ODSC_Lab1_Solution.ipynb

{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# ODSC Lab 1: Fair and Biased Coin Flip Solution"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "One of the powers of quantum is that we can easily generate coin flips by coding a simple circuit and measuring the probabilities of our qubit states. Let's assign a 'Heads' coin flip to our qubit outcome '0' and assign a 'Tails' flip to our qubit outcome '1'. In this lab, we will construct quantum circuit that create a fair 50-50 distribution, a 70-30 biased distrubtion, and a 0-100 completely biased distribution. Before trying the labs yourself, please make use of the Pyquil helper notation below! "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Necessary Imports"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "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": 6,
   "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 thetea 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": [
    "## Circuit 1: Fair Coin"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "In order to generate a fair coin distribution, we want the probability of getting a 'Heads' to be the same as the probability of getting a 'Tails'. "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "{'0': 0.4999999999999999, '1': 0.4999999999999999}\n"
     ]
    }
   ],
   "source": [
    "n_qubits = 1 #Sets the number of qubits or coin flips \n",
    "main_program = Program() #Creates a quantum circuit program \n",
    "for i in range(n_qubits): \n",
    "    main_program += H(i) #Performs a Hadamard Gate on the ith 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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     "image/png": 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"
    }
   },
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "![image.png](attachment:df678ff3-dd04-4bb9-8262-1f3793daeabd.png)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Circuit 2: 70-30 Biased Coin"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "In order to generate a 70-30 biased coin distribution, we want the probability of getting a 'Heads' to be 0.7 and the probability of getting a 'Tails' to be 0.3. In order to find the apropriate angle for our rotation, we need to do a bit of a guess and check. Have a try!  "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "(1+0j)|1>\n",
      "{'0': 3.749399456654644e-33, '1': 1.0}\n"
     ]
    }
   ],
   "source": [
    "main_program = Program() #Creates a quantum circuit program \n",
    "for i in range(n_qubits): \n",
    "    main_program += RY(theta/2.71,i) #Performs a Y-Rotation with the angle theta/2.71 on the ith qubit\n",
    "wave_function =  qvm.wavefunction(main_program) #Get the output quantum circuit wavefunction\n",
    "print(wave_function)\n",
    "probs_dist    =  wave_function.get_outcome_probs()  #Get the probability distribution\n",
    "print(probs_dist)\n"
   ]
  },
  {
   "attachments": {
    "593abfd9-3203-4f53-b9d4-35eaba35bd05.png": {
     "image/png": 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"
    }
   },
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "![Screen Shot 2020-10-15 at 9.17.15 PM.png](attachment:593abfd9-3203-4f53-b9d4-35eaba35bd05.png)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Circuit 3: 0-100 Biased Coin"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "In order to generate a 100-0 biased coin distribution, we want the probability of getting a 'Heads' to be 1 and the probability of getting a 'Tails'."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 54,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "{'0': 0.0, '1': 1.0}\n"
     ]
    }
   ],
   "source": [
    "main_program = Program() #Creates a quantum circuit program \n",
    "for i in range(n_qubits): \n",
    "    main_program += X(i) #Performs a Not Gate on the ith 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": {
    "0fa1266d-8560-4646-86b4-fa5250d8dcfd.png": {
     "image/png": 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"
    }
   },
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "![Screen Shot 2020-10-15 at 9.17.40 PM.png](attachment:0fa1266d-8560-4646-86b4-fa5250d8dcfd.png)"
   ]
  }
 ],
 "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": "3.8.5"
  }
 },
 "nbformat": 4,
 "nbformat_minor": 4
}