Lab1.ipynb

{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Lab 1: Fair and Biased Coin Flip"
   ]
  },
  {
   "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": 2,
   "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",
    "import matplotlib.pyplot as plt\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": [
    "## 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": 3,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "{'0': 0.4999999999999999, '1': 0.4999999999999999}\n"
     ]
    }
   ],
   "source": [
    "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": [
    "![Screen Shot 2020-10-15 at 9.14.41 PM.png](attachment:343bbd3a-24c4-448e-898c-4785ac497e83.png)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Exercise: 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": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "main_program = Program() #Creates a quantum circuit program\n",
    "\n",
    "# Your code here\n",
    "# Hint: The Hadamard gate won't help, you may need an RX or RY gate with a certain angle"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Exercise: Circuit 3: 0-100 Biased Coin"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "In order to generate a 0-100 biased coin distribution, we want the probability of getting a 'Heads' to be 0 and the probability of getting a 'Tails' to be 1."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {},
   "outputs": [],
   "source": [
    "main_program = Program() #Creates a quantum circuit program \n",
    "\n",
    "# Your code here\n",
    "# Hint: You can use an RX or RY gate with a particular angle, but there may be an easier way with a simpler gate!"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
  }
 ],
 "metadata": {
  "kernelspec": {
   "display_name": "Python 3",
   "language": "python",
   "name": "python3"
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  "language_info": {
   "codemirror_mode": {
    "name": "ipython",
    "version": 3
   },
   "file_extension": ".py",
   "mimetype": "text/x-python",
   "name": "python",
   "nbconvert_exporter": "python",
   "pygments_lexer": "ipython3",
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