4adO3fSvXt3nJ2dadGiBdu3b7fY5+7duwkODsbV1RVnZ2c6duzI+vXrjc93796dUaNGGR8nJSWZjWRLtllTpzW0Wi3Lly+nVatWODk50blzZxISEmjTpg1Tpkwx7ldcXIyzszPR0dEmxxcVFeHk5MTSpUtNag0ICGDr1q34+fnh7OyMn58fBw8eNDm25HV37NiRGTNmoNPpqFu3LiqVih49epR5DV9++SUajYaQkBBjm6OjI6NHj+bAgQPcuvWQ75Wrud+L4JvTFdvnwVMV258l32dW7JzxqRzTZWSVzcHBgRUrVvDtt9/Sp08fCgoKWLt2LRMmTKBbt2507tyZ4OBg5s2bx/fff0/dunWZM2cOaWlpTJ482Ri2X3zxBaNHj1ak5kob4aamptKrVy8mT56Mi4sLp0+f5vXXXwdg7dq1FBUVcerUKa5fv05ubi5z586lTp06vPHGG0yYMIFBgwZRp04dY39RUVEsXLiQcePGMX36dGrWrMnhw4fJy8sD9EF28uRJhg8fbjwmOTmZyZMnm9RVsq2sOq0VGRlJbGwsUVFR+Pv788MPPzBmzBh+/fVX5syZY9wvIyOD27dv06VLF5PjT506xd27d03ak5KSsLOzY92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+      ],
       "text/plain": [
-       "<Figure size 442.492x264.88 with 1 Axes>"
+       "<Figure size 614.573x367.889 with 1 Axes>"
       ]
      },
-     "execution_count": 10,
+     "execution_count": 14,
      "metadata": {},
      "output_type": "execute_result"
     }
    ],
    "source": [
-    "qc_init = QuantumCircuit(cq)\n",
+    "qc_init = QuantumCircuit(cq, lq, sb)\n",
     "qc_init.x(cq)\n",
     "\n",
-    "(qc_init+qc).draw()"
+    "qc_init.compose(qc).draw()"
    ]
   },
   {
    "cell_type": "code",
-   "execution_count": 11,
+   "execution_count": 15,
    "metadata": {},
    "outputs": [
     {
@@ -405,8 +469,22 @@
     }
    ],
    "source": [
-    "counts = execute(qc_init+qc, Aer.get_backend('qasm_simulator')).result().get_counts()\n",
-    "print('Results:',counts)"
+    "counts = execute(qc_init.compose(qc), Aer.get_backend('qasm_simulator')).result().get_counts()\n",
+    "print('Results:', counts)"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 16,
+   "metadata": {
+    "tags": [
+     "sanity-check"
+    ]
+   },
+   "outputs": [],
+   "source": [
+    "# The outcome in this case is also always 0.\n",
+    "assert list(counts.keys()) == ['0']"
    ]
   },
   {
@@ -418,32 +496,34 @@
   },
   {
    "cell_type": "code",
-   "execution_count": 12,
+   "execution_count": 17,
    "metadata": {},
    "outputs": [
     {
      "data": {
-      "image/png": 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+      ],
       "text/plain": [
-       "<Figure size 502.692x264.88 with 1 Axes>"
+       "<Figure size 698.184x367.889 with 1 Axes>"
       ]
      },
-     "execution_count": 12,
+     "execution_count": 17,
      "metadata": {},
      "output_type": "execute_result"
     }
    ],
    "source": [
-    "qc_init = QuantumCircuit(cq)\n",
+    "qc_init = QuantumCircuit(cq, lq, sb)\n",
     "qc_init.h(cq[0])\n",
-    "qc_init.cx(cq[0],cq[1])\n",
+    "qc_init.cx(cq[0], cq[1])\n",
     "\n",
-    "(qc_init+qc).draw()"
+    "qc_init.compose(qc).draw()"
    ]
   },
   {
    "cell_type": "code",
-   "execution_count": 13,
+   "execution_count": 18,
    "metadata": {},
    "outputs": [
     {
@@ -455,8 +535,23 @@
     }
    ],
    "source": [
-    "counts = execute(qc_init+qc, Aer.get_backend('qasm_simulator')).result().get_counts()\n",
-    "print('Results:',counts)"
+    "counts = execute(qc_init.compose(qc), Aer.get_backend('qasm_simulator')).result().get_counts()\n",
+    "print('Results:', counts)"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 19,
+   "metadata": {
+    "tags": [
+     "sanity-check"
+    ]
+   },
+   "outputs": [],
+   "source": [
+    "# ...we can expect the same to be true also for\n",
+    "# any superposition of |00⟩ and |11⟩...\n",
+    "assert list(counts.keys()) == ['0']"
    ]
   },
   {
@@ -468,33 +563,35 @@
   },
   {
    "cell_type": "code",
-   "execution_count": 14,
+   "execution_count": 20,
    "metadata": {},
    "outputs": [
     {
      "data": {
-      "image/png": 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+      ],
       "text/plain": [
-       "<Figure size 562.892x264.88 with 1 Axes>"
+       "<Figure size 781.795x367.889 with 1 Axes>"
       ]
      },
-     "execution_count": 14,
+     "execution_count": 20,
      "metadata": {},
      "output_type": "execute_result"
     }
    ],
    "source": [
-    "qc_init = QuantumCircuit(cq)\n",
+    "qc_init = QuantumCircuit(cq, lq, sb)\n",
     "qc_init.h(cq[0])\n",
     "qc_init.cx(cq[0],cq[1])\n",
     "qc_init.x(cq[0])\n",
     "\n",
-    "(qc_init+qc).draw()"
+    "qc_init.compose(qc).draw()"
    ]
   },
   {
    "cell_type": "code",
-   "execution_count": 15,
+   "execution_count": 21,
    "metadata": {},
    "outputs": [
     {
@@ -506,10 +603,24 @@
     }
    ],
    "source": [
-    "counts = execute(qc_init+qc, Aer.get_backend('qasm_simulator')).result().get_counts()\n",
+    "counts = execute(qc_init.compose(qc), Aer.get_backend('qasm_simulator')).result().get_counts()\n",
     "print('Results:',counts)"
    ]
   },
+  {
+   "cell_type": "code",
+   "execution_count": 22,
+   "metadata": {
+    "tags": [
+     "sanity-check"
+    ]
+   },
+   "outputs": [],
+   "source": [
+    "# In such cases the output is always '1'.\n",
+    "assert list(counts.keys()) == ['1']"
+   ]
+  },
   {
    "cell_type": "markdown",
    "metadata": {},
@@ -542,7 +653,7 @@
   },
   {
    "cell_type": "code",
-   "execution_count": 16,
+   "execution_count": 23,
    "metadata": {},
    "outputs": [],
    "source": [
@@ -560,14 +671,14 @@
   },
   {
    "cell_type": "code",
-   "execution_count": 17,
+   "execution_count": 24,
    "metadata": {},
    "outputs": [],
    "source": [
     "n = 3\n",
     "T = 1\n",
     "\n",
-    "code = RepetitionCode(n,T)"
+    "code = RepetitionCode(n, T)"
    ]
   },
   {
@@ -586,17 +697,19 @@
   },
   {
    "cell_type": "code",
-   "execution_count": 18,
+   "execution_count": 25,
    "metadata": {},
    "outputs": [
     {
      "data": {
-      "image/png": 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+      ],
       "text/plain": [
-       "<Figure size 771.475x445.48 with 1 Axes>"
+       "<Figure size 1071.49x618.722 with 1 Axes>"
       ]
      },
-     "execution_count": 18,
+     "execution_count": 25,
      "metadata": {},
      "output_type": "execute_result"
     }
@@ -607,17 +720,19 @@
   },
   {
    "cell_type": "code",
-   "execution_count": 19,
+   "execution_count": 26,
    "metadata": {},
    "outputs": [
     {
      "data": {
-      "image/png": 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+      ],
       "text/plain": [
-       "<Figure size 831.675x445.48 with 1 Axes>"
+       "<Figure size 1155.1x618.722 with 1 Axes>"
       ]
      },
-     "execution_count": 19,
+     "execution_count": 26,
      "metadata": {},
      "output_type": "execute_result"
     }
@@ -639,7 +754,7 @@
   },
   {
    "cell_type": "code",
-   "execution_count": 20,
+   "execution_count": 27,
    "metadata": {
     "scrolled": false
    },
@@ -648,16 +763,13 @@
      "name": "stdout",
      "output_type": "stream",
      "text": [
-      "Logical 0 : {'000 00': 1024} \n",
-      "\n",
-      "Logical 1 : {'111 00': 1024} \n",
-      "\n"
+      "Logical 0: {'000 00': 1024}\n",
+      "Logical 1: {'111 00': 1024}\n"
      ]
     }
    ],
    "source": [
     "def get_raw_results(code,noise_model=None):\n",
-    "\n",
     "    circuits = code.get_circuit_list()\n",
     "    raw_results = {}\n",
     "    for log in range(2):\n",
@@ -667,7 +779,7 @@
     "\n",
     "raw_results = get_raw_results(code)\n",
     "for log in raw_results:\n",
-    "    print('Logical',log,':',raw_results[log],'\\n')"
+    "    print(f'Logical {log}: {raw_results[log]}')"
    ]
   },
   {
@@ -681,26 +793,24 @@
   },
   {
    "cell_type": "code",
-   "execution_count": 21,
+   "execution_count": 28,
    "metadata": {},
    "outputs": [
     {
      "name": "stdout",
      "output_type": "stream",
      "text": [
-      "Logical 0 : {'000 00 00 00 00': 1024} \n",
-      "\n",
-      "Logical 1 : {'111 00 00 00 00': 1024} \n",
-      "\n"
+      "Logical 0: {'000 00 00 00 00': 1024}\n",
+      "Logical 1: {'111 00 00 00 00': 1024}\n"
      ]
     }
    ],
    "source": [
-    "code = RepetitionCode(n,4)\n",
+    "code = RepetitionCode(n, 4)\n",
     "\n",
     "raw_results = get_raw_results(code)\n",
     "for log in raw_results:\n",
-    "    print('Logical',log,':',raw_results[log],'\\n')"
+    "    print(f'Logical {log}: {raw_results[log]}')"
    ]
   },
   {
@@ -712,26 +822,24 @@
   },
   {
    "cell_type": "code",
-   "execution_count": 22,
+   "execution_count": 29,
    "metadata": {},
    "outputs": [
     {
      "name": "stdout",
      "output_type": "stream",
      "text": [
-      "Logical 0 : {'00000 0000 0000 0000 0000': 1024} \n",
-      "\n",
-      "Logical 1 : {'11111 0000 0000 0000 0000': 1024} \n",
-      "\n"
+      "Logical 0: {'00000 0000 0000 0000 0000': 1024}\n",
+      "Logical 1: {'11111 0000 0000 0000 0000': 1024}\n"
      ]
     }
    ],
    "source": [
-    "code = RepetitionCode(5,4)\n",
+    "code = RepetitionCode(5, 4)\n",
     "\n",
     "raw_results = get_raw_results(code)\n",
     "for log in raw_results:\n",
-    "    print('Logical',log,':',raw_results[log],'\\n')"
+    "    print(f'Logical {log}: {raw_results[log]}')"
    ]
   },
   {
@@ -745,7 +853,7 @@
   },
   {
    "cell_type": "code",
-   "execution_count": 23,
+   "execution_count": 30,
    "metadata": {
     "scrolled": true
    },
@@ -754,21 +862,21 @@
      "name": "stdout",
      "output_type": "stream",
      "text": [
-      "Logical 0 : {'000 00': 660, '000 01': 75, '100 00': 47, '100 01': 6, '100 10': 4, '101 00': 7, '110 00': 4, '110 01': 2, '111 00': 2, '000 10': 66, '000 11': 3, '001 00': 55, '001 01': 6, '001 10': 2, '010 00': 49, '010 01': 22, '010 10': 6, '010 11': 2, '011 00': 3, '011 01': 2, '011 11': 1} \n",
+      "Logical 0: {'011 11': 1, '010 11': 1, '001 01': 3, '101 00': 6, '100 10': 12, '110 00': 1, '100 00': 49, '110 01': 1, '010 10': 12, '100 01': 5, '010 00': 52, '010 01': 26, '000 00': 650, '000 10': 64, '011 00': 5, '001 00': 52, '000 11': 6, '000 01': 74, '001 10': 4}\n",
       "\n",
-      "Logical 1 : {'100 00': 3, '100 01': 1, '100 10': 2, '100 11': 1, '101 00': 55, '101 01': 20, '101 10': 5, '101 11': 15, '110 00': 56, '110 01': 21, '110 10': 7, '110 11': 3, '111 00': 613, '111 01': 61, '111 10': 65, '111 11': 7, '001 00': 3, '001 01': 3, '001 10': 1, '001 11': 2, '010 00': 3, '010 01': 2, '010 10': 4, '011 00': 34, '011 01': 10, '011 10': 24, '011 11': 3} \n",
+      "Logical 1: {'011 11': 2, '100 10': 1, '010 10': 1, '101 10': 9, '110 11': 1, '000 11': 1, '001 11': 1, '101 00': 47, '100 01': 3, '010 00': 4, '101 01': 31, '001 01': 1, '110 00': 40, '110 01': 24, '100 00': 5, '001 10': 2, '111 01': 74, '001 00': 3, '011 00': 59, '100 11': 7, '011 10': 25, '111 00': 574, '011 01': 5, '101 11': 20, '010 11': 4, '110 10': 6, '111 11': 5, '111 10': 69}\n",
       "\n"
      ]
     }
    ],
    "source": [
-    "code = RepetitionCode(3,1)\n",
+    "code = RepetitionCode(3, 1)\n",
     "\n",
-    "noise_model = get_noise(0.05,0.05)\n",
+    "noise_model = get_noise(0.05, 0.05)\n",
     "\n",
     "raw_results = get_raw_results(code,noise_model)\n",
     "for log in raw_results:\n",
-    "    print('Logical',log,':',raw_results[log],'\\n')"
+    "    print(f'Logical {log}: {raw_results[log]}\\n')"
    ]
   },
   {
@@ -792,7 +900,7 @@
   },
   {
    "cell_type": "code",
-   "execution_count": 24,
+   "execution_count": 31,
    "metadata": {},
    "outputs": [],
    "source": [
@@ -812,20 +920,20 @@
   },
   {
    "cell_type": "code",
-   "execution_count": 25,
+   "execution_count": 32,
    "metadata": {},
    "outputs": [
     {
      "name": "stdout",
      "output_type": "stream",
      "text": [
-      "P = {'0': 0.0194, '1': 0.028}\n"
+      "P = {'0': 0.0219, '1': 0.0253}\n"
      ]
     }
    ],
    "source": [
-    "P = lookuptable_decoding(raw_results,table_results)\n",
-    "print('P =',P)"
+    "P = lookuptable_decoding(raw_results, table_results)\n",
+    "print('P =', P)"
    ]
   },
   {
@@ -850,7 +958,7 @@
   },
   {
    "cell_type": "code",
-   "execution_count": 26,
+   "execution_count": 33,
    "metadata": {},
    "outputs": [
     {
@@ -859,24 +967,24 @@
      "text": [
       "\n",
       "Logical 0:\n",
-      "raw results        {'000 00 00': 476, '000 01 00': 58, '000 10 00': 55}\n",
-      "processed results  {'0 0  00 00 00': 476, '0 0  00 01 01': 58, '0 0  00 10 10': 55}\n",
+      "raw results        {'000 10 00': 55, '000 00 00': 463, '000 01 01': 56, '000 01 00': 52}\n",
+      "processed results  {'0 0  00 10 10': 55, '0 0  00 00 00': 463, '0 0  01 01 00': 56, '0 0  00 01 01': 52}\n",
       "\n",
       "Logical 1:\n",
-      "raw results        {'111 00 00': 427, '111 00 01': 51}\n",
-      "processed results  {'1 1  00 00 00': 427, '1 1  01 01 00': 51}\n"
+      "raw results        {'111 10 00': 53, '111 00 00': 437, '111 01 00': 59, '111 10 10': 54}\n",
+      "processed results  {'1 1  00 10 10': 53, '1 1  00 00 00': 437, '1 1  00 01 01': 59, '1 1  10 10 00': 54}\n"
      ]
     }
    ],
    "source": [
-    "code = RepetitionCode(3,2)\n",
+    "code = RepetitionCode(3, 2)\n",
     "\n",
-    "raw_results = get_raw_results(code,noise_model)\n",
+    "raw_results = get_raw_results(code, noise_model)\n",
     "\n",
-    "results = code.process_results( raw_results )\n",
+    "results = code.process_results(raw_results)\n",
     "\n",
-    "for log in ['0','1']:\n",
-    "    print('\\nLogical ' + log + ':')\n",
+    "for log in ['0', '1']:\n",
+    "    print(f'\\nLogical {log}:')\n",
     "    print('raw results       ', {string:raw_results[log][string] for string in raw_results[log] if raw_results[log][string]>=50 })\n",
     "    print('processed results ', {string:results[log][string] for string in results[log] if results[log][string]>=50 })"
    ]
@@ -943,7 +1051,7 @@
   },
   {
    "cell_type": "code",
-   "execution_count": 27,
+   "execution_count": 34,
    "metadata": {},
    "outputs": [],
    "source": [
@@ -960,7 +1068,7 @@
   },
   {
    "cell_type": "code",
-   "execution_count": 28,
+   "execution_count": 35,
    "metadata": {
     "tags": [
      "uses-hardware"
@@ -982,7 +1090,7 @@
   },
   {
    "cell_type": "code",
-   "execution_count": 29,
+   "execution_count": 36,
    "metadata": {
     "tags": [
      "uses-hardware"
@@ -1019,7 +1127,7 @@
   },
   {
    "cell_type": "code",
-   "execution_count": 30,
+   "execution_count": 37,
    "metadata": {
     "tags": [
      "uses-hardware"
@@ -1042,7 +1150,7 @@
   },
   {
    "cell_type": "code",
-   "execution_count": 31,
+   "execution_count": 38,
    "metadata": {
     "tags": [
      "uses-hardware"
@@ -1068,7 +1176,7 @@
   },
   {
    "cell_type": "code",
-   "execution_count": 32,
+   "execution_count": 39,
    "metadata": {
     "tags": [
      "uses-hardware"
@@ -1094,7 +1202,7 @@
   },
   {
    "cell_type": "code",
-   "execution_count": 33,
+   "execution_count": 40,
    "metadata": {
     "tags": [
      "uses-hardware"
@@ -1122,7 +1230,7 @@
   },
   {
    "cell_type": "code",
-   "execution_count": 34,
+   "execution_count": 41,
    "metadata": {
     "tags": [
      "uses-hardware"
@@ -1152,7 +1260,7 @@
   },
   {
    "cell_type": "code",
-   "execution_count": 35,
+   "execution_count": 42,
    "metadata": {
     "tags": [
      "uses-hardware"
@@ -1177,7 +1285,7 @@
   },
   {
    "cell_type": "code",
-   "execution_count": 36,
+   "execution_count": 43,
    "metadata": {
     "tags": [
      "uses-hardware"
@@ -1200,7 +1308,7 @@
   },
   {
    "cell_type": "code",
-   "execution_count": 37,
+   "execution_count": 44,
    "metadata": {
     "tags": [
      "uses-hardware"
@@ -1223,7 +1331,7 @@
   },
   {
    "cell_type": "code",
-   "execution_count": null,
+   "execution_count": 45,
    "metadata": {
     "tags": [
      "uses-hardware"
@@ -1256,13 +1364,26 @@
   },
   {
    "cell_type": "code",
-   "execution_count": null,
+   "execution_count": 46,
    "metadata": {
     "tags": [
      "uses-hardware"
     ]
    },
-   "outputs": [],
+   "outputs": [
+    {
+     "data": {
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+      ],
+      "text/plain": [
+       "<Figure size 640x480 with 1 Axes>"
+      ]
+     },
+     "metadata": {},
+     "output_type": "display_data"
+    }
+   ],
    "source": [
     "import matplotlib.pyplot as plt\n",
     "import numpy as np\n",
@@ -1299,7 +1420,7 @@
   },
   {
    "cell_type": "code",
-   "execution_count": null,
+   "execution_count": 47,
    "metadata": {
     "tags": [
      "uses-hardware"
@@ -1334,13 +1455,31 @@
   },
   {
    "cell_type": "code",
-   "execution_count": null,
+   "execution_count": 48,
    "metadata": {
     "tags": [
      "uses-hardware"
     ]
    },
-   "outputs": [],
+   "outputs": [
+    {
+     "data": {
+      "text/plain": [
+       "{'count': 29.0,\n",
+       " 'mean': 0.1839624024105741,\n",
+       " 'std': 0.08969281029182916,\n",
+       " 'min': 0.032099523882660116,\n",
+       " '25%': 0.1202076677316294,\n",
+       " '50%': 0.15479808944854537,\n",
+       " '75%': 0.2585266030013643,\n",
+       " 'max': 0.3820573455512858}"
+      ]
+     },
+     "execution_count": 48,
+     "metadata": {},
+     "output_type": "execute_result"
+    }
+   ],
    "source": [
     "import pandas as pd\n",
     "\n",
@@ -1356,13 +1495,31 @@
   },
   {
    "cell_type": "code",
-   "execution_count": null,
+   "execution_count": 49,
    "metadata": {
     "tags": [
      "uses-hardware"
     ]
    },
-   "outputs": [],
+   "outputs": [
+    {
+     "data": {
+      "text/plain": [
+       "{'count': 43.0,\n",
+       " 'mean': 0.03964249408264689,\n",
+       " 'std': 0.02227486941657462,\n",
+       " 'min': 0.01279016878072245,\n",
+       " '25%': 0.025389504940856747,\n",
+       " '50%': 0.03296455263355727,\n",
+       " '75%': 0.046331516695269206,\n",
+       " 'max': 0.11660000000000004}"
+      ]
+     },
+     "execution_count": 49,
+     "metadata": {},
+     "output_type": "execute_result"
+    }
+   ],
    "source": [
     "if step_3:\n",
     "\n",
@@ -1390,11 +1547,22 @@
   },
   {
    "cell_type": "code",
-   "execution_count": null,
+   "execution_count": 50,
    "metadata": {
     "scrolled": true
    },
-   "outputs": [],
+   "outputs": [
+    {
+     "data": {
+      "text/plain": [
+       "{'qiskit-terra': '0.21.2', 'qiskit-aer': '0.10.4', 'qiskit-ignis': None, 'qiskit-ibmq-provider': '0.19.2', 'qiskit': '0.37.2', 'qiskit-nature': '0.4.1', 'qiskit-finance': '0.3.2', 'qiskit-optimization': '0.4.0', 'qiskit-machine-learning': '0.4.0'}"
+      ]
+     },
+     "execution_count": 50,
+     "metadata": {},
+     "output_type": "execute_result"
+    }
+   ],
    "source": [
     "import qiskit\n",
     "qiskit.__qiskit_version__"
@@ -1417,7 +1585,7 @@
    "name": "python",
    "nbconvert_exporter": "python",
    "pygments_lexer": "ipython3",
-   "version": "3.9"
+   "version": "3.8.5"
   }
  },
  "nbformat": 4,
diff --git a/translations/ja/v2/toc.yaml b/translations/ja/v2/toc.yaml
index 23b349ae3..9c91b91e4 100755
--- a/translations/ja/v2/toc.yaml
+++ b/translations/ja/v2/toc.yaml
@@ -98,15 +98,18 @@
     - id: quantum-counting
       uuid: f1a9942c-b519-11ec-b909-0242ac120002
       url: /ch-algorithms/quantum-counting
-    - id: quantum-teleportation
-      uuid: f4f4a090-b519-11ec-b909-0242ac120002
-      url: /ch-algorithms/teleportation
-    - id: superdense-coding
-      uuid: f87cbd42-b519-11ec-b909-0242ac120002
-      url: /ch-algorithms/superdense-coding
+    - id: quantum-walk-search-algorithm
+      uuid: b2ef52cf-0ce4-494a-8142-fd00e0e8330d
+      url: /ch-algorithms/quantum-walk-search-algorithm
     - id: quantum-key-distribution
       uuid: fbafa8c6-b519-11ec-b909-0242ac120002
       url: /ch-algorithms/quantum-key-distribution
+    - id: superdense-coding
+      uuid: f87cbd42-b519-11ec-b909-0242ac120002
+      url: /ch-algorithms/superdense-coding
+    - id: quantum-teleportation
+      uuid: f4f4a090-b519-11ec-b909-0242ac120002
+      url: /ch-algorithms/teleportation
 
 
 - title: 量子アプリケーションのアルゴリズム
@@ -200,10 +203,10 @@
 - title: Appendix(付録)
   url: /ch-appendix
   sections:
-    - id: an-introduction-to-linear-algebra-for-quantum-computing
+    - id: linear_algebra
       uuid: 5d2bb2ca-b51a-11ec-b909-0242ac120002
       url: /ch-appendix/linear_algebra
-    - id: basic-qiskit-syntax
+    - id: qiskit
       uuid: 5fd6b79a-b51a-11ec-b909-0242ac120002
       url: /ch-appendix/qiskit
 
@@ -211,7 +214,7 @@
 - title: ゲームとデモ
   url: /ch-demos
   sections:
-    - id: bonus-level-sandbox
+    - id: hello-qiskit
       uuid: 63a85f0e-b51a-11ec-b909-0242ac120002
       url: /ch-ex/hello-qiskit
     - id: estimating-pi-using-quantum-phase-estimation-algorithm
diff --git a/translations/ko/toc.yaml b/translations/ko/toc.yaml
index 3abd079f3..611a907c0 100644
--- a/translations/ko/toc.yaml
+++ b/translations/ko/toc.yaml
@@ -124,129 +124,3 @@
   title: "\uC591\uC790 \uCEF4\uD4E8\uD305 \uC18C\uAC1C"
   type: learning-path
   url: /v2_introduction
-
-
-#---- Below was added by machine ---
-
-- sections:
-  - id: introduction
-    title: Introduction
-    url: v2/ch-states/introduction
-    uuid: afcc559e-b519-11ec-b909-0242ac120002
-  - id: the-atoms-of-computation
-    title: The Atoms of Computation
-    url: v2/ch-states/atoms-computation
-    uuid: b31c20ee-b519-11ec-b909-0242ac120002
-  - id: representing-qubit-states
-    title: Representing Qubit States
-    url: v2/ch-states/representing-qubit-states
-    uuid: b81717ca-b519-11ec-b909-0242ac120002
-  - id: single-qubit-gates
-    title: Single Qubit Gates
-    url: v2/ch-states/single-qubit-gates
-    uuid: bd4334a4-b519-11ec-b909-0242ac120002
-  - id: the-case-for-quantum-computers
-    title: The Case for Quantum
-    url: v2/ch-states/case-for-quantum
-    uuid: c0bbff62-b519-11ec-b909-0242ac120002
-  title: "\uC591\uC790 \uC0C1\uD0DC\uC640 \uD050\uBE44\uD2B8"
-  url: /v2_ch-states
-- sections:
-  - id: multiple-qubits-and-entangled-states
-    title: Multiple Qubits and Entangled States
-    url: v2/ch-gates/multiple-qubits-entangled-states
-    uuid: c7cfe502-b519-11ec-b909-0242ac120002
-  - id: basic-circuit-identities
-    title: More Circuit Identities
-    url: v2/ch-gates/more-circuit-identities
-    uuid: ce8ce75a-b519-11ec-b909-0242ac120002
-  title: "\uB2E4\uC911 \uD050\uBE44\uD2B8\uC640 \uC5BD\uD798"
-  url: /v2_ch-gates
-- sections:
-  - id: quantum-circuits
-    title: Defining Quantum Circuits
-    url: v2/ch-algorithms/defining-quantum-circuits
-    uuid: d90cbfb6-b519-11ec-b909-0242ac120002
-  - id: deutsch-jozsa-algorithm
-    title: Deutsch-Jozsa Algorithm
-    url: v2/ch-algorithms/deutsch-jozsa
-    uuid: dc9d3aa2-b519-11ec-b909-0242ac120002
-  - id: bernstein-vazirani-algorithm
-    title: Bernstein-Vazirani Algorithm
-    url: v2/ch-algorithms/bernstein-vazirani
-    uuid: df6e5a40-b519-11ec-b909-0242ac120002
-  - id: quantum-fourier-transform
-    title: Quantum Fourier Transform
-    url: v2/ch-algorithms/quantum-fourier-transform
-    uuid: e59ba472-b519-11ec-b909-0242ac120002
-  - id: quantum-phase-estimation
-    title: Quantum Phase Estimation
-    url: v2/ch-algorithms/quantum-phase-estimation
-    uuid: e8ac9ec8-b519-11ec-b909-0242ac120002
-  - id: shors-algorithm
-    title: Shor's Algorithm
-    url: v2/ch-algorithms/shor
-    uuid: ec157cd8-b519-11ec-b909-0242ac120002
-  - id: hidden-shift-problem
-    title: The Hidden Shift Problem
-    url: v2/ch-algorithms/hidden-shift-problem
-    uuid: 56d13a0c-cae7-11ec-9d64-0242ac120002
-  - id: quantum-counting
-    title: Quantum Counting
-    url: v2/ch-algorithms/quantum-counting
-    uuid: f1a9942c-b519-11ec-b909-0242ac120002
-  - id: quantum-teleportation
-    title: Quantum Teleportation
-    url: v2/ch-algorithms/teleportation
-    uuid: f4f4a090-b519-11ec-b909-0242ac120002
-  - id: superdense-coding
-    title: Superdense Coding
-    url: v2/ch-algorithms/superdense-coding
-    uuid: f87cbd42-b519-11ec-b909-0242ac120002
-  - id: quantum-key-distribution
-    title: Quantum Key Distribution
-    url: v2/ch-algorithms/quantum-key-distribution
-    uuid: fbafa8c6-b519-11ec-b909-0242ac120002
-  - id: quantum-walk-search-algorithm
-    title: Quantum Walk Search Algorithm
-    url: v2/ch-algorithms/quantum-walk-search-algorithm
-    uuid: 5083763f-3db8-461f-8e7d-675533924165
-  title: "\uC591\uC790 \uD504\uB85C\uD1A0\uCF5C\uB4E4\uACFC \uC54C\uACE0\uB9AC\uB4EC\
-    \uB4E4"
-  url: /v2_ch-algorithms
-- sections:
-  - id: phase-kickback
-    title: Phase Kickback
-    url: v2/algorithms/phase-kickback
-    uuid: 976e9da6-6c33-4e2f-b7eb-f9588f700f07
-  title: "\uC591\uC790 \uC54C\uACE0\uB9AC\uB4EC"
-  type: learning-path
-  url: /v2_algorithms
-- sections:
-  - id: why-quantum-computing
-    title: Why quantum computing?
-    url: v2/intro/why-quantum-computing
-    uuid: bdb1d662-e0f6-428a-8830-befe6b47f320
-  - id: what-is-quantum
-    title: What is quantum?
-    url: v2/intro/what-is-quantum
-    uuid: 7c765036-aebc-11ec-b909-0242ac120002
-  - id: describing-quantum-computers
-    title: Describing quantum computers
-    url: v2/intro/describing-quantum-computers
-    uuid: 9260a554-aebc-11ec-b909-0242ac120002
-  - id: entangled-states
-    title: Entangled states
-    url: v2/intro/entangled-states
-    uuid: 98aa9032-aebc-11ec-b909-0242ac120002
-  - id: grovers-search-algorithm
-    title: Grovers algorithm
-    url: v2/intro/grover-intro
-    uuid: a2e6f7a2-aebc-11ec-b909-0242ac120002
-  - id: project
-    title: Project
-    url: v2/intro/project
-    uuid: e25c3f90-aebd-11ec-b909-0242ac120002
-  title: "\uC591\uC790 \uCEF4\uD4E8\uD305 \uC18C\uAC1C"
-  type: learning-path
-  url: /v2_introduction
diff --git a/translations/ko/v1/ch-gates/introduction.ipynb b/translations/ko/v1/ch-gates/introduction.ipynb
deleted file mode 100644
index e696daca9..000000000
--- a/translations/ko/v1/ch-gates/introduction.ipynb
+++ /dev/null
@@ -1,52 +0,0 @@
-{
-  "cells": [
-    {
-      "cell_type": "markdown",
-      "metadata": {
-        "tags": [
-          "remove_cell"
-        ]
-      },
-      "source": [
-        "# 소개"
-      ]
-    },
-    {
-      "cell_type": "markdown",
-      "metadata": {
-      },
-      "source": [
-        "단일 큐비트와 단일 큐비트 게이트로 몇가지 흥미로운 효과들을 보았으나, 양자 컴퓨팅의 진정한 힘은 큐비트들 사이에 상호작용으로 실현됩니다. 이 섹션에서 다중 큐비트 게이트를 소개하고 다중 큐비트 시스템의 흥미로운 동작을 살펴볼 것입니다.\n",
-        "\n",
-        "일반적으로, 하드웨어에서 직접 구현할 수 있는 게이트들은 하나 혹은 두 개의 큐비트들에서 동작합니다. 회로에서, 우리는 많은 큐비트에서 동작하는 복잡한 게이트를 사용하는 것을 좋아합니다. 다행히도, 이것은 문제가 되지 않을 것입니다. 하드웨어로 부터 우리가 받은 하나의 큐비트 게이트와 두 개의 큐비트 게이트로, 다른 어떤 게이트도 만들 수 있습니다.\n",
-        "\n",
-        "이 장에서는 먼저 가장 기본적인 다중-큐비트 게이트와 이를 설명하고 분석하는데 사용하는 수학을 소개합니다. 그런 다음 이러한 게이트가 양자 알고리즘을 만드는데 사용될 수 있다는 것을 증명하는 방법을 보여줄 예정입니다. 그런 다음 이 장은 양자 게이트의 소규모 사용을 보는 것으로 마무리됩니다. 예를 들어, 단일 큐비트 및 2개의 큐비트 연산으로 부터 토폴리 같은 3개의 큐비트 게이트를 구축하는 방법을 봅니다.\n",
-        "\n",
-        "![A Toffoli made from single- and two-qubit gates](https://github.com/Qiskit/platypus/blob/main/translations/ko/ch-gates/images/basic3.png?raw=true)\n",
-        "\n",
-        "이것은 3개의 큐비트(q0, q1, q2)를 가진 토폴리입니다. 이 회로 예제에서, q0는 q2와 연결되지만, q1과는 연결되지 않습니다."
-      ]
-    }
-  ],
-  "metadata": {
-    "kernelspec": {
-      "display_name": "Python 3",
-      "language": "python",
-      "name": "python3"
-    },
-    "language_info": {
-      "codemirror_mode": {
-        "name": "ipython",
-        "version": 3
-      },
-      "file_extension": ".py",
-      "mimetype": "text/x-python",
-      "name": "python",
-      "nbconvert_exporter": "python",
-      "pygments_lexer": "ipython3",
-      "version": "3.9"
-    }
-  },
-  "nbformat": 4,
-  "nbformat_minor": 2
-}
diff --git a/translations/pt/toc.yaml b/translations/pt/toc.yaml
index 63a84f2da..faf4e471f 100644
--- a/translations/pt/toc.yaml
+++ b/translations/pt/toc.yaml
@@ -1,57 +1,5 @@
 
 
-#---- Below was added by machine ---
-
-- sections:
-  - id: environment-setup-guide-to-work-with-qiskit-textbook
-    url: v2/ch-prerequisites/setting-the-environment
-    uuid: a5a86ddc-b519-11ec-b909-0242ac120002
-  - id: introduction-to-python-and-jupyter-notebooks
-    url: v2/ch-prerequisites/python-and-jupyter-notebooks
-    uuid: a9f5ea68-b519-11ec-b909-0242ac120002
-  title: "Pr\xE9-requisitos"
-  url: /v2_ch-prerequisites
-- sections:
-  - id: phase-kickback
-    url: v2/algorithms/phase-kickback
-    uuid: 976e9da6-6c33-4e2f-b7eb-f9588f700f07
-  title: "Algoritmos Qu\xE2nticos"
-  type: learning-path
-  url: /v2_algorithms
-- sections:
-  - id: why-quantum-computing
-    url: v2/intro/why-quantum-computing
-    uuid: bdb1d662-e0f6-428a-8830-befe6b47f320
-  - id: the-atoms-of-computation
-    url: v2/intro/atoms-of-computation
-    uuid: 37ef477e-ab8d-11ec-b909-0242ac120002
-  - id: what-is-quantum
-    url: v2/intro/what-is-quantum
-    uuid: 7c765036-aebc-11ec-b909-0242ac120002
-  - id: describing-quantum-computers
-    url: v2/intro/describing-quantum-computers
-    uuid: 9260a554-aebc-11ec-b909-0242ac120002
-  - id: entangled-states
-    url: v2/intro/entangled-states
-    uuid: 98aa9032-aebc-11ec-b909-0242ac120002
-  - id: visualizing-entanglement
-    url: v2/intro/visualizing-entanglement
-    uuid: 9e8ceb9e-aebc-11ec-b909-0242ac120002
-  - id: project
-    url: v2/intro/project
-    uuid: e25c3f90-aebd-11ec-b909-0242ac120002
-  title: "Introdu\xE7\xE3o \xE0 computa\xE7\xE3o qu\xE2ntica"
-  type: learning-path
-  url: /v2_introduction
-- sections:
-  - id: data-encoding
-    url: v2/quantum-machine-learning/encoding
-    uuid: 1d09af66-b51b-11ec-b909-0242ac120002
-  title: "Aprendizado de m\xE1quina qu\xE2ntico"
-  type: learning-path
-  url: /v2_machine-learning
-
-
 #---- Below was added by machine ---
 
 - sections: