Tr2-6.5-6.6.ipynb

{
 "cells": [
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {},
   "outputs": [],
   "source": [
    "import numpy as np\n",
    "from math import *\n",
    "from qiskit import *\n",
    "from qiskit.tools.visualization import iplot_state_city\n",
    "import matplotlib.pyplot as plt"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {},
   "outputs": [],
   "source": [
    "#Helper function to find state\n",
    "def getStateVector(circ):\n",
    "    result = execute(circ, backend = BasicAer.get_backend('statevector_simulator')).result()\n",
    "    psi = result.get_statevector(circ)\n",
    "    return psi"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "***Exercise 6.5:***\n",
    "\n",
    "$\\newcommand{\\ket}[1]{\\left|{#1}\\right\\rangle}\n",
    "\\newcommand{\\bra}[1]{\\left\\langle{#1}\\right|}$\n",
    "a. Prove the following theorem:\n",
    "\n",
    "    When any one of Alice’s or Bob’s spin operators acts on a\n",
    "    product state, the result is still a product state.\n",
    "    \n",
    "The code below shows that the effect of applying an X-gate to flip the second qubit (Bob) still results in a product state vector with four components. Symbolically, this is:\n",
    "\n",
    "$\\tau _{x} \\ket{00} = \\ket{01}$\n",
    "\n",
    "b. Show that in a product state, the expectation value of any component of σ or τ is exactly the same as it would be in the individual single-spin states.\n",
    "\n",
    "So if we had an individual spin state , $\\ket{0}$ , the expectation value of $\\tau_{x}$ would be $\\bra{0} \\tau_{x}\\ket{0}$.\n",
    "\n",
    "$\\bra{0} \\tau_{x} \\ket{0} = \\begin{bmatrix}   \n",
    "                    1&&0 \n",
    "                    \\end{bmatrix} \\begin{bmatrix}   \n",
    "                                    0&&1\\\\\n",
    "                                     1&&0\n",
    "                                    \\end{bmatrix} \\begin{bmatrix}   \n",
    "                                                        1 \\\\\n",
    "                                                        0 \n",
    "                                                        \\end{bmatrix} = 0$\n",
    "                                                        \n",
    "Similarly, for the product state $\\ket{00}$, the expecation value of $\\tau_{x}$ would be $\\bra{00} \\tau_{x} \\ket{00}$.\n",
    "\n",
    "$\\bra{00} \\tau_{x} \\ket{ 00} = \\begin{bmatrix}   \n",
    "                    1&&0&&0&&0\n",
    "                    \\end{bmatrix} (I \\otimes \\tau_{x}) \\begin{bmatrix}   \n",
    "                                                        1 \\\\\n",
    "                                                        0 \\\\\n",
    "                                                        0 \\\\\n",
    "                                                        0 \n",
    "                                                        \\end{bmatrix} = \\begin{bmatrix}   \n",
    "                                                                        1&&0&&0&&0\n",
    "                                                                        \\end{bmatrix} \\begin{bmatrix}   \n",
    "                                                                                        0 \\\\\n",
    "                                                                                        1 \\\\\n",
    "                                                                                        0 \\\\\n",
    "                                                                                        0 \n",
    "                                                                                        \\end{bmatrix} =0$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "State Vector: [0.70710678+0.j 0.70710678+0.j 0.        +0.j 0.        +0.j]\n",
      "{'01': 42, '00': 58}\n"
     ]
    },
    {
     "data": {
      "image/png": "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\n",
      "text/plain": [
       "<Figure size 421.4x258.86 with 1 Axes>"
      ]
     },
     "execution_count": 4,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "q = QuantumRegister(2)\n",
    "c = ClassicalRegister(2)\n",
    "\n",
    "circ = QuantumCircuit(q,c) #the state is |00>\n",
    "circ.h(0) #flips the second qubit so the state becomes |01>\n",
    "\n",
    "print('State Vector:' , getStateVector(circ))\n",
    "\n",
    "meas = QuantumCircuit(q,c)\n",
    "meas.measure(q,c)\n",
    "\n",
    "final = circ + meas\n",
    "shots = 100\n",
    "job = execute(final, backend = Aer.get_backend('qasm_simulator'), shots = shots)\n",
    "counts = job.result().get_counts(final)\n",
    "print(counts)\n",
    "final.draw(output='mpl')"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "***Exercise 6.6:***\n",
    "\n",
    "$\\newcommand{\\ket}[1]{\\left|{#1}\\right\\rangle}\n",
    "\\newcommand{\\bra}[1]{\\left\\langle{#1}\\right|}$\n",
    "Assume Charlie has prepared the two spins in the singlet state. This time, Bob measures $τ_{y}$ and Alice measures $σ_{x}$. What is the expectation value of $σ_{x} τ_{y}$? What does this say about the correlation between the two measurements?\n",
    "\n",
    "Below we first prepare a singlet state described by:\n",
    "\n",
    "$\\ket{sing} = \\frac {1}{\\sqrt{2}} (\\ket{01} - \\ket{10})$\n",
    "\n",
    "Then, we applied the a Y-gate to the second qubit and then an X-gate to the first qubit, and found the expectation value of $\\bra{sing}σ_{x}τ_{y}\\ket{sing}$.\n",
    "\n",
    "$\\bra{sing}σ_{x}τ_{y}\\ket{sing} = \\begin{bmatrix}   \n",
    "                            0&&\\frac{1}{\\sqrt{2}}&&-\\frac{1}{\\sqrt{2}}&&0\n",
    "                            \\end{bmatrix} (σ_{x} \\otimes \\tau_{y}) \\begin{bmatrix}   \n",
    "                                                                0 \\\\\n",
    "                                                                \\frac{1}{\\sqrt{2}}\\\\\n",
    "                                                                -\\frac{1}{\\sqrt{2}} \\\\\n",
    "                                                                0 \n",
    "                                                                \\end{bmatrix} = \\begin{bmatrix}   \n",
    "                                                                                0&&\\frac{1}{\\sqrt{2}}&&-\\frac{1}{\\sqrt{2}}&&0\n",
    "                                                                                    \\end{bmatrix} \\begin{bmatrix}   \n",
    "                                                                                                0 \\\\\n",
    "                                                                                                -\\frac{i}{\\sqrt{2}} \\\\\n",
    "                                                                                                -\\frac{i}{\\sqrt{2}} \\\\\n",
    "                                                                                                0 \n",
    "                                                                                                \\end{bmatrix} = 0$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {
    "scrolled": true
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "State Vector: [ 0.        +0.j  0.70710678+0.j -0.70710678+0.j  0.        +0.j]\n",
      "Counts: {'10': 49, '01': 51}\n"
     ]
    },
    {
     "data": {
      "text/html": [
       "\n",
       "    <p>\n",
       "        <div id=\"content_15628593440327086\" style=\"position: absolute; z-index: 1;\">\n",
       "            <div id=\"cities_15628593440327086\"></div>\n",
       "        </div>\n",
       "    </p>\n",
       "    \n",
       "    <script>\n",
       "        requirejs.config({\n",
       "            paths: {\n",
       "                qVisualization: \"https://qvisualization.mybluemix.net/q-visualizations\"\n",
       "            }\n",
       "        });\n",
       "\n",
       "        require([\"qVisualization\"], function(qVisualizations) {\n",
       "            data = {\n",
       "                real: [[0.0, 0.0, 0.0, 0.0], [0.0, 0.5000000000000001, -0.5000000000000001, 0.0], [0.0, -0.5000000000000001, 0.5000000000000001, 0.0], [0.0, 0.0, 0.0, 0.0]],\n",
       "                titleReal: \"Real.[rho]\",\n",
       "                imaginary: [[0.0, 0.0, -0.0, 0.0], [0.0, 0.0, -0.0, 0.0], [0.0, 0.0, 0.0, 0.0], [0.0, 0.0, -0.0, 0.0]],\n",
       "                titleImaginary: \"Im.[rho]\",\n",
       "                qbits: 4\n",
       "            };\n",
       "            qVisualizations.plotState(\"cities_15628593440327086\",\n",
       "                                      \"cities\",\n",
       "                                      data,\n",
       "                                      {});\n",
       "        });\n",
       "    </script>\n",
       "    "
      ],
      "text/plain": [
       "<IPython.core.display.HTML object>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "#Preparing Singlet State\n",
    "q = QuantumRegister(2)\n",
    "c = ClassicalRegister(2)\n",
    "\n",
    "circS = QuantumCircuit(q,c)\n",
    "circS.x(1) \n",
    "circS.x(0)\n",
    "circS.h(1)\n",
    "circS.cx(q[1],q[0])\n",
    "\n",
    "print('State Vector:' , getStateVector(circS))\n",
    "\n",
    "#Measuring Singlet State\n",
    "circ2 = QuantumCircuit(q,c)\n",
    "circ2.measure(q,c)\n",
    "\n",
    "singlet = circS + circ2\n",
    "\n",
    "shots = 100\n",
    "job = execute(singlet, backend = Aer.get_backend('qasm_simulator'), shots = shots)\n",
    "counts = job.result().get_counts(singlet)\n",
    "\n",
    "print('Counts:' , counts)\n",
    "iplot_state_city(getStateVector(circS))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Counts: {'10': 9, '01': 11}\n",
      "State Vector After tao_y and sigma_x: [0.+0.j         0.-0.70710678j 0.-0.70710678j 0.+0.j        ]\n",
      "Singlet State: [ 0.        +0.j  0.70710678+0.j -0.70710678+0.j  0.        +0.j]\n",
      "Expectation Value, <sigma_x.tao_y>: 0j\n"
     ]
    }
   ],
   "source": [
    "#Apply the X and Y gates\n",
    "circ3 = QuantumCircuit(q,c)\n",
    "circ3.y(0)\n",
    "circ3.x(1)\n",
    "\n",
    "tao_y = circS + circ3\n",
    "\n",
    "#Measure\n",
    "circ4 = QuantumCircuit(q,c)\n",
    "circ4.measure(q,c)\n",
    "\n",
    "tao_y_meas = circS + circ3 + circ4\n",
    "\n",
    "shots = 20\n",
    "job = execute(tao_y_meas, backend = Aer.get_backend('qasm_simulator'), shots = shots)\n",
    "counts = job.result().get_counts(tao_y_meas)\n",
    "print('Counts:' , counts)\n",
    "\n",
    "print('State Vector After tao_y and sigma_x:' , getStateVector(tao_y))\n",
    "print('Singlet State:' , getStateVector(circS))\n",
    "\n",
    "#Take inner product\n",
    "print('Expectation Value, <sigma_x.tao_y>:' , np.transpose(getStateVector(circS))@getStateVector(tao_y))\n"
   ]
  }
 ],
 "metadata": {
  "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": "3.7.3"
  }
 },
 "nbformat": 4,
 "nbformat_minor": 2
}