lessons/misc/quantum-computing/grovers-algorthim-2-qubits.ipynb

{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Simulating Grover's Search Algorithm with 2 Qubits"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "import numpy as np\n",
    "from matplotlib import pyplot as plt\n",
    "%matplotlib inline"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Define the zero and one vectors\n",
    "Define the initial state $\\psi$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 22,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[[1]\n",
      " [0]\n",
      " [0]\n",
      " [0]]\n"
     ]
    }
   ],
   "source": [
    "zero = np.matrix([[1],[0]]);\n",
    "one = np.matrix([[0],[1]]);\n",
    "\n",
    "psi = np.kron(zero,zero);\n",
    "print(psi)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Define the gates we will use:\n",
    "\n",
    "$\n",
    "\\text{Id} = \\begin{pmatrix} \n",
    "1 & 0  \\\\\n",
    "0 & 1  \n",
    "\\end{pmatrix},\n",
    "\\quad\n",
    "X = \\begin{pmatrix} \n",
    "0 & 1   \\\\\n",
    "1 & 0  \n",
    "\\end{pmatrix},\n",
    "\\quad\n",
    "Z = \\begin{pmatrix} \n",
    "1 &  0 \\\\\n",
    "0 & -1 \n",
    "\\end{pmatrix},\n",
    "\\quad\n",
    "H = \\frac{1}{\\sqrt{2}}\\begin{pmatrix} \n",
    "1 & 1   \\\\\n",
    "1 & -1  \n",
    "\\end{pmatrix},\n",
    "\\quad\n",
    "\\text{CNOT} = \\begin{pmatrix} \n",
    "1 & 0 & 0 & 0 \\\\\n",
    "0 & 1 & 0 & 0 \\\\  \n",
    "0 & 0 & 0 & 1 \\\\\n",
    "0 & 0 & 1 & 0\n",
    "\\end{pmatrix},\n",
    "\\quad\n",
    "CZ = (\\text{Id} \\otimes H) \\text{ CNOT } (\\text{Id} \\otimes H) \n",
    "$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 23,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[[ 1.  0.  0.  0.]\n",
      " [ 0.  1.  0.  0.]\n",
      " [ 0.  0.  1.  0.]\n",
      " [ 0.  0.  0. -1.]]\n"
     ]
    }
   ],
   "source": [
    "Id = np.matrix([[1,0],[0,1]]);\n",
    "X = np.matrix([[0,1],[1,0]]);\n",
    "Z = np.matrix([[1,0],[0,-1]]);\n",
    "H = np.sqrt(0.5) * np.matrix([[1,1],[1,-1]]);\n",
    "\n",
    "CNOT = np.matrix([[1,0,0,0],[0,1,0,0],[0,0,0,1],[0,0,1,0]]);\n",
    "CZ = np.kron(Id,H).dot(CNOT).dot(np.kron(Id,H));\n",
    "print(CZ)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Define the oracle for Grover's algorithm  (take search answer to be \"10\")\n",
    "\n",
    "$\n",
    "\\text{oracle} = \\begin{pmatrix} \n",
    "1 & 0 & 0 & 0  \\\\\n",
    "0 & 1 & 0 & 0 \\\\\n",
    "0 & 0 & -1 & 0 \\\\\n",
    "0 & 0 & 0 & 1\n",
    "\\end{pmatrix}\n",
    "= (Z \\otimes \\text{Id})  CZ\n",
    "$\n",
    "\n",
    "Use different combinations of $Z \\otimes \\text{Id}$ to change where search answer is. "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 24,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[[ 1.  0.  0.  0.]\n",
      " [ 0.  1.  0.  0.]\n",
      " [ 0.  0. -1.  0.]\n",
      " [ 0.  0.  0.  1.]]\n"
     ]
    }
   ],
   "source": [
    "oracle = np.kron(Z,Id).dot(CZ);\n",
    "print(oracle)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Act the H gates on the input vector and apply the oracle "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 26,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[[ 0.5]\n",
      " [ 0.5]\n",
      " [-0.5]\n",
      " [ 0.5]]\n"
     ]
    }
   ],
   "source": [
    "psi0 = np.kron(H,H).dot(psi);\n",
    "psi1 = oracle.dot(psi0);\n",
    "print(psi1)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Remember that when we measure the result (\"00\", \"01\", \"10\", \"11\") is chosen randomly with probabilities given by the vector elements squared."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 27,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[[ 0.25]\n",
      " [ 0.25]\n",
      " [ 0.25]\n",
      " [ 0.25]]\n"
     ]
    }
   ],
   "source": [
    "print(np.multiply(psi1,psi1))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "There is no difference between any of the probabilities. It's still just a 25% chance of getting the right answer. \n",
    "\n",
    "We need some of gates after the oracle before measuring to converge on the right answer.  \n",
    "\n",
    "These gates do the operation $W = \\frac{1}{2}\\begin{pmatrix} \n",
    "-1 & 1 & 1 & 1  \\\\\n",
    "1 & -1 & 1 & 1 \\\\\n",
    "1 & 1 & -1 & 1 \\\\\n",
    "1 & 1 & 1 & -1\n",
    "\\end{pmatrix}\n",
    "=\n",
    "(H \\otimes H)(Z \\otimes Z) CZ (H \\otimes H)\n",
    "$\n",
    "\n",
    "Notice that if the matrix W is multiplied by the vector after the oracle, W $\\frac{1}{2}\\begin{pmatrix} \n",
    "1 \\\\\n",
    "1 \\\\\n",
    "-1 \\\\\n",
    "1\n",
    "\\end{pmatrix} \n",
    "= \\begin{pmatrix} \n",
    "0 \\\\\n",
    "0 \\\\\n",
    "1 \\\\\n",
    "0\n",
    "\\end{pmatrix}  $,\n",
    "every vector element decreases, except the correct answer element which increases. This would be true if if we chose a different place for the search result originally."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 29,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[[-0.5  0.5  0.5  0.5]\n",
      " [ 0.5 -0.5  0.5  0.5]\n",
      " [ 0.5  0.5 -0.5  0.5]\n",
      " [ 0.5  0.5  0.5 -0.5]]\n"
     ]
    }
   ],
   "source": [
    "W = np.kron(H,H).dot(np.kron(Z,Z)).dot(CZ).dot(np.kron(H,H));\n",
    "print(W)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 31,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[[  1.23259516e-32]\n",
      " [  3.08148791e-33]\n",
      " [  1.00000000e+00]\n",
      " [  3.08148791e-33]]\n"
     ]
    }
   ],
   "source": [
    "psif = W.dot(psi1);\n",
    "print(np.multiply(psif,psif))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 44,
   "metadata": {},
   "outputs": [
    {
     "data": {
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      "text/plain": [
       "<matplotlib.figure.Figure at 0x10ecb0ba8>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "x = [0,1,2,3];\n",
    "xb = [0.25,1.25,2.25,3.25];\n",
    "labels=['00', '01', '10', '11'];\n",
    "plt.axis([-0.5,3.5,-1.25,1.25]);\n",
    "plt.xticks(x,labels);\n",
    "plt.bar(x, np.ravel(psi0), 1/1.5, color=\"red\");\n",
    "plt.bar(xb, np.ravel(np.multiply(psi0,psi0)), 1/2., color=\"blue\");"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 45,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAX8AAAD8CAYAAACfF6SlAAAABHNCSVQICAgIfAhkiAAAAAlwSFlz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\nA9M+lMGwYYbcAAAAAElFTkSuQmCC\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x10ec82b00>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "labels=['00', '01', '10', '11'];\n",
    "plt.axis([-0.5,3.5,-1.25,1.25]);\n",
    "plt.xticks(x,labels);\n",
    "plt.bar(x, np.ravel(psi1), 1/1.5, color=\"red\");\n",
    "plt.bar(xb, np.ravel(np.multiply(psi1,psi1)), 1/2., color=\"blue\");"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 46,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAX8AAAD8CAYAAACfF6SlAAAABHNCSVQICAgIfAhkiAAAAAlwSFlz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\n/fp5hKcupA8AAAAASUVORK5CYII=\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x10ea345f8>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "labels=['00', '01', '10', '11'];\n",
    "plt.axis([-0.5,3.5,-1.25,1.25]);\n",
    "plt.xticks(x,labels);\n",
    "plt.bar(x, np.ravel(psif), 1/1.5, color=\"red\");\n",
    "plt.bar(xb, np.ravel(np.multiply(psif,psif)), 1/2., color=\"blue\");"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": []
  }
 ],
 "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": "3.6.2"
  }
 },
 "nbformat": 4,
 "nbformat_minor": 2
}