BV_algorithm.ipynb

{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Bernstein-Vazirani Algorithm"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Let us assume a black box which has n bits. How many attempts does a classical computer vs QC takes to figure out those 'n' bits?"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Bernstein-Vazirani Problem"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We are given a blackbox function f, which takes as input string of bits (x), and retruns either 0 or 1.\n",
    "\n",
    "$f(\\{x_0,x_1,x_2,...\\}) \\rightarrow 0 \\textrm{ or } 1 \\textrm{ where } x_n \\textrm{ is }0 \\textrm{ or } 1$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Instead of the function being balanced or constant as in the Deutsch-Josza problem, now the function is guaranteed to return the bitwise product of the input with some string, s.\n",
    "In other words, given an input x:\n",
    "\n",
    "$  \n",
    "f\n",
    "(\n",
    "x\n",
    ")\n",
    "=\n",
    "s\n",
    "⋅\n",
    "x\n",
    "(mod 2)$\n",
    "\n",
    "We are expected to find  s.\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## The Classical Solution:"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Classically, the oracle returns:\n",
    "$f_s(x) = s \\cdot x \\mod 2 $\n",
    " \n",
    "given an input x. Thus, the hidden bit string  \n",
    "s can be revealed by querying the oracle with the sequence of inputs:\n",
    "\n",
    "Input(x)\n",
    "\n",
    "100...0\n",
    "\n",
    "010...0\n",
    "\n",
    "001...0\n",
    "\n",
    "000...1\n",
    "\n",
    "Where each query reveals a different bit of s. For example, with x = 1000...0 one can obtain the least significant bit of s, with x = 0100...0 we can find the next least significant, and so on. This means we would need to call the function $f_s(x) $, n times.\n",
    "\n",
    "For example, if s=101000, then you need to query this six times to find s using classical approach. \n",
    "\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "#101\n",
    "000 -put AND gate ->010\n",
    "--- 3 step "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## The Quantum Solution"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Using a quantum computer, we can solve this problem with 100% confidence after only one call to the function $f(x)$. The quantum Bernstein-Vazirani algorithm to find the hidden bit string is very simple:\n",
    "\n",
    "1) Initialise the inputs qubits to the  $|0\\rangle^{\\otimes n}$ state, and output qubit to $|−⟩$.\n",
    "\n",
    "2) Apply Hadamard gates to the input register\n",
    "\n",
    "3) Query the oracle\n",
    "\n",
    "4) Apply Hadamard gates to the input register\n",
    "\n",
    "5) Measure"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "To explain the algorithm, let’s look more closely at what happens when we apply a H-gate to each qubit. If we have an n-qubit state, |a⟩, and apply the H-gates, we will see the transformation:\n",
    "        \n",
    "$|a\\rangle \\xrightarrow{H^{\\otimes n}} \\frac{1}{\\sqrt{2^n}} \\sum_{x\\in \\{0,1\\}^n} (-1)^{a\\cdot x}|x\\rangle.$\n",
    "\n",
    "Explanation:\n",
    "\n",
    "$H|0\\rangle = \\tfrac{1}{\\sqrt{2}}(|0\\rangle + |1\\rangle)$\n",
    "\n",
    "$H|1\\rangle = \\tfrac{1}{\\sqrt{2}}(|0\\rangle - |1\\rangle)$\n",
    "\n",
    "Using summation notation above equation can be written as:\n",
    "\n",
    "$H|a\\rangle = \\frac{1}{\\sqrt{2}}\\sum_{x\\in \\{0,1\\}} (-1)^{a\\cdot x}|x\\rangle.$\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "In particular, when we start with a quantum register|00…0⟩ and apply  n Hadamard gates to it, we have the familiar quantum superposition:\n",
    "\n",
    "$|00\\dots 0\\rangle \\xrightarrow{H^{\\otimes n}} \\frac{1}{\\sqrt{2^n}} \\sum_{x\\in \\{0,1\\}^n} |x\\rangle $\n",
    "\n",
    "The algorithm to reveal the hidden bit string follows naturally by querying the quantum oracle  fs with the quantum superposition obtained from the Hadamard transformation of  |00…0⟩. Namely,\n",
    "\n",
    "$|00\\dots 0\\rangle \\xrightarrow{H^{\\otimes n}} \\frac{1}{\\sqrt{2^n}} \\sum_{x\\in \\{0,1\\}^n} |x\\rangle \\xrightarrow{f_a} \\frac{1}{\\sqrt{2^n}} \\sum_{x\\in \\{0,1\\}^n} (-1)^{a\\cdot x}|x\\rangle$\n",
    "\n",
    "Because the inverse of the n Hadamard gates is again the n Hadamard gates, we can obtain a by\n",
    "\n",
    "$\\frac{1}{\\sqrt{2^n}} \\sum_{x\\in \\{0,1\\}^n} (-1)^{a\\cdot x}|x\\rangle \\xrightarrow{H^{\\otimes n}} |a\\rangle$\n",
    "\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Let's start with an example"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Here we are trying to find string hidden inside the oracle."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "n=2\n",
    "\n",
    "s=11 # This is the secret number. "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "1) The register of two qubits is initialized to zero:\n",
    "$|\n",
    "ψ\n",
    "0\n",
    "⟩\n",
    "=\n",
    "|\n",
    "0\n",
    "0\n",
    "⟩$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "2) Apply a Hadamard gate to both qubits:\n",
    "$\\lvert \\psi_1 \\rangle = \\frac{1}{2} \\left( \\lvert 0 0 \\rangle + \\lvert 0 1 \\rangle + \\lvert 1 0 \\rangle + \\lvert 1 1 \\rangle \\right)$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "3) For the string  \n",
    "s=11, the quantum oracle performs the operation:\n",
    "$|x \\rangle \\xrightarrow{f_s} (-1)^{x\\cdot 11} |x \\rangle.$\n",
    "\n",
    "$\\lvert \\psi_2 \\rangle = \\frac{1}{2} \\left( (-1)^{00\\cdot 11}|00\\rangle + (-1)^{01\\cdot 11}|01\\rangle + (-1)^{10\\cdot 11}|10\\rangle + (-1)^{11\\cdot 11}|11\\rangle \\right)$\n",
    " \n",
    "$\\lvert \\psi_2 \\rangle = \\frac{1}{2} \\left( \\lvert 0 0 \\rangle - \\lvert 0 1 \\rangle - \\lvert 1 0 \\rangle + \\lvert 1 1 \\rangle \\right)$\n",
    " \n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "4) Apply a hadamard gate to both qubits:\n",
    " \n",
    "$\\lvert \\psi_3 \\rangle = \\lvert 1 1 \\rangle$  \n",
    "\n",
    "5) Measure to find the secret string s=11."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Qiskit Implementation"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {},
   "outputs": [],
   "source": [
    "# initialization\n",
    "import matplotlib.pyplot as plt\n",
    "import numpy as np\n",
    "\n",
    "# importing Qiskit\n",
    "from qiskit import IBMQ, BasicAer\n",
    "from qiskit.providers.ibmq import least_busy\n",
    "from qiskit import QuantumCircuit, ClassicalRegister, QuantumRegister, execute\n",
    "\n",
    "# import basic plot tools\n",
    "from qiskit.visualization import plot_histogram"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {},
   "outputs": [],
   "source": [
    "n = 5# number of qubits used to represent s\n",
    "s = '10011'   # the hidden binary string"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<pre style=\"word-wrap: normal;white-space: pre;background: #fff0;line-height: 1.1;font-family: &quot;Courier New&quot;,Courier,monospace\">     ┌───┐      ░                      ░ ┌───┐┌─┐            \n",
       "q_0: ┤ H ├──────░───■──────────────────░─┤ H ├┤M├────────────\n",
       "     ├───┤      ░   │                  ░ ├───┤└╥┘┌─┐         \n",
       "q_1: ┤ H ├──────░───┼─────────■────────░─┤ H ├─╫─┤M├─────────\n",
       "     ├───┤      ░   │  ┌───┐  │        ░ ├───┤ ║ └╥┘┌─┐      \n",
       "q_2: ┤ H ├──────░───┼──┤ I ├──┼────────░─┤ H ├─╫──╫─┤M├──────\n",
       "     ├───┤      ░   │  ├───┤  │        ░ ├───┤ ║  ║ └╥┘┌─┐   \n",
       "q_3: ┤ H ├──────░───┼──┤ I ├──┼────────░─┤ H ├─╫──╫──╫─┤M├───\n",
       "     ├───┤      ░   │  └───┘  │        ░ ├───┤ ║  ║  ║ └╥┘┌─┐\n",
       "q_4: ┤ H ├──────░───┼─────────┼────■───░─┤ H ├─╫──╫──╫──╫─┤M├\n",
       "     ├───┤┌───┐ ░ ┌─┴─┐     ┌─┴─┐┌─┴─┐ ░ └───┘ ║  ║  ║  ║ └╥┘\n",
       "q_5: ┤ H ├┤ Z ├─░─┤ X ├─────┤ X ├┤ X ├─░───────╫──╫──╫──╫──╫─\n",
       "     └───┘└───┘ ░ └───┘     └───┘└───┘ ░       ║  ║  ║  ║  ║ \n",
       "c_0: ══════════════════════════════════════════╩══╬══╬══╬══╬═\n",
       "                                                  ║  ║  ║  ║ \n",
       "c_1: ═════════════════════════════════════════════╩══╬══╬══╬═\n",
       "                                                     ║  ║  ║ \n",
       "c_2: ════════════════════════════════════════════════╩══╬══╬═\n",
       "                                                        ║  ║ \n",
       "c_3: ═══════════════════════════════════════════════════╩══╬═\n",
       "                                                           ║ \n",
       "c_4: ══════════════════════════════════════════════════════╩═\n",
       "                                                             </pre>"
      ],
      "text/plain": [
       "     ┌───┐      ░                      ░ ┌───┐┌─┐            \n",
       "q_0: ┤ H ├──────░───■──────────────────░─┤ H ├┤M├────────────\n",
       "     ├───┤      ░   │                  ░ ├───┤└╥┘┌─┐         \n",
       "q_1: ┤ H ├──────░───┼─────────■────────░─┤ H ├─╫─┤M├─────────\n",
       "     ├───┤      ░   │  ┌───┐  │        ░ ├───┤ ║ └╥┘┌─┐      \n",
       "q_2: ┤ H ├──────░───┼──┤ I ├──┼────────░─┤ H ├─╫──╫─┤M├──────\n",
       "     ├───┤      ░   │  ├───┤  │        ░ ├───┤ ║  ║ └╥┘┌─┐   \n",
       "q_3: ┤ H ├──────░───┼──┤ I ├──┼────────░─┤ H ├─╫──╫──╫─┤M├───\n",
       "     ├───┤      ░   │  └───┘  │        ░ ├───┤ ║  ║  ║ └╥┘┌─┐\n",
       "q_4: ┤ H ├──────░───┼─────────┼────■───░─┤ H ├─╫──╫──╫──╫─┤M├\n",
       "     ├───┤┌───┐ ░ ┌─┴─┐     ┌─┴─┐┌─┴─┐ ░ └───┘ ║  ║  ║  ║ └╥┘\n",
       "q_5: ┤ H ├┤ Z ├─░─┤ X ├─────┤ X ├┤ X ├─░───────╫──╫──╫──╫──╫─\n",
       "     └───┘└───┘ ░ └───┘     └───┘└───┘ ░       ║  ║  ║  ║  ║ \n",
       "c_0: ══════════════════════════════════════════╩══╬══╬══╬══╬═\n",
       "                                                  ║  ║  ║  ║ \n",
       "c_1: ═════════════════════════════════════════════╩══╬══╬══╬═\n",
       "                                                     ║  ║  ║ \n",
       "c_2: ════════════════════════════════════════════════╩══╬══╬═\n",
       "                                                        ║  ║ \n",
       "c_3: ═══════════════════════════════════════════════════╩══╬═\n",
       "                                                           ║ \n",
       "c_4: ══════════════════════════════════════════════════════╩═\n",
       "                                                             "
      ]
     },
     "execution_count": 9,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# We need a circuit with n qubits, plus one ancilla qubit\n",
    "# Also need n classical bits to write the output to\n",
    "bv_circuit = QuantumCircuit(n+1, n)\n",
    "# put ancilla in state |->\n",
    "bv_circuit.h(n)\n",
    "bv_circuit.z(n)\n",
    "# Apply Hadamard gates before querying the oracle\n",
    "for i in range(n):\n",
    "    bv_circuit.h(i)\n",
    " \n",
    "# Apply barrier \n",
    "bv_circuit.barrier()\n",
    "\n",
    "# Apply the inner-product oracle\n",
    "s = s[::-1] # reverse s to fit qiskit's qubit ordering\n",
    "for q in range(n):\n",
    "    if s[q] == '0':\n",
    "        bv_circuit.i(q)\n",
    "    else:\n",
    "        bv_circuit.cx(q, n)\n",
    "        \n",
    "# Apply barrier \n",
    "bv_circuit.barrier()\n",
    "\n",
    "#Apply Hadamard gates after querying the oracle\n",
    "for i in range(n):\n",
    "    bv_circuit.h(i)\n",
    "\n",
    "# Measurement\n",
    "for i in range(n):\n",
    "    bv_circuit.measure(i, i)\n",
    "\n",
    "bv_circuit.draw()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/plain": [
       "<Figure size 504x360 with 1 Axes>"
      ]
     },
     "execution_count": 10,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# use local simulator\n",
    "backend = BasicAer.get_backend('qasm_simulator')\n",
    "shots = 1024\n",
    "results = execute(bv_circuit, backend=backend, shots=shots).result()\n",
    "answer = results.get_counts()\n",
    "\n",
    "plot_histogram(answer)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "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": "3.7.6"
  }
 },
 "nbformat": 4,
 "nbformat_minor": 4
}