4.a-Quantum-Fourier-Transform.ipynb

{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Quantum Fourier Transform \n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {},
   "outputs": [],
   "source": [
    "from qiskit import *\n",
    "from math import pi\n",
    "import numpy as np\n",
    "from qiskit.visualization import *\n",
    "import matplotlib.pyplot as plt\n",
    "from qutip import *"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "\n",
    "\n",
    "QFT is defined as\n",
    "\n",
    "$ QFT:|x> = \\frac{1}{\\sqrt{N}}\\sum_{k=0}^{N-1} \\omega_{x}^{k}|k>$\n",
    "\n",
    "where $\\omega_{x}^{k}$ is $N^{th}$ ($N = 2^{n}$) root of unity: $e^{\\frac{2\\pi i}{2^{n}}}$.\n",
    "\n",
    "QFT: $F_{N} =  \\frac{1}{\\sqrt{N}} \\begin{bmatrix}\n",
    "    1  &         1   &        1       &     1          &   \\cdots   &  1   \\\\\n",
    "    1  &  \\omega_{n} & \\omega_{n}^{2} & \\omega_{n}^{3} &   \\cdots   & \\omega_{n} ^{N-1}\\\\\n",
    "    1  &  \\omega_{n}^{2} & \\omega_{n}^{4} & \\omega_{n}^{6} &   \\cdots   & \\omega_{n} ^{2(N-1)}\\\\\n",
    "    1  &  \\omega_{n}^{3} & \\omega_{n}^{6} & \\omega_{n}^{9} &   \\cdots   & \\omega_{n} ^{3(N-1)}\\\\\n",
    "   \\vdots  & \\vdots  & \\vdots         & \\vdots         &    \\dots   & \\vdots \\\\\n",
    "   1  &  \\omega_{n}^{(N-1)} & \\omega_{n}^{2(N-1)} & \\omega_{n}^{3(N-1)} &   \\cdots   & \\omega_{n} ^{(N-1((N-1)}\\\\\n",
    "\\end{bmatrix}$\n",
    "\n",
    "\n",
    "\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### 1. QFT ( 1 qubit)\n",
    "\n",
    "\n",
    "\n",
    "For single qubit circuit ($n = 1, N = 2^{1} = 2)$\n",
    "\n",
    "\n",
    "$\\omega_n = e^{\\frac{2\\pi i}{2^{n}}} = -1$.\n",
    "\n",
    "QFT = $\\frac{1}{\\sqrt{2}} \\begin{bmatrix}\n",
    "    1  &   1 \\\\\n",
    "    1  &  -1\n",
    "\\end{bmatrix}$\n",
    "    \n",
    "It is very simple, QFT in single qubit id just a Hadamate operation.\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### 2. QFT (2 qubits)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "\n",
    "    \n",
    "For two qubit circuit, ($n =2, N = 2^{2} = 4)$\n",
    "\n",
    "\n",
    "$\\omega_{n} = e^{\\frac{2\\pi i}{2^{n}}} = i$\n",
    "\n",
    "$QFT_2 = \\frac{1}{{2}} \\begin{bmatrix}\n",
    "    1  &   1  &  1  &   1\\\\\n",
    "    1  &   i  &  -1  &  -i\\\\\n",
    "    1  &   -1  &  1  &   -1\\\\\n",
    "    1  &  -i  &  -1  &  i\n",
    "\\end{bmatrix}$\n",
    " \n",
    "Our task is to represent this matrix in terms of fundamental gate metrix."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### 2.2 Circuit"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 599.592x144.48 with 1 Axes>"
      ]
     },
     "execution_count": 6,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "qc = QuantumCircuit(2)\n",
    "qc.h(1)\n",
    "qc.barrier()\n",
    "qc.cu1(np.pi/2, 0, 1)\n",
    "qc.barrier()\n",
    "qc.h(0)\n",
    "qc.barrier()\n",
    "qc.swap(0,1)\n",
    "qc.draw('mpl')"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### 2.3  State vector\n",
    "\n",
    "Lets observe what happens to quantum state as we pass through these quantum gates one by one: \n",
    "\n",
    "\n",
    "\n",
    "<ul>    \n",
    "<li> Initial state: |00>\n",
    "<li> After Hadamard Gate on qubit 2 : $\\frac{1}{\\sqrt{2}}|00> + \\frac{1}{\\sqrt{2}}|10> $\n",
    "<li> After phase shift Gate : $\\frac{1}{\\sqrt{2}}|00> + \\frac{1}{\\sqrt{2}}|10> $\n",
    "<li> After Hadamard Gate on qubit 1 : $\\frac{1}{{2}}|00> + \\frac{1}{{2}}|01>  +  \\frac{1}{{2}}|10> - \\frac{1}{{2}}|11>$\n",
    "<li> After SWAP:  $\\frac{1}{{2}}|00> + \\frac{1}{{2}}|10>  +  \\frac{1}{{2}}|01> - \\frac{1}{{2}}|11>$\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[0.5+0.j 0.5+0.j 0.5+0.j 0.5+0.j]\n"
     ]
    }
   ],
   "source": [
    "backend = Aer.get_backend('statevector_simulator')\n",
    "out = execute(qc,backend).result().get_statevector()\n",
    "print(out)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### 2.5 State vector as tensor product"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "\n",
    "    \n",
    " Since this circuit consists of pure superposition brought up by hadamard gate, it is possible to obtain final state vector by taking direct product of single qubit states. Results from operation of H gate on single qubit can be multiplied with direct product operation to get final state.\n",
    "    \n",
    "$ \\frac{1}{\\sqrt{2}} (|0 \\rangle + | 1\\rangle) \\otimes \\frac{1}{\\sqrt{2}}(|0 \\rangle + | 1\\rangle) = \\frac{1}{2}(|00 \\rangle + |01 \\rangle + |10 \\rangle + |11 \\rangle $\n",
    "    \n",
    "\n",
    "i.e., $\\frac{1}{\\sqrt{2}}\\begin{bmatrix}\n",
    "    1   \\\\\n",
    "    1 \n",
    "\\end{bmatrix} \\otimes \\frac{1}{\\sqrt{2}} \\begin{bmatrix}\n",
    "    1   \\\\\n",
    "    1 \n",
    "\\end{bmatrix} = \\frac{1}{2} \\begin{bmatrix}\n",
    "    1   \\\\\n",
    "    1 \\\\\n",
    "    1 \\\\\n",
    "    1\n",
    "\\end{bmatrix}$\n",
    "\n",
    "Where $|00 \\rangle , |01 \\rangle , |10 \\rangle $ and $ |11 \\rangle $ are basis states for two qubit system.\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### 2.6 Matrix Element"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### How to realize it the quantum circuit?\n",
    "\n",
    "\n",
    "    \n",
    "    \n",
    "$ I = \\begin{bmatrix}\n",
    "    1  &   0 \\\\\n",
    "    0  &   1\n",
    "\\end{bmatrix};H = \\frac{1}{\\sqrt{2}} \\begin{bmatrix}\n",
    "    1  &   0 \\\\\n",
    "    0  &   1\n",
    "\\end{bmatrix}; C_{u1} = \\begin{pmatrix}\n",
    "1 & 0 & 0 & 0\\\\\n",
    "0 & 1 & 0 & 0\\\\\n",
    "0 & 0 & 1 & 0\\\\\n",
    "0 & 0 & 0 & i\n",
    "\\end{pmatrix}; SWAP =\\begin{pmatrix}\n",
    "1 & 0 & 0 & 0\\\\\n",
    "0 & 0 & 1 & 0\\\\\n",
    "0 & 1 & 0 & 0\\\\\n",
    "0 & 0 & 0 & 1\n",
    "\\end{pmatrix}$\n",
    "\n",
    "\n",
    "At first barrier: $ U_1 = I \\otimes H $\n",
    "    \n",
    "At second barrier: $ U_2 =  C _{u1} \\times (I \\otimes H) $\n",
    "    \n",
    "At third barrier: $ U_3 = (H \\otimes I) \\times C _{u1} \\times (I \\otimes H) $    \n",
    "\n",
    "At fourth barrier: $U_4 =  SWAP \\times (H \\otimes I) \\times C _{u1} \\times (I \\otimes H) $     "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[[ 0.5+0.j   0.5-0.j   0.5-0.j   0.5-0.j ]\n",
      " [ 0.5+0.j   0. +0.5j -0.5+0.j  -0. -0.5j]\n",
      " [ 0.5+0.j  -0.5+0.j   0.5-0.j  -0.5+0.j ]\n",
      " [ 0.5+0.j  -0. -0.5j -0.5+0.j   0. +0.5j]]\n"
     ]
    }
   ],
   "source": [
    "backend = Aer.get_backend('unitary_simulator')\n",
    "job = execute(qc, backend)\n",
    "result = job.result()\n",
    "print(result.get_unitary(qc, decimals=3))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "- Implementing Numpy"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 99,
   "metadata": {},
   "outputs": [],
   "source": [
    "I = np.eye(2,2)\n",
    "H = 1/np.sqrt(2)*np.array([[1,1],[1,-1]])\n",
    "H_kron_I = np.kron(H,I)\n",
    "CU1 = np.array([[1,0,0,0],[0,1,0,0],[0,0,1,0],[0,0,0,0.+1.j]])\n",
    "I_kron_H = np.kron(I,H)\n",
    "SWAP = np.array([[1,0,0,0],[0,0,1,0],[0,1,0,0],[0,0,0,1]])"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 100,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([[ 0.5+0.j ,  0.5+0.j ,  0.5+0.j ,  0.5+0.j ],\n",
       "       [ 0.5+0.j ,  0. +0.5j, -0.5+0.j ,  0. -0.5j],\n",
       "       [ 0.5+0.j , -0.5+0.j ,  0.5+0.j , -0.5+0.j ],\n",
       "       [ 0.5+0.j ,  0. -0.5j, -0.5+0.j ,  0. +0.5j]])"
      ]
     },
     "execution_count": 100,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "U = np.dot(H_kron_I,np.dot(CU1,np.dot(I_kron_H,SWAP)))\n",
    "U"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 101,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([0.5+0.j, 0.5+0.j, 0.5+0.j, 0.5+0.j])"
      ]
     },
     "execution_count": 101,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "ket = np.array([1,0,0,0])\n",
    "np.dot(U,ket)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### 3. QFT (3 qubits)\n",
    "\n",
    "#### 3.1 Circuit Diagram"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "metadata": {},
   "outputs": [],
   "source": [
    "def qft3():\n",
    "    n = 3\n",
    "    q = QuantumRegister(n)\n",
    "    c = ClassicalRegister(n)\n",
    "    qc = QuantumCircuit(q,c)\n",
    "    qc.h(q[2])\n",
    "    qc.barrier()\n",
    "    qc.cu1(np.pi/2, q[1], q[2])\n",
    "    qc.barrier()\n",
    "    qc.h(q[1])\n",
    "    qc.barrier()\n",
    "    qc.cu1(np.pi/4, q[0], q[2])\n",
    "    qc.barrier()\n",
    "    qc.cu1(np.pi/2, q[0], q[1])\n",
    "    qc.barrier()\n",
    "    qc.h(q[0])\n",
    "    qc.barrier()\n",
    "    qc.swap(q[0], q[2])\n",
    "    return q,c,qc"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 1031.83x264.88 with 1 Axes>"
      ]
     },
     "execution_count": 14,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "q,c,qc = qft3()\n",
    "qc.barrier()\n",
    "qc.draw(output='mpl')"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### 3.2 State vector\n",
    "\n",
    "\n",
    "    \n",
    "<ul>        \n",
    "<li> Initial state: |000>\n",
    "<li> After Hadamard Gate on 3rd qubit : $\\frac{1}{\\sqrt{2}}(|000> + |001>) $\n",
    "<li> After CU1 gate on qubit (2,3) : $\\frac{1}{\\sqrt{2}}(|000> + |001>) $\n",
    "<li> After CU1 gate on qubit (1,3) : $\\frac{1}{\\sqrt{2}}(|000> + |001>) $\n",
    "<li> After Hadamard Gate on qubit 2 : $\\frac{1}{{2}}(|000> + |010> +  |001> -|011> )$\n",
    "<li> After CU1 gate on qubit (0,1): $\\frac{1}{{2}}(|000> + |010> +  |001> -|011> )$\n",
    "<li> After Hadamard Gate on qubit 3 : $\\frac{1}{2\\sqrt{2}}(|000> + |001> +  |010> + |011> +  |000> - |001> - |010> + |011)$\n",
    "    \n",
    "    "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 16,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[0.35355339+0.j 0.35355339+0.j 0.35355339+0.j 0.35355339+0.j\n",
      " 0.35355339+0.j 0.35355339+0.j 0.35355339+0.j 0.35355339+0.j]\n"
     ]
    }
   ],
   "source": [
    "backend = Aer.get_backend('statevector_simulator')\n",
    "out = execute(qc,backend).result().get_statevector()\n",
    "print(out)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Measurement"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### References\n",
    "-----\n",
    "1. https://www.youtube.com/watch?v=bntew-yoMzk "
   ]
  }
 ],
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