exercises/exercise_6.ipynb

{
 "cells": [
  {
   "cell_type": "code",
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   "metadata": {
    "colab": {},
    "colab_type": "code",
    "id": "CshpxnyNQQNu"
   },
   "outputs": [],
   "source": [
    "from qiskit import *\n",
    "from qiskit.tools.visualization import plot_histogram\n",
    "import numpy as np"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "colab_type": "text",
    "id": "pmm5uV8cQQN6"
   },
   "source": [
    "# Exercise 4\n",
    "\n",
    "* Hints: 08/04/2020\n",
    "* Hand-in: 15/04/2020\n",
    "* Solutions: 22/04/2020\n",
    "* *There will be no penalty for late hand-in, but handing in on-time will allow theTAs to address your problems.*\n",
    "\n",
    "## 1\n",
    "\n",
    "Show that the Hadamard gate can be written in the following two forms\n",
    "\n",
    "$$H = \\frac{X+Z}{\\sqrt{2}} \\sim \\exp\\left(i \\frac{\\pi}{2} \\, \\frac{X+Z}{\\sqrt{2}}\\right).$$\n",
    "\n",
    "Here $\\sim$ is used to denote that the equality is valid up to a global phase, and hence that the resulting gates are physically equivalent.\n",
    "\n",
    "Hint: it might even be easiest to prove that $e^{i\\frac{\\pi}{2} M} \\sim M$ for any matrix whose eigenvalues are all $\\pm 1$, and that such matrices uniquely satisfy $M^2=I$."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "colab_type": "text",
    "id": "CJyxxSDUQQN9"
   },
   "source": [
    "## 2\n",
    "\n",
    "The Hadamard can be constructed from `rx` and `rz` operations as\n",
    "\n",
    "$$ R_x(\\theta) = e^{i\\frac{\\theta}{2} X}, ~~~ R_z(\\theta) = e^{i\\frac{\\theta}{2} Z},\\\\ H \\equiv \\lim_{n\\rightarrow\\infty} \\left( ~R_x\\left(\\frac{\\theta}{n}\\right) ~~R_z \\left(\\frac{\\theta}{n}\\right) ~\\right)^n.$$\n",
    "\n",
    "For some suitably chosen $\\theta$. When implemented for finite $n$, the resulting gate will be an approximation to the Hadamard whose error decreases with $n$.\n",
    "\n",
    "The following shows an example of this implemented with Qiskit with an incorrectly chosen value of $\\theta$ (and with the global phase ignored).\n",
    "\n",
    "* Determine the correct value of $\\theta$.\n",
    "\n",
    "* Show that the error (when using the correct value of $\\theta$) decreases quadratically with $n$."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {
    "colab": {},
    "colab_type": "code",
    "id": "4yqeQMlZQQN_"
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   "outputs": [
    {
     "data": {
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\n",
      "text/plain": [
       "<Figure size 504x360 with 1 Axes>"
      ]
     },
     "execution_count": 7,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "q = QuantumRegister(1)\n",
    "c = ClassicalRegister(1)\n",
    "\n",
    "error = {}\n",
    "for n in range(1,11):\n",
    "\n",
    "    # Create a blank circuit\n",
    "    qc = QuantumCircuit(q,c)\n",
    "    \n",
    "    # Implement an approximate Hadamard\n",
    "    theta = np.pi # here we incorrectly choose theta=pi\n",
    "    for j in range(n):\n",
    "        qc.rx(theta/n,q[0])\n",
    "        qc.rz(theta/n,q[0])\n",
    "      \n",
    "    # We need to measure how good the above approximation is. Here's a simple way to do this.\n",
    "    # Step 1: Use a real hadamard to cancel the above approximation.\n",
    "    # For a good approximatuon, the qubit will return to state 0. For a bad one, it will end up as some superposition.\n",
    "    qc.h(q[0])\n",
    "    \n",
    "    # Step 2: Run the circuit, and see how many times we get the outcome 1.\n",
    "    # Since it should return 0 with certainty, the fraction of 1s is a measure of the error.\n",
    "    qc.measure(q,c)\n",
    "    shots = 20000\n",
    "    job = execute(qc, Aer.get_backend('qasm_simulator'),shots=shots)\n",
    "    try:\n",
    "        error[n] = (job.result().get_counts()['1']/shots)\n",
    "    except:\n",
    "        pass\n",
    "        \n",
    "plot_histogram(error)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "colab_type": "text",
    "id": "Yr5vuU_eQQOU"
   },
   "source": [
    "## 3\n",
    "\n",
    "An improved version of the approximation can be found from,\n",
    "\n",
    "$$H \\equiv \\lim_{n\\rightarrow\\infty} \\left( ~ R_z \\left(\\frac{\\theta}{2n}\\right)~~ R_x\\left(\\frac{\\theta}{n}\\right) ~~ R_z \\left(\\frac{\\theta}{2n}\\right) ~\\right)^n.$$\n",
    "\n",
    "Implement this, and investigate the scaling of the error."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 0,
   "metadata": {
    "colab": {},
    "colab_type": "code",
    "id": "TeQnxYGDQ2ro"
   },
   "outputs": [],
   "source": []
  }
 ],
 "metadata": {
  "colab": {
   "collapsed_sections": [],
   "name": "Exercise_4.ipynb",
   "provenance": [],
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  "kernelspec": {
   "display_name": "Python 3",
   "language": "python",
   "name": "python3"
  },
  "language_info": {
   "codemirror_mode": {
    "name": "ipython",
    "version": 3
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   "file_extension": ".py",
   "mimetype": "text/x-python",
   "name": "python",
   "nbconvert_exporter": "python",
   "pygments_lexer": "ipython3",
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