From e6ed39edcbff36049e490f24c7dc9006303859c1 Mon Sep 17 00:00:00 2001 From: James Wootton Date: Tue, 12 Apr 2022 10:09:47 +0200 Subject: [PATCH] ex6 --- exercises/Exercise6.ipynb | 243 ++++++++++++++++++++++++++++++++++++++ exercises/circuits.png | Bin 0 -> 120218 bytes 2 files changed, 243 insertions(+) create mode 100644 exercises/Exercise6.ipynb create mode 100644 exercises/circuits.png diff --git a/exercises/Exercise6.ipynb b/exercises/Exercise6.ipynb new file mode 100644 index 0000000..1c75460 --- /dev/null +++ b/exercises/Exercise6.ipynb @@ -0,0 +1,243 @@ +{ + "cells": [ + { + "cell_type": "markdown", + "metadata": { + "colab_type": "text", + "id": "pmm5uV8cQQN6" + }, + "source": [ + "# Exercise 6\n", + "\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": 2, + "metadata": {}, + "outputs": [], + "source": [ + "from qiskit import *\n", + "from qiskit.tools.visualization import plot_histogram\n", + "import numpy as np" + ] + }, + { + "cell_type": "code", + "execution_count": 3, + "metadata": { + "colab": {}, + "colab_type": "code", + "id": "4yqeQMlZQQN_" + }, + "outputs": [ + { + "data": { + "image/png": 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" + ] + }, + "execution_count": 3, + "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": "markdown", + "metadata": {}, + "source": [ + "## 4\n", + "\n", + "Write a circuit to implement the following unitary with perfect precision, using only `rx`, `rz` and Clifford gates\n", + "\n", + "$$U = \\exp\\left[i \\frac{\\theta}{2} \\, \\left( X\\otimes X + Z\\otimes Z \\right) \\right].$$\n", + "\n", + "Apply this to the initial state $|10\\rangle$ and determine the final state for the following values of $\\theta$.\n", + "\n", + "* (a) $\\theta=\\pi/4 $\n", + "* (b) $\\theta=\\pi/2 $\n", + "* (c) $\\theta=\\pi $\n", + "\n", + "You can use the following as a starting point." + ] + }, + { + "cell_type": "code", + "execution_count": 4, + "metadata": {}, + "outputs": [ + { + "name": "stdout", + "output_type": "stream", + "text": [ + "Statevector([0.+0.j, 0.+0.j, 1.+0.j, 0.+0.j],\n", + " dims=(2, 2))\n" + ] + } + ], + "source": [ + "theta = np.pi/2\n", + "\n", + "q = QuantumRegister(2)\n", + "\n", + "# Create a blank circuit\n", + "qc = QuantumCircuit(q)\n", + "\n", + "# prepare the |10> state\n", + "qc.x(1)\n", + "\n", + "# do things!\n", + "\n", + "# get the final statevector\n", + "job = Aer.get_backend('statevector_simulator').run(qc)\n", + "statevector = job.result().get_statevector()\n", + "print(statevector)" + ] + }, + { + "cell_type": "markdown", + "metadata": { + "colab": {}, + "colab_type": "code", + "id": "TeQnxYGDQ2ro" + }, + "source": [ + "## 5. Effects of garbage in quantum circuits\n", + "\n", + "The CX gate performs the mapping \n", + "$(i,j) \\rightarrow (i,i+j \\mod 2)$ from an input on two bits to an output on two bits.\n", + "\n", + "It is interesting to note (though not relevant to the problem) that this mapping is not of the form $(z,0) \\rightarrow (z,f(z))$, as considered last week, since the function $f(i,j) = i+j \\mod 2$ does not require the whole input to be present for reversibility.\n", + "\n", + "Here are two circuits with a CX gate.\n", + "\n", + "![](circuits.png)\n", + "\n", + "* (a) The first circuit is composed of both a quantum part and an irreversible classical part. Determine the final output bit .\n", + "\n", + "* (b) Show that the second circuit effectively acts as a CX between the first and third qubits, but with the additional effect of producting a garbage qubit on the second.\n", + "\n", + "* (c) Replace the true CX in the first circuit with the garbage producing one from the second. Show that this changes the output bit.\n", + "\n", + "* (d) Show how the garbage producing CX can be corrected by uncomputation." + ] + }, + { + "cell_type": "code", + "execution_count": null, + "metadata": {}, + "outputs": [], + "source": [] + } + ], + "metadata": { + "colab": { + "collapsed_sections": [], + "name": "Exercise_4.ipynb", + "provenance": [], + "version": "0.3.2" + }, + "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": "3.9.1" + } + }, + "nbformat": 4, + "nbformat_minor": 1 +} diff --git a/exercises/circuits.png b/exercises/circuits.png new file mode 100644 index 0000000000000000000000000000000000000000..67c98ee060c9836f565bfe7e7b5cef96242107a2 GIT binary patch literal 120218 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