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Superdense coding and teleportation in Cirq
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| { | |
| "nbformat": 4, | |
| "nbformat_minor": 0, | |
| "metadata": { | |
| "colab": { | |
| "name": "Superdense coding and teleportation in Cirq", | |
| "provenance": [], | |
| "collapsed_sections": [], | |
| "authorship_tag": "ABX9TyPbmxRn1Y4EacORjavzzM3G", | |
| "include_colab_link": true | |
| }, | |
| "kernelspec": { | |
| "name": "python3", | |
| "display_name": "Python 3" | |
| } | |
| }, | |
| "cells": [ | |
| { | |
| "cell_type": "markdown", | |
| "metadata": { | |
| "id": "view-in-github", | |
| "colab_type": "text" | |
| }, | |
| "source": [ | |
| "<a href=\"https://colab.research.google.com/gist/csferrie/fd97d7fc037110af21b5a26c7f74609e/superdense-coding-and-teleportation-in-cirq.ipynb\" target=\"_parent\"><img src=\"https://colab.research.google.com/assets/colab-badge.svg\" alt=\"Open In Colab\"/></a>" | |
| ] | |
| }, | |
| { | |
| "cell_type": "markdown", | |
| "metadata": { | |
| "id": "xV7VUIMwH2IT", | |
| "colab_type": "text" | |
| }, | |
| "source": [ | |
| "# Superdense coding and teleportation in Cirq\n", | |
| "\n", | |
| "This exercise is part of [Lab 6](https://medium.com/@csferrie/superdense-coding-and-quantum-teleportation-in-three-quantum-programming-languages-3fe7032d1dbc) of [Introduction to Quantum Computing](https://medium.com/@csferrie/introduction-to-quantum-computing-df9e1182a831).\n", | |
| "\n", | |
| "This notebook provides implementation of superdense coding and teleportation in the [Cirq](https://cirq.readthedocs.io/en/stable/) quantum programming language." | |
| ] | |
| }, | |
| { | |
| "cell_type": "markdown", | |
| "metadata": { | |
| "id": "zp0yynx1I9Jl", | |
| "colab_type": "text" | |
| }, | |
| "source": [ | |
| "# Cirq installation\n", | |
| "\n", | |
| "The first step is to install cirq. Installation instructions are available in the documentation for Cirq here: https://cirq.readthedocs.io/en/stable/index.html." | |
| ] | |
| }, | |
| { | |
| "cell_type": "code", | |
| "metadata": { | |
| "id": "MFG2OUCoyJY1", | |
| "colab_type": "code", | |
| "colab": { | |
| "base_uri": "https://localhost:8080/", | |
| "height": 34 | |
| }, | |
| "outputId": "8a51cc1b-3e0e-4ac1-91e5-57c54a577536" | |
| }, | |
| "source": [ | |
| "# install cirq\n", | |
| "!pip install cirq --quiet" | |
| ], | |
| "execution_count": null, | |
| "outputs": [ | |
| { | |
| "output_type": "stream", | |
| "text": [ | |
| "\u001b[K |████████████████████████████████| 1.4MB 2.6MB/s \n", | |
| "\u001b[?25h" | |
| ], | |
| "name": "stdout" | |
| } | |
| ] | |
| }, | |
| { | |
| "cell_type": "markdown", | |
| "metadata": { | |
| "id": "M7uLB5v1JtHm", | |
| "colab_type": "text" | |
| }, | |
| "source": [ | |
| "Obviously we need to import cirq. But we will also need numpy later." | |
| ] | |
| }, | |
| { | |
| "cell_type": "code", | |
| "metadata": { | |
| "id": "zRiG_m_fTIRG", | |
| "colab_type": "code", | |
| "colab": {} | |
| }, | |
| "source": [ | |
| "import cirq\n", | |
| "import numpy as np" | |
| ], | |
| "execution_count": null, | |
| "outputs": [] | |
| }, | |
| { | |
| "cell_type": "markdown", | |
| "metadata": { | |
| "id": "r1NMu4QBTTV2", | |
| "colab_type": "text" | |
| }, | |
| "source": [ | |
| "# Superdense coding\n", | |
| "\n", | |
| "[Superdense coding](https://en.wikipedia.org/wiki/Superdense_coding) is a protocol that allows the tranmission of two classical bits of information using 1 qubit of information. \n", | |
| "\n", | |
| "This, the simplest implementation in Cirq, accompanies the [lecture](https://medium.com/@csferrie/my-first-quantum-protocol-de336d290322) on the same topic.\n", | |
| "\n", | |
| "Let's get started. The first thing Alice and Bob need to do is create entanglement.\n", | |
| "\n", | |
| "" | |
| ] | |
| }, | |
| { | |
| "cell_type": "code", | |
| "metadata": { | |
| "id": "lCK97MrEWMjn", | |
| "colab_type": "code", | |
| "colab": { | |
| "base_uri": "https://localhost:8080/", | |
| "height": 68 | |
| }, | |
| "outputId": "ef4f782a-270e-4e4a-cdd8-f37f0895702a" | |
| }, | |
| "source": [ | |
| "# create two qubits\n", | |
| "alice = cirq.NamedQubit('alice')\n", | |
| "bob = cirq.NamedQubit('bob')\n", | |
| "\n", | |
| "# create cirquit\n", | |
| "superdense = cirq.Circuit()\n", | |
| "\n", | |
| "# add your gates (aka \"moments\")\n", | |
| "superdense.append([\n", | |
| " cirq.H(alice),\n", | |
| " cirq.CNOT(alice,bob)\n", | |
| " ])\n", | |
| "\n", | |
| "print(superdense)" | |
| ], | |
| "execution_count": null, | |
| "outputs": [ | |
| { | |
| "output_type": "stream", | |
| "text": [ | |
| "alice: ───H───@───\n", | |
| " │\n", | |
| "bob: ─────────X───\n" | |
| ], | |
| "name": "stdout" | |
| } | |
| ] | |
| }, | |
| { | |
| "cell_type": "markdown", | |
| "metadata": { | |
| "id": "CFm4NQNbVMAa", | |
| "colab_type": "text" | |
| }, | |
| "source": [ | |
| "Now Alice needs to encode her message. There are four possibilities. \n", | |
| "\n", | |
| "If Alice wants to send\n", | |
| "* 00, she does nothing;\n", | |
| "* 01, she performs an $X$ gate;\n", | |
| "* 10, she performs a $Z$ gate;\n", | |
| "* 11, she performs $XZ$.\n", | |
| "\n", | |
| "Let's say Alice choses 11. \n" | |
| ] | |
| }, | |
| { | |
| "cell_type": "code", | |
| "metadata": { | |
| "id": "sCAKLdh3V26d", | |
| "colab_type": "code", | |
| "colab": { | |
| "base_uri": "https://localhost:8080/", | |
| "height": 68 | |
| }, | |
| "outputId": "f14bdc17-a6e9-422b-d4cf-ebd2ddc4bbb8" | |
| }, | |
| "source": [ | |
| "superdense.append([\n", | |
| " cirq.X(alice),\n", | |
| " cirq.Z(alice)\n", | |
| " ])\n", | |
| "\n", | |
| "print(superdense)" | |
| ], | |
| "execution_count": null, | |
| "outputs": [ | |
| { | |
| "output_type": "stream", | |
| "text": [ | |
| "alice: ───H───@───Z───\n", | |
| " │\n", | |
| "bob: ─────────X───────\n" | |
| ], | |
| "name": "stdout" | |
| } | |
| ] | |
| }, | |
| { | |
| "cell_type": "markdown", | |
| "metadata": { | |
| "id": "tsEySC0hWVN0", | |
| "colab_type": "text" | |
| }, | |
| "source": [ | |
| "Now Alice sends her qubit to Bob. Bob unentangles them." | |
| ] | |
| }, | |
| { | |
| "cell_type": "code", | |
| "metadata": { | |
| "id": "4fg_HryHWdJN", | |
| "colab_type": "code", | |
| "colab": { | |
| "base_uri": "https://localhost:8080/", | |
| "height": 68 | |
| }, | |
| "outputId": "56474866-7035-473e-c656-23c8da4b2751" | |
| }, | |
| "source": [ | |
| "superdense.append([\n", | |
| " cirq.CNOT(alice,bob), \n", | |
| " cirq.H(alice)\n", | |
| " ])\n", | |
| "\n", | |
| "print(superdense)" | |
| ], | |
| "execution_count": null, | |
| "outputs": [ | |
| { | |
| "output_type": "stream", | |
| "text": [ | |
| "alice: ───H───@───Z───@───H───\n", | |
| " │ │\n", | |
| "bob: ─────────X───────X───────\n" | |
| ], | |
| "name": "stdout" | |
| } | |
| ] | |
| }, | |
| { | |
| "cell_type": "markdown", | |
| "metadata": { | |
| "id": "R2CSMU5gXQGk", | |
| "colab_type": "text" | |
| }, | |
| "source": [ | |
| "Finally Bob measures both qubits." | |
| ] | |
| }, | |
| { | |
| "cell_type": "code", | |
| "metadata": { | |
| "id": "72XXxbakXTW6", | |
| "colab_type": "code", | |
| "colab": { | |
| "base_uri": "https://localhost:8080/", | |
| "height": 68 | |
| }, | |
| "outputId": "d1d3631f-a0b3-4b7a-b9ec-0100cce39998" | |
| }, | |
| "source": [ | |
| "superdense.append(cirq.measure(alice, bob, key='received'))\n", | |
| "\n", | |
| "print(superdense)" | |
| ], | |
| "execution_count": null, | |
| "outputs": [ | |
| { | |
| "output_type": "stream", | |
| "text": [ | |
| "alice: ───H───@───Z───@───H───M('received')───\n", | |
| " │ │ │\n", | |
| "bob: ─────────X───────X───────M───────────────\n" | |
| ], | |
| "name": "stdout" | |
| } | |
| ] | |
| }, | |
| { | |
| "cell_type": "markdown", | |
| "metadata": { | |
| "id": "otFJsni1YnMW", | |
| "colab_type": "text" | |
| }, | |
| "source": [ | |
| "Let's simulate this circuit." | |
| ] | |
| }, | |
| { | |
| "cell_type": "code", | |
| "metadata": { | |
| "id": "QiBXRNvKYCFg", | |
| "colab_type": "code", | |
| "colab": { | |
| "base_uri": "https://localhost:8080/", | |
| "height": 34 | |
| }, | |
| "outputId": "b758bae9-8a6c-40b1-bcfe-cd279c9e490f" | |
| }, | |
| "source": [ | |
| "simulator = cirq.Simulator()\n", | |
| "message = simulator.run(superdense, repetitions=1)\n", | |
| "\n", | |
| "print(message)" | |
| ], | |
| "execution_count": null, | |
| "outputs": [ | |
| { | |
| "output_type": "stream", | |
| "text": [ | |
| "received=1, 0\n" | |
| ], | |
| "name": "stdout" | |
| } | |
| ] | |
| }, | |
| { | |
| "cell_type": "markdown", | |
| "metadata": { | |
| "id": "jVofPW1XuIoP", | |
| "colab_type": "text" | |
| }, | |
| "source": [ | |
| "Huzzah! It works. Now you try to create a better implementation that doesn't involve hard-coding Alice's message in." | |
| ] | |
| }, | |
| { | |
| "cell_type": "markdown", | |
| "metadata": { | |
| "colab_type": "text", | |
| "id": "-5vVz9psB-2l" | |
| }, | |
| "source": [ | |
| "# Teleportation\n", | |
| "\n", | |
| "[Quantum teleportation](https://en.wikipedia.org/wiki/Quantum_teleportation) is a protocol that allows the tranmission of 1 qubit of information using two classical bits of information.\n", | |
| "\n", | |
| "This, the simplest implementation in cirq, accompanies the [lecture](https://medium.com/@csferrie/my-first-quantum-protocol-de336d290322) on the same topic.\n", | |
| "\n", | |
| "\n", | |
| "\n" | |
| ] | |
| }, | |
| { | |
| "cell_type": "markdown", | |
| "metadata": { | |
| "id": "guo2WRncDRa1", | |
| "colab_type": "text" | |
| }, | |
| "source": [ | |
| "The first thing we'll do is create the state Alcie wants to teleport. Let's pick a \"random\" state. Here I've chose a $Y$-axis rotation of $\\pi/4$. " | |
| ] | |
| }, | |
| { | |
| "cell_type": "code", | |
| "metadata": { | |
| "colab_type": "code", | |
| "id": "_uBy17t7CS_e", | |
| "colab": { | |
| "base_uri": "https://localhost:8080/", | |
| "height": 34 | |
| }, | |
| "outputId": "0ba023eb-0ab5-4344-b6d6-f8d7d5843b36" | |
| }, | |
| "source": [ | |
| "msg = cirq.NamedQubit('msg')\n", | |
| "\n", | |
| "teleportation = cirq.Circuit()\n", | |
| "\n", | |
| "teleportation.append(cirq.ry(np.pi/4)(msg))\n", | |
| "\n", | |
| "print(teleportation)" | |
| ], | |
| "execution_count": null, | |
| "outputs": [ | |
| { | |
| "output_type": "stream", | |
| "text": [ | |
| "msg: ───Ry(0.25π)───\n" | |
| ], | |
| "name": "stdout" | |
| } | |
| ] | |
| }, | |
| { | |
| "cell_type": "markdown", | |
| "metadata": { | |
| "id": "zNyyhOKzbemY", | |
| "colab_type": "text" | |
| }, | |
| "source": [ | |
| "Let's take a look at the state and compare it to the state when the protocol is complete later." | |
| ] | |
| }, | |
| { | |
| "cell_type": "code", | |
| "metadata": { | |
| "id": "OPPFg2XZbarl", | |
| "colab_type": "code", | |
| "colab": { | |
| "base_uri": "https://localhost:8080/", | |
| "height": 34 | |
| }, | |
| "outputId": "7a579495-b0b7-499a-c031-653233420fb7" | |
| }, | |
| "source": [ | |
| "simulator = cirq.Simulator()\n", | |
| "\n", | |
| "alice_message = simulator.simulate(teleportation)\n", | |
| "\n", | |
| "print(alice_message.final_state)" | |
| ], | |
| "execution_count": null, | |
| "outputs": [ | |
| { | |
| "output_type": "stream", | |
| "text": [ | |
| "[0.9238795 +0.j 0.38268343+0.j]\n" | |
| ], | |
| "name": "stdout" | |
| } | |
| ] | |
| }, | |
| { | |
| "cell_type": "markdown", | |
| "metadata": { | |
| "id": "N898apnLiG9c", | |
| "colab_type": "text" | |
| }, | |
| "source": [ | |
| "Alice and Bob need to entangle their qubits. This is done with the $H$ and $CNOT$ gates." | |
| ] | |
| }, | |
| { | |
| "cell_type": "code", | |
| "metadata": { | |
| "id": "HpdwhUQBb_82", | |
| "colab_type": "code", | |
| "colab": { | |
| "base_uri": "https://localhost:8080/", | |
| "height": 102 | |
| }, | |
| "outputId": "9e848391-b9fd-48b8-8fab-31f84f10a8ad" | |
| }, | |
| "source": [ | |
| "alice = cirq.NamedQubit('alice')\n", | |
| "bob = cirq.NamedQubit('bob')\n", | |
| "\n", | |
| "teleportation.append([\n", | |
| " cirq.H(alice),\n", | |
| " cirq.CNOT(alice, bob)\n", | |
| " ])\n", | |
| "\n", | |
| "print(teleportation)" | |
| ], | |
| "execution_count": null, | |
| "outputs": [ | |
| { | |
| "output_type": "stream", | |
| "text": [ | |
| "alice: ───H───────────@───\n", | |
| " │\n", | |
| "bob: ─────────────────X───\n", | |
| "\n", | |
| "msg: ─────Ry(0.25π)───────\n" | |
| ], | |
| "name": "stdout" | |
| } | |
| ] | |
| }, | |
| { | |
| "cell_type": "markdown", | |
| "metadata": { | |
| "id": "roS4qY3sAsh_", | |
| "colab_type": "text" | |
| }, | |
| "source": [ | |
| "Alice needs to perform an \"entangled\" measurement. She can do this by running the standard $H + CNOT$ circuit in reverse." | |
| ] | |
| }, | |
| { | |
| "cell_type": "code", | |
| "metadata": { | |
| "id": "Hw8Rcz8kcOjz", | |
| "colab_type": "code", | |
| "colab": { | |
| "base_uri": "https://localhost:8080/", | |
| "height": 102 | |
| }, | |
| "outputId": "5c08de0e-37ac-49e9-e013-f9db73fa25e1" | |
| }, | |
| "source": [ | |
| "teleportation.append([\n", | |
| " cirq.CNOT(msg, alice),\n", | |
| " cirq.H(msg)\n", | |
| " ])\n", | |
| "\n", | |
| "print(teleportation)" | |
| ], | |
| "execution_count": null, | |
| "outputs": [ | |
| { | |
| "output_type": "stream", | |
| "text": [ | |
| "alice: ───H───────────@───X───────\n", | |
| " │ │\n", | |
| "bob: ─────────────────X───┼───────\n", | |
| " │\n", | |
| "msg: ─────Ry(0.25π)───────@───H───\n" | |
| ], | |
| "name": "stdout" | |
| } | |
| ] | |
| }, | |
| { | |
| "cell_type": "markdown", | |
| "metadata": { | |
| "id": "c_JZhWI1A-5I", | |
| "colab_type": "text" | |
| }, | |
| "source": [ | |
| "Alice measures her two qubits. In cirq we can control on these registers which will assumed to be classical after the measurement." | |
| ] | |
| }, | |
| { | |
| "cell_type": "code", | |
| "metadata": { | |
| "id": "_mstAC3BcjV3", | |
| "colab_type": "code", | |
| "colab": { | |
| "base_uri": "https://localhost:8080/", | |
| "height": 102 | |
| }, | |
| "outputId": "a2cf03c6-a633-4bee-fd4e-d5d4075a9da1" | |
| }, | |
| "source": [ | |
| "teleportation.append(cirq.measure(msg, alice))\n", | |
| "\n", | |
| "print(teleportation)" | |
| ], | |
| "execution_count": null, | |
| "outputs": [ | |
| { | |
| "output_type": "stream", | |
| "text": [ | |
| "alice: ───H───────────@───X───────M───\n", | |
| " │ │ │\n", | |
| "bob: ─────────────────X───┼───────┼───\n", | |
| " │ │\n", | |
| "msg: ─────Ry(0.25π)───────@───H───M───\n" | |
| ], | |
| "name": "stdout" | |
| } | |
| ] | |
| }, | |
| { | |
| "cell_type": "markdown", | |
| "metadata": { | |
| "id": "jaE40zTMBRdm", | |
| "colab_type": "text" | |
| }, | |
| "source": [ | |
| "Assuming Bob now has received the two bits from Alice's measurement, he performs the conditional operations $X$ and $Z$." | |
| ] | |
| }, | |
| { | |
| "cell_type": "code", | |
| "metadata": { | |
| "id": "BB-RquG6c2co", | |
| "colab_type": "code", | |
| "colab": { | |
| "base_uri": "https://localhost:8080/", | |
| "height": 102 | |
| }, | |
| "outputId": "28e3e394-c5dc-44b6-a3e2-b039e12b4e59" | |
| }, | |
| "source": [ | |
| "teleportation.append([\n", | |
| " cirq.CNOT(alice, bob),\n", | |
| " cirq.CZ(msg, bob)\n", | |
| " ])\n", | |
| "\n", | |
| "print(teleportation)" | |
| ], | |
| "execution_count": null, | |
| "outputs": [ | |
| { | |
| "output_type": "stream", | |
| "text": [ | |
| "alice: ───H───────────@───X───────M───@───────\n", | |
| " │ │ │ │\n", | |
| "bob: ─────────────────X───┼───────┼───X───@───\n", | |
| " │ │ │\n", | |
| "msg: ─────Ry(0.25π)───────@───H───M───────@───\n" | |
| ], | |
| "name": "stdout" | |
| } | |
| ] | |
| }, | |
| { | |
| "cell_type": "markdown", | |
| "metadata": { | |
| "id": "tIW5Wn9LBbZQ", | |
| "colab_type": "text" | |
| }, | |
| "source": [ | |
| "In principle, that is it! Let's run the simulation again to see if Alice's state has been teleported." | |
| ] | |
| }, | |
| { | |
| "cell_type": "code", | |
| "metadata": { | |
| "id": "ZvKsTw4YdT1x", | |
| "colab_type": "code", | |
| "colab": { | |
| "base_uri": "https://localhost:8080/", | |
| "height": 34 | |
| }, | |
| "outputId": "5d4911b1-8ded-4169-9eca-523a5cb739a9" | |
| }, | |
| "source": [ | |
| "bob_received = simulator.simulate(teleportation)\n", | |
| "print(bob_received.final_state[[1,3]]) # the state is a 3-qubit state, you need to play around to find the correct indices of Bob's qubit" | |
| ], | |
| "execution_count": null, | |
| "outputs": [ | |
| { | |
| "output_type": "stream", | |
| "text": [ | |
| "[0.92387944+0.j 0.38268343-0.j]\n" | |
| ], | |
| "name": "stdout" | |
| } | |
| ] | |
| }, | |
| { | |
| "cell_type": "markdown", | |
| "metadata": { | |
| "id": "2i3dkPcHBh2b", | |
| "colab_type": "text" | |
| }, | |
| "source": [ | |
| "Excellent! Now it's your turn. Here we wrote an ugly script if it's put all together. Why don't you create your own functions to perform teleportation of any state?" | |
| ] | |
| } | |
| ] | |
| } |
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