Deutsch-Josza_Algorithm.ipynb

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   "source": [
    "# Deutsch-Josza Algorithm"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Deutsch-Josza Algorithm  is the first algorithm that brought clear quantum advantage over classical computing. It was introduced by David Deutsch and Richard Jozsa in 1992.[1] In Deutsch’s problem we are given a black box which computes a one-bit boolean function. That is, a function which takes in one bit and outputs one bit. We can represent the function f as $f\\{0,1\\} \\rightarrow \\{0,1\\}$ \n"
   ]
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   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The function takes n-digit binary values as input and produces either a 0 or a 1 as output for each such value. We are promised that the function is either constant (0 on all outputs or 1 on all outputs) or balanced (returns 1 for half of the input domain and 0 for the other half). The task then is to determine if $f$ is constant or balanced by using the oracle."
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  {
   "cell_type": "markdown",
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   "source": [
    "Let us look at the classical version of the problem for two bit input."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 15,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Input:  [0, 1]\n",
      "Output: [0, 0]\n"
     ]
    }
   ],
   "source": [
    "# consider a case for constant function\n",
    "# For constant output\n",
    "constant=0\n",
    "I=[0,1]\n",
    "def constant_f(constant,I=[0,1]): # function to create a constant output\n",
    "    O_constant =[constant for _ in I]\n",
    "    return O_constant\n",
    "\n",
    "# Output for constant function with constant: 0\n",
    "O_constant=constant_f(0)\n",
    "\n",
    "print(\"Input: \", I)\n",
    "print(\"Output:\", O_constant)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 16,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Input:  [0, 1]\n",
      "Output: [1, 0]\n"
     ]
    }
   ],
   "source": [
    "# consider a case for balanced function\n",
    "\n",
    "def balanced_f(reverse= False, I=[0,1]):  # function to create a balanced function\n",
    "    O_balanced=I\n",
    "    if reverse:\n",
    "        O_balanced= []\n",
    "    return O_balanced\n",
    "\n",
    "# Output for balanced function with negation or reverse= True\n",
    "O_balanced= balanced_f(reverse=True)\n",
    "\n",
    "print(\"Input: \", I)\n",
    "print(\"Output:\", O_balanced)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
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     "text": [
      "Constant\n"
     ]
    }
   ],
   "source": [
    "# Determine which fuction: constant or balanced is used.\n",
    "def determine_function(Output):\n",
    "    if Output[0]==Output[1]:\n",
    "        print(\"Constant\")\n",
    "    else:\n",
    "        print(\"Balanced\")\n",
    "\n",
    "# Check the function for our case\n",
    "determine_function(O)\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Deutsch-Josza Algorithm in quantum computing"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {},
   "outputs": [],
   "source": [
    "#importing libraries and functions\n",
    "%matplotlib inline\n",
    "from math import pi\n",
    "import numpy as np\n",
    "from qiskit import *\n",
    "from qiskit.visualization import *\n",
    "from qiskit.compiler import transpile, assemble\n",
    "from qiskit.tools.jupyter import *\n",
    "from qiskit import Aer\n",
    "from qiskit.providers.aer import unitary_simulator\n",
    "backend = Aer.get_backend('unitary_simulator')\n",
    "simulator = Aer.get_backend('qasm_simulator')\n",
    "statevector= Aer.get_backend('statevector_simulator')\n",
    "%config InlineBackend.figure_format = 'svg'"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 17,
   "metadata": {},
   "outputs": [],
   "source": [
    "\n",
    "def blackbox(n):  # Create a blackbox that outputs either balanced or constant function randomly\n",
    "    output = np.random.randint(2)\n",
    "    # We need to make a QuantumCircuit object to return\n",
    "    # This circuit has n+1 qubits: the size of the input,\n",
    "    # plus one output qubit\n",
    "    qc = QuantumCircuit(n+1) #circuit for oracle\n",
    "    if output==0:      #for constant oracle case\n",
    "         # First decide what the fixed output of the oracle will be\n",
    "        # (either always 0 or always 1)\n",
    "        constant = np.random.randint(2)\n",
    "        if constant == 1:\n",
    "            qc.x(n)\n",
    "\n",
    "    \n",
    "    if output==1:      #for balanced oracle case\n",
    "        for qubit in range(n):\n",
    "            qc.cx(qubit, n)\n",
    "    \n",
    "    return qc.to_gate()\n",
    "def dj_algorithm(n):\n",
    "    qc=QuantumCircuit(n+1,n+1)\n",
    "    qc.x(n)\n",
    "    for qubit in range(n+1):\n",
    "        qc.h(qubit)\n",
    "    qc.append(blackbox(n),range(n+1))\n",
    "    for qubit in range(n):\n",
    "        qc.h(qubit)\n",
    "    for i in range(n):\n",
    "        qc.measure(i,i)\n",
    "   \n",
    "    return qc"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 30,
   "metadata": {},
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     },
     "metadata": {},
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    },
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "{'00000': 1000}\n"
     ]
    }
   ],
   "source": [
    "counts=execute(qc,simulator,shots=1000).result().get_counts(qc) # simulator = Aer.get_backend('qasm_simulator')\n",
    "display(plot_histogram(counts))\n",
    "print(counts)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 32,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Constant\n"
     ]
    }
   ],
   "source": [
    "for key in counts.keys():\n",
    "    key=int(key)\n",
    "    if key==0:\n",
    "         print('Constant')\n",
    "    else:\n",
    "        print('balanced')"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## References"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "1. https://qiskit.org/textbook/ch-algorithms/deutsch-jozsa.html\n",
    "2. Hidary, Jack. (2019). Quantum Computing: An Applied Approach. 10.1007/978-3-030-23922-0.\n",
    "3. https://en.wikipedia.org/wiki/Deutsch%E2%80%93Jozsa_algorithm\n",
    "4. Michael A Nielsen and Isaac Chuang.Quantum  computation  and  quantum  infor-mation.  Cambridge  University  Press,  2002.isbn:  978-1-107-00217-3.url:www.cambridge.org%5C9781107002173."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
  }
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