01_adiabatic_process.ipynb

{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Simulate Adiabatic Process with Qiskit"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "This article uses qiskit to simulate Adiabatic Process to find an optimizal solution of a Max-Cut problem."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Adiabatic Theorem"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Adiabatic Theorem says that if a system is in a ground state (or highest energy state) and the Hamiltonian changes slowly to a new one, the system will stay on ground state (or highest energy state) of the new Hamiltonian."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Given,\n",
    "$$\n",
    "\\begin{align*}\n",
    "X |+\\rangle &= |+\\rangle  \\\\\n",
    "X |-\\rangle &= -|-\\rangle\n",
    "\\end{align*}\n",
    "$$\n",
    "\n",
    "For a Mixer Hamiltonian $H_M = \\prod_i X_i$,\n",
    "\n",
    "$|+\\dots+\\rangle$ is the corresponding highest energy state."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Define a Cost Hamiltonian $H_C$.  \n",
    "\n",
    "and time vary Hamiltonian where $t\\in[0, T]$\n",
    "\n",
    "$$\n",
    "\\hat{H}(t) = (1 - \\frac{t}{T})H_M + \\frac{t}{T}H_C              \\tag{1}\n",
    "$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Note: use $\\hat{H}$ for hamiltonian to distinguish from Hadamard gate $H$."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "According to Adiabatic Theorem, if applying $\\hat{H}(t)$ to $|+\\dots+\\rangle$ with a big enough $T$, the final state is the highest energy state of $H_C$."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Taking T as a big enough integer, the highest energy state of $H_C$ can be approximated as\n",
    "\n",
    "$$\n",
    "\\hat{H}(T) \\hat{H}(T-1) \\dots \\hat{H}(1) \\hat{H}(0) |+ \\dots + \\rangle      \\tag{2}\n",
    "$$\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Schrodinger equation,\n",
    "\n",
    "$$\n",
    "\\begin{align*}\n",
    "\\hat{H}|\\psi(t)\\rangle &= i \\hbar \\frac{\\partial}{\\partial t} |\\psi(t)\\rangle \\\\\n",
    "|\\psi(t) &= e^{-i \\hat{H} t / \\hbar} | \\psi(0) \\rangle                  \\tag{3}\n",
    "\\end{align*}\n",
    "$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Apply equation (3) to (2), highest energy state $|\\psi \\rangle$ of $H_C$ becomes\n",
    "\n",
    "$$\n",
    "| \\psi \\rangle = e^{-i \\hat{H}(T) / \\hbar} e^{-i \\hat{H}(T-1) / \\hbar} \\dots e^{-i \\hat{H}(1) / \\hbar} e^{-i \\hat{H}(0) / \\hbar} | + \\dots + \\rangle        \\tag{4}\n",
    "$$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {},
   "outputs": [],
   "source": [
    "# setup\n",
    "\n",
    "# General imports\n",
    "import numpy as np\n",
    "\n",
    "# Pre-defined ansatz circuit, operator class and visualization tools\n",
    "import qiskit\n",
    "from qiskit.circuit.library import QAOAAnsatz\n",
    "from qiskit.quantum_info import SparsePauliOp\n",
    "from qiskit.visualization import plot_distribution\n",
    "\n",
    "# Qiskit Runtime\n",
    "from qiskit_ibm_runtime import QiskitRuntimeService, Session\n",
    "from qiskit_ibm_runtime import EstimatorV2 as Estimator\n",
    "from qiskit_ibm_runtime import SamplerV2 as Sampler\n",
    "\n",
    "# rustworkx graph library\n",
    "import rustworkx as rx\n",
    "from rustworkx.visualization import mpl_draw"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Max-Cut example\n",
    "\n",
    "5 nodes with node 0 connecting to every other nodes"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {},
   "outputs": [
    {
     "data": {
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",
      "text/plain": [
       "<Figure size 640x480 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "num_nodes = 5\n",
    "# The edge syntax is (start, end, weight)\n",
    "edges = [(0, 1, 1), (0, 2, 1), (0, 3, 1), (0, 4, 1)]\n",
    "\n",
    "G = rx.PyGraph()\n",
    "G.add_nodes_from(range(num_nodes))\n",
    "G.add_edges_from(edges)\n",
    "mpl_draw(\n",
    "    G, pos=rx.bipartite_layout(G, {0}), with_labels=True, node_color=\"#EE5396\", font_color=\"#F4F4F4\"\n",
    ")"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The goal is the separate the nodes into 2 sets such that the number of edges from one set to another is maximum.\n",
    "\n",
    "The solution is represented as 5 qubits and the state 0 or 1 label the node in set 0 or set 1.  For example, $|00110 \\rangle$ means nodes {0, 4, 5} in set 0 and nodes {1, 2} in set 1."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Define Cost Hamiltonian $H_C$,\n",
    "\n",
    "$$\n",
    "H_C = - IIIZZ - IIZIZ - IZIIZ - ZIIIZ           \\tag{5}\n",
    "$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "If nodes {0, 2, 3, 4} in set 0 and node {1} in set 1 i.e. $|00010 \\rangle$,\n",
    "\n",
    "$\\langle 00010| (-IIIZZ) |00010 \\rangle = 1$ which positively contribute to $H_C$ because edge01 (edge01 from node 0 to node 1) is in this cut configuration.\n",
    "\n",
    "$\\langle 00010| (-IIZIZ) |00010 \\rangle = -1$ which negatively contribute to $H_C$ because edge02 (edge from node 0 to node 2) is not in this cut configuration.\n",
    "\n",
    "A quantum state maximized $H_C$ is the solution of this Max-Cut problem."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Define Mixer Hamiltonian $H_M$,\n",
    "\n",
    "$$\n",
    "H_M = XXXXX         \\tag{6}\n",
    "$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The Max-Cut solution quantum state can be found by substituting equations (5) and (6) to (1) and (2)\n",
    "\n",
    "$$\n",
    "\\begin{align*}\n",
    "| \\psi \\rangle &= e^{-i \\hat{H}(T) / \\hbar} e^{-i \\hat{H}(T-1) / \\hbar} \\dots e^{-i \\hat{H}(1) / \\hbar} e^{-i \\hat{H}(0) / \\hbar} | +++++ \\rangle   \\\\\n",
    "&= e^{-i H_C / \\hbar}\n",
    "e^{-i (\\frac{1}{T}XXXXX + \\frac{T-1}{T}H_C) / \\hbar}\n",
    "\\dots\n",
    "e^{-i (\\frac{T-1}{T}XXXXX + \\frac{1}{T}H_C) / \\hbar}\n",
    "e^{-i XXXXX / \\hbar}\n",
    "|+++++>                                             \\\\\n",
    "&= e^{-i H_C / \\hbar}\n",
    "e^{-i (\\frac{1}{T}XXXXX + \\frac{T-1}{T}H_C) / \\hbar}\n",
    "\\dots\n",
    "e^{-i (\\frac{T-1}{T}XXXXX + \\frac{1}{T}H_C) / \\hbar}\n",
    "|+++++>                                             \\tag{7}\n",
    "\\end{align*}  \n",
    "$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Trotter product formula\n",
    "\n",
    "For $A$ and $B$ do not commute with large $N$,\n",
    "\n",
    "$$\n",
    "e^{A+B} \\approx (e^{A/N} e^{B/N})^N             \\tag{8}\n",
    "$$\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Apply equation (8) to (7) with large $T$ and $N$,\n",
    "\n",
    "$$\n",
    "| \\psi \\rangle = \n",
    "e^{-i H_C / \\hbar}\n",
    "(e^{-i (\\frac{1}{TN}XXXXX)} e^{-i (\\frac{T-1}{TN}H_C) / \\hbar})^N\n",
    "\\dots\n",
    "(e^{-i (\\frac{T-1}{T}XXXXX)} e^{-i (\\frac{1}{T}H_C) / \\hbar})^N\n",
    "|+++++>                                                         \\tag{9}\n",
    "$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Exponential of Pauli Matrices\n",
    "\n",
    "Definition of exponential,\n",
    "\n",
    "$$\n",
    "e^M = \\sum_{k=0}^{\\inf} \\frac{1}{k!} M^k\n",
    "$$\n",
    "\n",
    "and Pauli matrics,\n",
    "\n",
    "$$\n",
    "X = \n",
    "\\begin{bmatrix}\n",
    "0 & 1 \\\\\n",
    "1 & 0\n",
    "\\end{bmatrix} \n",
    ", Y = \n",
    "\\begin{bmatrix}\n",
    "0 & -i \\\\\n",
    "i & 0\n",
    "\\end{bmatrix}\n",
    ", Z = \n",
    "\\begin{bmatrix}\n",
    "1 & 0 \\\\\n",
    "0 & -1\n",
    "\\end{bmatrix}\n",
    "$$\n",
    "\n",
    "With some calculations,\n",
    "\n",
    "$$\n",
    "\\begin{align*}\n",
    "\n",
    "e^{-i\\theta X} &= R_X(2 \\theta) =\n",
    "\\begin{pmatrix}\n",
    "\\cos(\\theta)    & -i \\sin(\\theta)    \\\\\n",
    "-i \\sin(\\theta) & \\cos(\\theta)\n",
    "\\end{pmatrix}                           \\\\\n",
    "                                        \\\\\n",
    "e^{-i\\theta Y} &= R_Y(2 \\theta) =\n",
    "\\begin{pmatrix}\n",
    "\\cos(\\theta)    & \\sin(\\theta)    \\\\\n",
    "\\sin(\\theta)    & \\cos(\\theta)\n",
    "\\end{pmatrix}                           \\\\\n",
    "                                        \\\\\n",
    "e^{-i\\theta Z} &= R_Z(2 \\theta) =\n",
    "\\begin{pmatrix}\n",
    "e^{-i \\theta}  & 0                 \\\\\n",
    "0               & e^{i \\theta}\n",
    "\\end{pmatrix}\n",
    "\n",
    "\\end{align*}\n",
    "$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Additionally, $e^{-i \\theta Z \\otimes Z} = R_{ZZ}(2 \\theta)$,\n",
    "\n",
    "$$\n",
    "\\begin{align*}\n",
    "\n",
    "e^{-i \\theta Z \\otimes Z} &= \n",
    "e^{-i \\theta \n",
    "\\begin{pmatrix}\n",
    "1   & 0     \\\\\n",
    "0   & -1\n",
    "\\end{pmatrix}\n",
    "\\otimes\n",
    "\\begin{pmatrix}\n",
    "1   & 0     \\\\\n",
    "0   & -1\n",
    "\\end{pmatrix}\n",
    "}                               \\\\\n",
    "\n",
    "&= e^{-i \\theta\n",
    "\\begin{pmatrix}\n",
    "1   & 0     & 0     & 0     \\\\\n",
    "0   & -1    & 0     & 0     \\\\\n",
    "0   & 0     & -1    & 0     \\\\\n",
    "0   & 0     & 0     & 1\n",
    "\\end{pmatrix}\n",
    "}                               \\\\\n",
    "\n",
    "&= e^{\n",
    "\\begin{pmatrix}\n",
    "-i \\theta   & 0     & 0     & 0     \\\\\n",
    "0   & i \\theta    & 0     & 0     \\\\\n",
    "0   & 0     & i \\theta    & 0     \\\\\n",
    "0   & 0     & 0     & -i \\theta\n",
    "\\end{pmatrix}\n",
    "}                               \\\\\n",
    "\n",
    "&= R_{ZZ}(2 \\theta)\n",
    "\n",
    "\\end{align*}\n",
    "$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Create a circuit\n",
    "\n",
    "Create a quantum circuit to implement equation (9) to calculate $|\\psi \\rangle$."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {},
   "outputs": [
    {
     "name": "stderr",
     "output_type": "stream",
     "text": [
      "/Users/thyung/opt/anaconda3/envs/qiskit1_env/lib/python3.11/site-packages/numpy/linalg/linalg.py:2180: RuntimeWarning: divide by zero encountered in det\n",
      "  r = _umath_linalg.det(a, signature=signature)\n",
      "/Users/thyung/opt/anaconda3/envs/qiskit1_env/lib/python3.11/site-packages/numpy/linalg/linalg.py:2180: RuntimeWarning: invalid value encountered in det\n",
      "  r = _umath_linalg.det(a, signature=signature)\n"
     ]
    }
   ],
   "source": [
    "from qiskit.transpiler.preset_passmanagers import generate_preset_pass_manager\n",
    "from qiskit_aer import AerSimulator\n",
    "\n",
    "backend = AerSimulator()\n",
    "backend.name\n",
    "target = backend.target\n",
    "pm = generate_preset_pass_manager(target=target, optimization_level=3)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {},
   "outputs": [
    {
     "data": {
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      "text/plain": [
       "<Figure size 2252.61x2876.22 with 1 Axes>"
      ]
     },
     "execution_count": 4,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# Try \n",
    "T=3\n",
    "N=3\n",
    "\n",
    "# Problem to Hamiltonian operator\n",
    "H_C = SparsePauliOp.from_list([(\"IIIZZ\", -1), (\"IIZIZ\", -1), (\"IZIIZ\", -1), (\"ZIIIZ\", -1)])\n",
    "# QAOA ansatz circuit repeating e^{H_C} and e^{H_M} \"reps\" times with gamma and beta parameters\n",
    "ansatz = QAOAAnsatz(H_C, reps=T*N)\n",
    "\n",
    "params = []\n",
    "\n",
    "# parameters for H_M part of ansatz\n",
    "for i in range(1, T+1):\n",
    "    for j in range(N):\n",
    "        params.append(2 * (1 - i * 1.0 / N) / T)  # beta R_X(2 \\theta) \n",
    "\n",
    "# parameters for H_C part of ansatz\n",
    "for i in range(1, T+1):\n",
    "    for j in range(N):\n",
    "        params.append(2 * (i * 1.0 / N) / T)      # gamma R_Z(2 \\theta)\n",
    "\n",
    "\n",
    "\n",
    "qc = ansatz.assign_parameters(params)\n",
    "qc.measure_all()\n",
    "qc_isa = pm.run(qc)\n",
    "\n",
    "qc_isa.decompose(reps=0).draw(output='mpl')"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {},
   "outputs": [
    {
     "name": "stderr",
     "output_type": "stream",
     "text": [
      "/Users/thyung/opt/anaconda3/envs/qiskit1_env/lib/python3.11/site-packages/qiskit_ibm_runtime/session.py:156: UserWarning: Session is not supported in local testing mode or when using a simulator.\n",
      "  warnings.warn(\n"
     ]
    },
    {
     "data": {
      "image/png": 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",
      "text/plain": [
       "<Figure size 700x500 with 1 Axes>"
      ]
     },
     "execution_count": 5,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "session = Session(backend=backend)\n",
    "sampler = Sampler(session=session)\n",
    "result = sampler.run([qc_isa]).result()\n",
    "samp_dist = result[0].data.meas.get_counts()\n",
    "session.close()\n",
    "plot_distribution(samp_dist)\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Max-Cut solution according to Adiabatic Process simulation\n",
    "\n",
    "$|00001 \\rangle$  and $|11110 \\rangle$ show high probabilities.  They represent the sets {0} and {1, 2, 3, 4} are the Max-Cut solution."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### References:\n",
    "\n",
    "1. Lecture 5.2 - Introduction to the Quantum Approximate Optimization Algorithm and Applications, https://www.youtube.com/watch?v=YpLzSQPrgSc\n",
    "2. Tutorial - Quantum approximate optimization algorithm, https://learning.quantum.ibm.com/tutorial/quantum-approximate-optimization-algorithm\n",
    "3. Trotter product formula https://encyclopediaofmath.org/wiki/Trotter_product_formula\n",
    "4. $R_{ZZ}$, https://docs.quantum.ibm.com/api/qiskit/qiskit.circuit.library.RZZGate"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "'1.0.2'"
      ]
     },
     "execution_count": 6,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "qiskit.__version__"
   ]
  }
 ],
 "metadata": {
  "kernelspec": {
   "display_name": "qiskit1_env",
   "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": "3.11.8"
  }
 },
 "nbformat": 4,
 "nbformat_minor": 2
}