notebook/hadamard_test.ipynb

{
 "cells": [
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {},
   "outputs": [],
   "source": [
    "import pennylane as qml\n",
    "from pennylane import numpy as np\n",
    "import matplotlib.pyplot as plt\n",
    "from enum import Enum"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {},
   "outputs": [],
   "source": [
    "qml.drawer.use_style('sketch')"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {},
   "outputs": [],
   "source": [
    "class ReOrIm(Enum):\n",
    "    real = 0\n",
    "    imag = 1"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {},
   "outputs": [],
   "source": [
    "dev = qml.device('default.qubit', wires=2)\n",
    "\n",
    "@qml.qnode(dev)\n",
    "def hadamard_test_circuit(theta, phi, re_or_im=ReOrIm.real):\n",
    "    # Prepare the superposition state |psi> = (|0>|a> + |1>|b>)/sqrt(2)\n",
    "    # 1. Apply the Hadamard gate to the first qubit\n",
    "    qml.Hadamard(wires=0)\n",
    "\n",
    "    # 2. Apply S^dagger to the first qubit if we are measuring the imaginary part\n",
    "    if re_or_im == ReOrIm.imag:\n",
    "        qml.adjoint(qml.S)(wires=0)\n",
    "        \n",
    "    # 3. Apply the controlled unitary U where U|0> = |a> and U|1> = |b>\n",
    "    qml.RY(2 * theta, wires=1) # |a> = Ry(theta)|0>\n",
    "    qml.CRY(2 * (phi-theta), wires=[0, 1])\n",
    "\n",
    "    # 4. Apply the Hadamard gate to the first qubit\n",
    "    qml.Hadamard(wires=0)\n",
    "    return qml.probs(wires=0)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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",
      "text/plain": [
       "<Figure size 600x300 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "qml.draw_mpl(hadamard_test_circuit)(np.pi/4, np.pi/3)\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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",
      "text/plain": [
       "<Figure size 700x300 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "qml.draw_mpl(hadamard_test_circuit)(np.pi/4, np.pi/3, re_or_im=ReOrIm.imag)\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**Explanation of the circuit**\n",
    "\n",
    "Assume $|a\\rangle = R_y(2\\theta)|0\\rangle$ and $|b\\rangle = R_y(2\\phi)|0\\rangle$.\n",
    "\n",
    "1) Apply Hadamard gate to first qubit (0) to create superposition\n",
    "$$\n",
    "H|0\\rangle = \\frac{1}{\\sqrt{2}}(|0\\rangle + |1\\rangle)\n",
    "$$\n",
    "\n",
    "2) Apply RY gate to second qubit (1) with angle $2\\theta$ to create state $|a\\rangle$\n",
    "$$\n",
    "|a\\rangle = R_y(2\\theta)|0\\rangle\n",
    "$$\n",
    "\n",
    "3) Apply CRY gate to first qubit (0) and second qubit (1) with angle $2(\\phi-\\theta)$ to create state $|b\\rangle$\n",
    "$$\n",
    "\\begin{aligned}\n",
    "|\\psi\\rangle &= CR_y(2(\\phi-\\theta))\\frac{1}{\\sqrt{2}}(|0\\rangle + |1\\rangle)|a\\rangle \\\\\n",
    "&= \\frac{1}{\\sqrt{2}}(|0\\rangle|a\\rangle + |1\\rangle R_y (2(\\phi-\\theta))|a\\rangle) \\\\\n",
    "&= \\frac{1}{\\sqrt{2}}(|0\\rangle|a\\rangle + |1\\rangle R_y(2\\phi)|0\\rangle) \\\\\n",
    "&= \\frac{1}{\\sqrt{2}}(|0\\rangle|a\\rangle + |1\\rangle|b\\rangle)\n",
    "\\end{aligned}\n",
    "$$\n",
    "\n",
    "4) Apply Hadamard gate to first qubit (0) again\n",
    "$$\n",
    "\\begin{aligned}\n",
    "H|\\psi\\rangle &= \\frac{1}{\\sqrt{2}}(H|0\\rangle|a\\rangle + H|1\\rangle|b\\rangle) \\\\\n",
    "&= \\frac{1}{2}(|0\\rangle|a\\rangle + |0\\rangle|b\\rangle + |1\\rangle|a\\rangle - |1\\rangle|b\\rangle) \\\\\n",
    "&= \\frac{1}{2}(|0\\rangle(|a\\rangle + |b\\rangle) + |1\\rangle(|a\\rangle - |b\\rangle))\n",
    "\\end{aligned}\n",
    "$$\n",
    "\n",
    "5) Measure first qubit (0).\n",
    "$$\n",
    "P(|x\\rangle = |0\\rangle) = \\frac{1}{4}(|a\\rangle + |b\\rangle)^2 = \\frac{1}{2}(1 + \\text{Re}(\\langle a|b\\rangle))\n",
    "$$\n",
    "\n",
    "For imaginary part of $\\langle a|b\\rangle$, we add $S^\\dagger$ gate to the ancilla qubit after step 1.\n",
    "\n",
    "$$\n",
    "S^\\dagger H|0\\rangle = \\frac{1}{\\sqrt{2}}(|0\\rangle - i|1\\rangle) \n",
    "$$\n",
    "\n",
    "Then repeat step 2 to 5.\n",
    "\n",
    "$$\n",
    "P(|x\\rangle = |0\\rangle) = \\frac{1}{4}(|a\\rangle - i|b\\rangle)^2 = \\frac{1}{2}(1 - \\text{Im}(\\langle a|b\\rangle))\n",
    "$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "For example, let's obtain $\\langle a | b \\rangle$ for $\\theta = \\pi/4$ and $\\phi = \\pi/3$.\n",
    "\n",
    "$$\n",
    "\\begin{aligned}\n",
    "|a\\rangle &= R_y(2\\times\\frac{\\pi}{4})|0\\rangle = \\frac{1}{\\sqrt{2}}|0\\rangle + \\frac{1}{\\sqrt{2}}|1\\rangle \\\\\n",
    "|b\\rangle &= R_y(2\\times\\frac{\\pi}{3})|0\\rangle = \\frac{1}{2}|0\\rangle + \\frac{\\sqrt{3}}{2}|1\\rangle \\\\\n",
    "\\langle a | b \\rangle &= \\frac{1}{2\\sqrt{2}} + \\frac{\\sqrt{3}}{2\\sqrt{2}} \\simeq 0.9659\n",
    "\\end{aligned}\n",
    "$$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Probability of measuring |0> when measuring the real part: 0.9830\n",
      "Probability of measuring |0> when measuring the imaginary part: 0.5000\n",
      "Real part of the inner product: 0.9659\n",
      "Imaginary part of the inner product: 0.0000\n",
      "Inner product of the two states: 0.9659+0.0000j\n"
     ]
    }
   ],
   "source": [
    "p_0_real = hadamard_test_circuit(np.pi/4, np.pi/3, re_or_im=ReOrIm.real)[0]\n",
    "p_0_imag = hadamard_test_circuit(np.pi/4, np.pi/3, re_or_im=ReOrIm.imag)[0]\n",
    "\n",
    "print(f\"Probability of measuring |0> when measuring the real part: {p_0_real:.4f}\")\n",
    "print(f\"Probability of measuring |0> when measuring the imaginary part: {p_0_imag:.4f}\")\n",
    "\n",
    "inner_prod_real = 2 * p_0_real - 1\n",
    "inner_prod_imag = 1 - 2 * p_0_imag\n",
    "\n",
    "print(f\"Real part of the inner product: {inner_prod_real:.4f}\")\n",
    "print(f\"Imaginary part of the inner product: {inner_prod_imag:.4f}\")\n",
    "\n",
    "# The inner product of the two states is the sum of the real and imaginary parts\n",
    "inner_prod = inner_prod_real + 1j * inner_prod_imag\n",
    "print(f\"Inner product of the two states: {inner_prod:.4f}\")"
   ]
  }
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