Lab2_Solution.ipynb

{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Lab 2: Quantum Entangler Solution"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "An additional power of quantum is that we can entangle two particles or pieces of information together such that having the knowledge of one piece will provide us with knowledge about the other. There are different ways in which we can generate entanglement using quantum gates, but the one requirement is that we utilize two-qubit gates. These gates are actually difficult to implement physically and therefore, remain a limitation of near-term quantum computing. Thus, it is important that we find ways to create short depth circuits that use less two-qubit gates. \n",
    "\n",
    "In this lab, we will construct a quantum circuit that creates a Bell State for two-qubits, which is what we know as perfect or maximum entanglement. For a challenge, you may also add on qubits and see how this entanglement becomes harder to generate with an increased number of qubits! "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Necessary Imports"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {},
   "outputs": [],
   "source": [
    "#load the necessary packages and libraries\n",
    "from pyquil.quil import Program\n",
    "import pyquil.api as api\n",
    "from pyquil.gates import *\n",
    "import matplotlib.pyplot as plt\n",
    "from random import *\n",
    "import numpy as np\n",
    "import math\n",
    "\n",
    "qvm = api.QVMConnection()  "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Pyquil Notation Tricks "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {},
   "outputs": [],
   "source": [
    "n_qubits = 1 #Sets the number of qubits or coin flips \n",
    "\n",
    "theta = math.pi #Sets the 1-qubit gate rotation angle theta to pi\n",
    "\n",
    "main_program = Program() #Creates a quantum circuit program \n",
    "\n",
    "main_program += H(0) #Performs a Hadamard Gate on the 0th qubit \n",
    "\n",
    "main_program += X(0) #Performs a Not Gate on the 0th qubit \n",
    "\n",
    "main_program += CNOT(0,1) #Performs a Controlled-Not Gate with 0th qubit as the control and the 1st qubit as the target\n",
    "\n",
    "main_program += RZ(theta, 0) #Performs a Pauli Z Rotation by angle theta on the 0th qubit \n",
    "\n",
    "main_program += RX(theta, 0) #Performs a Pauli X Rotation by angle theta on the 0th qubit \n",
    "\n",
    "main_program += RY(theta, 0) #Performs a Pauli Y Rotation by angle theta on the 0th qubit \n",
    "\n",
    "wave_function =  qvm.wavefunction(main_program) #Get the output quantum circuit wavefunction\n",
    "\n",
    "probs_dist    =  wave_function.get_outcome_probs()  #Get the probability distribution"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## 1. Circuit: Two qubit entangler "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "In order to entangle two qubits, we need to utilize a two qubit CNOT gate as well as our single qubit rotation gates. We are looking to generate the state with an equal probability on '00' and '11'. "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "{'00': 0.4999999999999999, '01': 0.0, '10': 0.0, '11': 0.4999999999999999}\n"
     ]
    }
   ],
   "source": [
    "main_program = Program() #Creates a quantum circuit program \n",
    "main_program += H(0) #Performs a Hadamard Gate on the 0th qubit\n",
    "main_program += CNOT(0,1) #Performs a CNOT Gate on the control 0th qubit to the target 1st qubit\n",
    "wave_function =  qvm.wavefunction(main_program) #Get the output quantum circuit wavefunction\n",
    "probs_dist    =  wave_function.get_outcome_probs()  #Get the probability distribution\n",
    "print(probs_dist)"
   ]
  },
  {
   "attachments": {
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"
    }
   },
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "![Screen Shot 2020-10-16 at 10.58.53 AM.png](attachment:8589c582-b789-4140-b000-42883dc3f3ce.png)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## 2. Exercise: Another two qubit entangler"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Build a circuit that generates the state with an equal probability on '01' and '10', and zero probability on '00' and '11'."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "{'00': 0.0, '01': 0.4999999999999999, '10': 0.4999999999999999, '11': 0.0}\n"
     ]
    }
   ],
   "source": [
    "# All we need to do is to add an NOT gate on any qubit at the end of the previous circuit\n",
    "\n",
    "main_program = Program() #Creates a quantum circuit program \n",
    "main_program += H(0) #Performs a Hadamard Gate on the 0th qubit\n",
    "main_program += CNOT(0,1) #Performs a CNOT Gate on the control 0th qubit to the target 1st qubit\n",
    "main_program += X(0)\n",
    "wave_function =  qvm.wavefunction(main_program) #Get the output quantum circuit wavefunction\n",
    "probs_dist    =  wave_function.get_outcome_probs()  #Get the probability distribution\n",
    "print(probs_dist)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
  }
 ],
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