shor_ibm_fixed.ipynb

{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {
    "tags": [
     "remove_cell"
    ]
   },
   "source": [
    "# Shor's Algorithm"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Shor’s algorithm is famous for factoring integers in polynomial time. Since the best-known classical algorithm requires superpolynomial time to factor the product of two primes, the widely used cryptosystem, RSA, relies on factoring being impossible for large enough integers.\n",
    "\n",
    "In this chapter we will focus on the quantum part of Shor’s algorithm, which actually solves the problem of _period finding_. Since a factoring problem can be turned into a period finding problem in polynomial time, an efficient period finding algorithm can be used to factor integers efficiently too. For now its enough to show that if we can compute the period of $a^x\\bmod N$ efficiently, then we can also efficiently factor. Since period finding is a worthy problem in its own right, we will first solve this, then discuss how this can be used to factor in section 5."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {
    "tags": [
     "thebelab-init"
    ]
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Imports Successful\n"
     ]
    }
   ],
   "source": [
    "import matplotlib.pyplot as plt\n",
    "import numpy as np\n",
    "from qiskit import QuantumCircuit, Aer, transpile, assemble\n",
    "from qiskit.visualization import plot_histogram\n",
    "from math import gcd\n",
    "from numpy.random import randint\n",
    "import pandas as pd\n",
    "from fractions import Fraction\n",
    "print(\"Imports Successful\")"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## 1. The Problem: Period Finding\n",
    "\n",
    "Let’s look at the periodic function:\n",
    "\n",
    "$$ f(x) = a^x \\bmod{N}$$\n",
    "\n",
    "<details>\n",
    "    <summary>Reminder: Modulo &amp; Modular Arithmetic (Click here to expand)</summary>\n",
    "\n",
    "The modulo operation (abbreviated to 'mod') simply means to find the remainder when dividing one number by another. For example:\n",
    "\n",
    "$$ 17 \\bmod 5 = 2 $$\n",
    "\n",
    "Since $17 \\div 5 = 3$ with remainder $2$. (i.e. $17 = (3\\times 5) + 2$). In Python, the modulo operation is denoted through the <code>%</code> symbol.\n",
    "\n",
    "This behaviour is used in <a href=\"https://en.wikipedia.org/wiki/Modular_arithmetic\">modular arithmetic</a>, where numbers 'wrap round' after reaching a certain value (the modulus). Using modular arithmetic, we could write:\n",
    "\n",
    "$$ 17 = 2 \\pmod 5$$\n",
    "\n",
    "Note that here the $\\pmod 5$ applies to the entire equation (since it is in parenthesis), unlike the equation above where it only applied to the left-hand side of the equation.\n",
    "</details>\n",
    "\n",
    "where $a$ and $N$ are positive integers, $a$ is less than $N$, and they have no common factors. The period, or order ($r$), is the smallest (non-zero) integer such that:\n",
    "\n",
    "$$a^r \\bmod N = 1 $$ \n",
    "\n",
    "We can see an example of this function plotted on the graph below. Note that the lines between points are to help see the periodicity and do not represent the intermediate values between the x-markers."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 114,
   "metadata": {
    "tags": [
     "remove_input"
    ]
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "a^( 0 * 12 ) mod 35  = 1\n",
      "a^( 1 * 12 ) mod 35  = 1\n",
      "a^( 2 * 12 ) mod 35  = 1\n"
     ]
    },
    {
     "data": {
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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "# Fix original notebook () by ch4rl13b: \n",
    "# \n",
    "#\n",
    "# Original test value\n",
    "# N = 35\n",
    "# a = 3\n",
    "\n",
    "N = 35\n",
    "a = 3\n",
    "\n",
    "# Calculate the plotting data\n",
    "xvals = np.arange(N)\n",
    "yvals = [np.mod(a**x, N) for x in xvals]\n",
    "\n",
    "# Use matplotlib to display it nicely\n",
    "fig, ax = plt.subplots()\n",
    "ax.plot(xvals, yvals, linewidth=1, linestyle='dotted', marker='x')\n",
    "ax.set(xlabel='$x$', ylabel='$%i^x$ mod $%i$' % (a, N),\n",
    "       title=\"Example of Periodic Function in Shor's Algorithm\")\n",
    "try: # plot r on the graph\n",
    "    p = yvals[1:].index(1) +1 \n",
    "    plt.annotate('', xy=(0,1), xytext=(p,1), arrowprops=dict(arrowstyle='<->'))\n",
    "    plt.annotate('$p=%i$' % p, xy=(p/3,1.5))\n",
    "except ValueError:\n",
    "    print('Could not find period, check a < N and have no common factors.')\n",
    "\n",
    "# Testing that a^(k*p) mod N = 1\n",
    "\n",
    "for k in range(3):\n",
    "    print(\"a^(\",k,\"*\",p,\") mod\",N,\" =\",np.mod(a**(k*p), N))\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## 2. The Solution\n",
    "\n",
    "Shor’s solution was to use [quantum phase estimation](./quantum-phase-estimation.html) on the unitary operator:\n",
    "\n",
    "$$ U|y\\rangle \\equiv |ay \\bmod N \\rangle $$\n",
    "\n",
    "To see how this is helpful, let’s work out what an eigenstate of U might look like. If we started in the state $|1\\rangle$, we can see that each successive application of U will multiply the state of our register by $a \\pmod N$, and after $r$ applications we will arrive at the state $|1\\rangle$ again. For example with $a = 3$ and $N = 35$:\n",
    "\n",
    "$$\\begin{aligned}\n",
    "U|1\\rangle &= |3\\rangle & \\\\\n",
    "U^2|1\\rangle &= |9\\rangle \\\\\n",
    "U^3|1\\rangle &= |27\\rangle \\\\\n",
    "& \\vdots \\\\\n",
    "U^{(r-1)}|1\\rangle &= |12\\rangle \\\\\n",
    "U^r|1\\rangle &= |1\\rangle \n",
    "\\end{aligned}$$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {
    "scrolled": false,
    "tags": [
     "remove_input"
    ]
   },
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "execution_count": 3,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "ax.set(xlabel='Number of applications of U', ylabel='End state of register',\n",
    "       title=\"Effect of Successive Applications of U\")\n",
    "fig"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "So a superposition of the states in this cycle ($|u_0\\rangle$) would be an eigenstate of $U$:\n",
    "\n",
    "$$|u_0\\rangle = \\tfrac{1}{\\sqrt{r}}\\sum_{k=0}^{r-1}{|a^k \\bmod N\\rangle} $$\n",
    "\n",
    "\n",
    "<details>\n",
    "    <summary>Click to Expand: Example with $a = 3$ and $N=35$</summary>\n",
    "\n",
    "$$\\begin{aligned}\n",
    "|u_0\\rangle &= \\tfrac{1}{\\sqrt{12}}(|1\\rangle + |3\\rangle + |9\\rangle \\dots + |4\\rangle + |12\\rangle) \\\\[10pt]\n",
    "U|u_0\\rangle &= \\tfrac{1}{\\sqrt{12}}(U|1\\rangle + U|3\\rangle + U|9\\rangle \\dots + U|4\\rangle + U|12\\rangle) \\\\[10pt]\n",
    " &= \\tfrac{1}{\\sqrt{12}}(|3\\rangle + |9\\rangle + |27\\rangle \\dots + |12\\rangle + |1\\rangle) \\\\[10pt]\n",
    " &= |u_0\\rangle\n",
    "\\end{aligned}$$\n",
    "</details>\n",
    "\n",
    "\n",
    "This eigenstate has an eigenvalue of 1, which isn’t very interesting. A more interesting eigenstate could be one in which the phase is different for each of these computational basis states. Specifically, let’s look at the case in which the phase of the $k$th state is proportional to $k$:\n",
    "\n",
    "$$\\begin{aligned}\n",
    "|u_1\\rangle &= \\tfrac{1}{\\sqrt{r}}\\sum_{k=0}^{r-1}{e^{-\\tfrac{2\\pi i k}{r}}|a^k \\bmod N\\rangle}\\\\[10pt]\n",
    "U|u_1\\rangle &= e^{\\tfrac{2\\pi i}{r}}|u_1\\rangle \n",
    "\\end{aligned}\n",
    "$$\n",
    "\n",
    "<details>\n",
    "    <summary>Click to Expand: Example with $a = 3$ and $N=35$</summary>\n",
    "\n",
    "$$\\begin{aligned}\n",
    "|u_1\\rangle &= \\tfrac{1}{\\sqrt{12}}(|1\\rangle + e^{-\\tfrac{2\\pi i}{12}}|3\\rangle + e^{-\\tfrac{4\\pi i}{12}}|9\\rangle \\dots + e^{-\\tfrac{20\\pi i}{12}}|4\\rangle + e^{-\\tfrac{22\\pi i}{12}}|12\\rangle) \\\\[10pt]\n",
    "U|u_1\\rangle &= \\tfrac{1}{\\sqrt{12}}(|3\\rangle + e^{-\\tfrac{2\\pi i}{12}}|9\\rangle + e^{-\\tfrac{4\\pi i}{12}}|27\\rangle \\dots + e^{-\\tfrac{20\\pi i}{12}}|12\\rangle + e^{-\\tfrac{22\\pi i}{12}}|1\\rangle) \\\\[10pt]\n",
    "U|u_1\\rangle &= e^{\\tfrac{2\\pi i}{12}}\\cdot\\tfrac{1}{\\sqrt{12}}(e^{\\tfrac{-2\\pi i}{12}}|3\\rangle + e^{-\\tfrac{4\\pi i}{12}}|9\\rangle + e^{-\\tfrac{6\\pi i}{12}}|27\\rangle \\dots + e^{-\\tfrac{22\\pi i}{12}}|12\\rangle + e^{-\\tfrac{24\\pi i}{12}}|1\\rangle) \\\\[10pt]\n",
    "U|u_1\\rangle &= e^{\\tfrac{2\\pi i}{12}}|u_1\\rangle\n",
    "\\end{aligned}$$\n",
    "\n",
    "(We can see $r = 12$ appears in the denominator of the phase.)\n",
    "</details>\n",
    "\n",
    "This is a particularly interesting eigenvalue as it contains $r$. In fact, $r$ has to be included to make sure the phase differences between the $r$ computational basis states are equal. This is not the only eigenstate with this behaviour; to generalise this further, we can multiply an integer, $s$, to this phase difference, which will show up in our eigenvalue:\n",
    "\n",
    "$$\\begin{aligned}\n",
    "|u_s\\rangle &= \\tfrac{1}{\\sqrt{r}}\\sum_{k=0}^{r-1}{e^{-\\tfrac{2\\pi i s k}{r}}|a^k \\bmod N\\rangle}\\\\[10pt]\n",
    "U|u_s\\rangle &= e^{\\tfrac{2\\pi i s}{r}}|u_s\\rangle \n",
    "\\end{aligned}\n",
    "$$\n",
    "\n",
    "<details>\n",
    "    <summary>Click to Expand: Example with $a = 3$ and $N=35$</summary>\n",
    "\n",
    "$$\\begin{aligned}\n",
    "|u_s\\rangle &= \\tfrac{1}{\\sqrt{12}}(|1\\rangle + e^{-\\tfrac{2\\pi i s}{12}}|3\\rangle + e^{-\\tfrac{4\\pi i s}{12}}|9\\rangle \\dots + e^{-\\tfrac{20\\pi i s}{12}}|4\\rangle + e^{-\\tfrac{22\\pi i s}{12}}|12\\rangle) \\\\[10pt]\n",
    "U|u_s\\rangle &= \\tfrac{1}{\\sqrt{12}}(|3\\rangle + e^{-\\tfrac{2\\pi i s}{12}}|9\\rangle + e^{-\\tfrac{4\\pi i s}{12}}|27\\rangle \\dots + e^{-\\tfrac{20\\pi i s}{12}}|12\\rangle + e^{-\\tfrac{22\\pi i s}{12}}|1\\rangle) \\\\[10pt]\n",
    "U|u_s\\rangle &= e^{\\tfrac{2\\pi i s}{12}}\\cdot\\tfrac{1}{\\sqrt{12}}(e^{-\\tfrac{2\\pi i s}{12}}|3\\rangle + e^{-\\tfrac{4\\pi i s}{12}}|9\\rangle + e^{-\\tfrac{6\\pi i s}{12}}|27\\rangle \\dots + e^{-\\tfrac{22\\pi i s}{12}}|12\\rangle + e^{-\\tfrac{24\\pi i s}{12}}|1\\rangle) \\\\[10pt]\n",
    "U|u_s\\rangle &= e^{\\tfrac{2\\pi i s}{12}}|u_s\\rangle\n",
    "\\end{aligned}$$\n",
    "\n",
    "</details>\n",
    "\n",
    "We now have a unique eigenstate for each integer value of $s$ where $$0 \\leq s \\leq r-1.$$ Very conveniently, if we sum up all these eigenstates, the different phases cancel out all computational basis states except $|1\\rangle$:\n",
    "\n",
    "$$ \\tfrac{1}{\\sqrt{r}}\\sum_{s=0}^{r-1} |u_s\\rangle = |1\\rangle$$\n",
    "\n",
    "<details>\n",
    "    <summary>Click to Expand: Example with $a = 7$ and $N=15$</summary>\n",
    "\n",
    "For this, we will look at a smaller example where $a = 7$ and $N=15$. In this case $r=4$:\n",
    "\n",
    "$$\\begin{aligned}\n",
    "\\tfrac{1}{2}(\\quad|u_0\\rangle &= \\tfrac{1}{2}(|1\\rangle \\hphantom{e^{-\\tfrac{2\\pi i}{12}}}+ |7\\rangle \\hphantom{e^{-\\tfrac{12\\pi i}{12}}} + |4\\rangle \\hphantom{e^{-\\tfrac{12\\pi i}{12}}} + |13\\rangle)\\dots \\\\[10pt]\n",
    "+ |u_1\\rangle &= \\tfrac{1}{2}(|1\\rangle + e^{-\\tfrac{2\\pi i}{4}}|7\\rangle + e^{-\\tfrac{\\hphantom{1}4\\pi i}{4}}|4\\rangle + e^{-\\tfrac{\\hphantom{1}6\\pi i}{4}}|13\\rangle)\\dots \\\\[10pt]\n",
    "+ |u_2\\rangle &= \\tfrac{1}{2}(|1\\rangle + e^{-\\tfrac{4\\pi i}{4}}|7\\rangle + e^{-\\tfrac{\\hphantom{1}8\\pi i}{4}}|4\\rangle + e^{-\\tfrac{12\\pi i}{4}}|13\\rangle)\\dots \\\\[10pt]\n",
    "+ |u_3\\rangle &= \\tfrac{1}{2}(|1\\rangle + e^{-\\tfrac{6\\pi i}{4}}|7\\rangle + e^{-\\tfrac{12\\pi i}{4}}|4\\rangle + e^{-\\tfrac{18\\pi i}{4}}|13\\rangle)\\quad) = |1\\rangle \\\\[10pt]\n",
    "\\end{aligned}$$\n",
    "\n",
    "</details>\n",
    "\n",
    "Since the computational basis state $|1\\rangle$ is a superposition of these eigenstates, which means if we do QPE on $U$ using the state $|1\\rangle$, we will measure a phase:\n",
    "\n",
    "$$\\phi = \\frac{s}{r}$$\n",
    "\n",
    "Where $s$ is a random integer between $0$ and $r-1$. We finally use the [continued fractions](https://en.wikipedia.org/wiki/Continued_fraction) algorithm on $\\phi$ to find $r$. The circuit diagram looks like this (note that this diagram uses Qiskit's qubit ordering convention):\n",
    "\n",
    "<img src=\"images/shor_circuit_1.svg\">\n",
    "\n",
    "We will next demonstrate Shor’s algorithm using Qiskit’s simulators. For this demonstration we will provide the circuits for $U$ without explanation, but in section 4 we will discuss how circuits for $U^{2^j}$ can be constructed efficiently."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## 3. Qiskit Implementation\n",
    "\n",
    "In this example we will solve the period finding problem for $a=7$ and $N=15$. We provide the circuits for $U$ where:\n",
    "\n",
    "$$U|y\\rangle = |ay\\bmod 15\\rangle $$\n",
    "\n",
    "without explanation. To create $U^x$, we will simply repeat the circuit $x$ times. In the next section we will discuss a general method for creating these circuits efficiently. The function `c_amod15` returns the controlled-U gate for `a`, repeated `power` times."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {
    "tags": [
     "thebelab-init"
    ]
   },
   "outputs": [],
   "source": [
    "def c_amod15(a, power):\n",
    "    \"\"\"Controlled multiplication by a mod 15\"\"\"\n",
    "    if a not in [2,7,8,11,13]:\n",
    "        raise ValueError(\"'a' must be 2,7,8,11 or 13\")\n",
    "    U = QuantumCircuit(4)        \n",
    "    for iteration in range(power):\n",
    "        if a in [2,13]:\n",
    "            U.swap(0,1)\n",
    "            U.swap(1,2)\n",
    "            U.swap(2,3)\n",
    "        if a in [7,8]:\n",
    "            U.swap(2,3)\n",
    "            U.swap(1,2)\n",
    "            U.swap(0,1)\n",
    "        if a == 11:\n",
    "            U.swap(1,3)\n",
    "            U.swap(0,2)\n",
    "        if a in [7,11,13]:\n",
    "            for q in range(4):\n",
    "                U.x(q)\n",
    "    U = U.to_gate()\n",
    "    U.name = \"%i^%i mod 15\" % (a, power)\n",
    "    c_U = U.control()\n",
    "    return c_U"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We will use 8 counting qubits:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {
    "tags": [
     "thebelab-init"
    ]
   },
   "outputs": [],
   "source": [
    "# Specify variables\n",
    "n_count = 8  # number of counting qubits\n",
    "a = 11"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We also import the circuit for the QFT (you can read more about the QFT in the [quantum Fourier transform chapter](./quantum-fourier-transform.html#generalqft)):"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {
    "tags": [
     "thebelab-init"
    ]
   },
   "outputs": [],
   "source": [
    "def qft_dagger(n):\n",
    "    \"\"\"n-qubit QFTdagger the first n qubits in circ\"\"\"\n",
    "    qc = QuantumCircuit(n)\n",
    "    # Don't forget the Swaps!\n",
    "    for qubit in range(n//2):\n",
    "        qc.swap(qubit, n-qubit-1)\n",
    "    for j in range(n):\n",
    "        for m in range(j):\n",
    "            qc.cp(-np.pi/float(2**(j-m)), m, j)\n",
    "        qc.h(j)\n",
    "    qc.name = \"QFT†\"\n",
    "    return qc"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "With these building blocks we can easily construct the circuit for Shor's algorithm:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {
    "scrolled": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "11\n"
     ]
    },
    {
     "data": {
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      "text/plain": [
       "<Figure size 2206.58x806.68 with 1 Axes>"
      ]
     },
     "execution_count": 7,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# Create QuantumCircuit with n_count counting qubits\n",
    "# plus 4 qubits for U to act on\n",
    "qc = QuantumCircuit(n_count + 4, n_count)\n",
    "\n",
    "# Initialize counting qubits\n",
    "# in state |+>\n",
    "for q in range(n_count):\n",
    "    qc.h(q)\n",
    "    \n",
    "# And auxiliary register in state |1>\n",
    "qc.x(3+n_count)\n",
    "print(3+n_count)\n",
    "\n",
    "# Do controlled-U operations\n",
    "for q in range(n_count):\n",
    "    qc.append(c_amod15(a, 2**q), \n",
    "             [q] + [i+n_count for i in range(4)])\n",
    "\n",
    "# Do inverse-QFT\n",
    "qc.append(qft_dagger(n_count), range(n_count))\n",
    "\n",
    "# Measure circuit\n",
    "qc.measure(range(n_count), range(n_count))\n",
    "qc.draw(fold=-1,filename='/home/ceb/qf_shor_circuit.png',output='mpl')  # -1 means 'do not fold' \n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Let's see what results we measure:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {},
   "outputs": [
    {
     "name": "stderr",
     "output_type": "stream",
     "text": [
      "/home/ceb/miniconda3/lib/python3.8/site-packages/qiskit/utils/deprecation.py:62: DeprecationWarning: Using a qobj for run() is deprecated as of qiskit-aer 0.9.0 and will be removed no sooner than 3 months from that release date. Transpiled circuits should now be passed directly using `backend.run(circuits, **run_options).\n",
      "  return func(*args, **kwargs)\n"
     ]
    },
    {
     "data": {
      "image/png": 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NMrOBotds820/7bCKJEnqoto9uUFEnB8R90XE0xExptz20Yh4a8eXJ0lS19HeyQ0uAj4BfAOIZrseBj7YcWVJktT1tPdO833AuZl5BbCl2faFwIQOq0qSpC6ovaH5EmBxK9ufBfrsejmSJHVd7Q3N5cCkVra/Dliy6+VIktR1tbf37H8BV0XEiyjaNKdExD8CHwHe3dHFSZLUlbR3nOa3I6I38BngRRQTHTwMXJiZN+2G+iRJ6jJ2ZpzmNcA1EXEgsFdmPtLxZUmS1PW0OzSbZOajHVmIJEld3Q5DMyLuAaZn5hMRcS+QbR2bmUd1ZHGSJHUlVe40bwEam71uMzQlSerOdhiamflvzV5ftlurkSSpC2vvNHq/joj9W9nePyJ+3WFVSZLUBbV3coMTgX1a2b4vcMIuVyNJUhdWqfdsRDSfBeioiHi82e+9gNcCazqyMEmSupqqQ04aKDoAJXBbK/s3ARd0VFGSJHVFVUNzNMW0ecuBlwN/brZvM/BIZm7t4NokSepSKoVmZj5Uvmz3otWSJHUXVSY3OA3438x8tnzdpsz8nw6rTJKkLqbKneYPgCHAI+XrtiRFpyBJkrqlKpMb7NXaa0mSehpDUJKkiqq2aVZim6YkqTur2qZZhW2akqRurV1tmpIk9WQGoiRJFTlOU5KkihynKUlSRY7TlCSpIkNQkqSK2h2aETEpIq6LiIby5/oW621KktQttSs0I+LtwO+Ag4GflT+DgfkR8Y6K1zg/IlZExDMRsSAiTtjOsdMj4s6IeCwiNkXEAxFxSSvHvTkilkREY/nnm9rzviRJqqLqeppN/gP4ZGZ+pvnGiPg48Gngu9s7OSLOAK4AzgfmlH/eGhHjM/NPrZyyEbgSuBd4Gjge+HpEPJ2ZXy2vOQW4CfhX4H+A04CbI+L4zJzXzvcnSVKb2vt49iDg+61svxkYVOH8i4FrM/OazLw/My8A1gLvb+3gzFyQmTdm5n2ZuSIzvwv8Amh+d3oR8JvM/I/ymv8BzCy3S5LUYdobmr8BTmxl+4nArO2dGBH7AC8Dbmux6zbgFVX+8og4pjy2+d81pZVr/qLqNSVJqqq9E7bfCnw2IuqBu8ptx1E8Er1sB5c6kGIc5/oW29cDJ+2ghtUUd7m9gX/LzKub7R7SxjWHtHGt84DzAIYOHcrMmTMBGDNmDP369WPRokUADBw4kAkTJjB79mwAevfuzdSpU1m4cCFPPvkkAPX19axfvx44ZHvlaw/V0NDAxo0bAZg8eTKrV69mzZo1AIwbN45evXqxZMkSAIYMGcLo0aOZO3cuAH369GHy5MnMmzePTZs2ATBlyhRWrFjBunXrABg/fjxbt25l6dKlAAwbNozhw4czb17RqtC3b1/q6+uZO3cujY2NAEydOpUHH3yQRx55BIAjjjiCxsZGli1bBsCIESMYPHgwDQ0NAPTv359JkyYxZ84ctmzZAsC0adO47777eOyxxwCYOHEiGzZsYPny5QCMGjWKAQMGsHDhQgAOOOAAJk6cyKxZs8hMIoLp06ezaNEinnjiCQAmTZrE448/zsqVK4Fd+zytWrUKgMMOO4y6ujoWL14MwKBBgxg7dixz5swBoK6ujilTpuzUvxPUtfO/Bu0J1q5d2yGfp+2JzNz+ARHPVaw3M7PNyQ0iYiiwBpiembObbb8UeHtmjtvOuaOBvhQB/XngQ5l5fblvM3BOZl7X7PizgGsyc7ufjPr6+mz6n8uuOPfyXb6EuqBrLqp1Bdpd/Mx2Tx31mY2IBZlZ39q+zpyw/VFgK0Vv2+YGA+t2UMOK8uW9ETGY4q72+nLbup25piRJ7dVpkxtk5mZgAXByi10nA3e241J78fxnK3M74JqSJO1Qe4ecEBEHAP8fMBLYp/m+zPzUDk7/EnB9RMwH7gDeBwwFri6vfV15nbPK3y8AVgBLy/OnAZcAX212zSuA2RHxMeBHwJuAVwJT2/veJEnannaFZkQcB/wUaKTomLOGYqKDRmAlsN3QzMybImIg8InyvMXA6zLzofKQkS1O6UXRhjkK2AL8EfgYZciW17wzIt5GMU70U+UxZzhGU5LU0dp7p/kF4L+BDwFPAq8CngJuAL5V5QLlpARfbWPfiS1+vxy4vMI1f8D2V2CRJGmXtbdN8yjgqiy63G4F6jJzPfBRdjzkRJKkPVp7Q3Nzs9frgZeUrzdStE1KktRttffx7ELgWOBBiqnqPl0OAXkHcE/HliZJUtfS3jvNfwEeLl9/Avgz8BXgAMpZdiRJ6q7adaeZmQ3NXv+ZYuiJJEk9QrvHaQJExCHA4eWvSzJzeceVJElS19TecZoDKYaWnAo897fN8RPg3Zn5WAfXJ0lSl9HeNs1vAodSrGe5b/kzDRgNXNOxpUmS1LW09/Hsa4FXZ+bcZtvuiIj3Ar/suLIkSep62nun+WeKGYBaehrw0awkqVtrb2h+Crg8IoY1bShff5EdzDsrSdKeboePZyPiXqD5StWjgZURsab8fRjwDDCIos1TkqRuqUqbphOhS5JEhdDMzH/rjEIkSerqdnZyg1cB4yke296XmTM7sihJkrqi9k5uMAz4IfAy/jYH7dCIaADelJkPt3myJEl7uPb2nr2SYh3NQzNzRGaOAA4rt13Z0cVJktSVtPfx7MnAiZm5omlDZi6PiAuBX3VoZZIkdTHtvdOE5w8/2d42SZK6lfaG5q+Ar0TEiKYNETESuBzvNCVJ3Vx7Q/NCYD9geUQ8FBEPAX8st13Y0cVJktSVtLdN8zHg5cCJwEvLbfdnppO1S5K6vcqhGRG9gL8CEzPzduD23VaVJEldUOXHs5m5FXgI2Gf3lSNJUtfV3jbNfwc+FxEH7o5iJEnqytrbpnkJxSonayJiNS3W1szMozqqMEmSupr2huYPKMZkxm6oRZKkLq1SaEbEi4AvAH8P7E0xJvOCzHx095UmSVLXUrVN89+AdwE/BW4ATgK+tptqkiSpS6r6ePY04D2ZeSNARPw3cEdE9Cp71UqS1O1VvdMcAfy26ZfMnA9sAYbujqIkSeqKqoZmL2Bzi21b2MlFrCVJ2hNVDb0AvhsRjc227QtcExFPN23IzFM7sjhJkrqSqqH5nVa2fbcjC5EkqaurFJqZefbuLkSSpK5uZxahliSpRzI0JUmqyNCUJKkiQ1OSpIoMTUmSKjI0JUmqyNCUJKkiQ1OSpIoMTUmSKjI0JUmqyNCUJKkiQ1OSpIoMTUmSKjI0JUmqyNCUJKkiQ1OSpIoMTUmSKjI0JUmqyNCUJKkiQ1OSpIo6PTQj4vyIWBERz0TEgog4YTvHHhwR34uIByJia0Rc28ox74qIbOVn3936RiRJPU6nhmZEnAFcAXwGOAa4E7g1Ika2cUod8CjwOWDedi79NHBw85/MfKaj6pYkCTr/TvNi4NrMvCYz78/MC4C1wPtbOzgzV2bmhZl5LfD4dq6bmbmu+U/Hly5J6uk6LTQjYh/gZcBtLXbdBrxiFy/fJyIeiojVEfGTiDhmF68nSdIL9O7Ev+tAoBewvsX29cBJu3DdpcC7gUVAP+BDwB0RMTEzl7U8OCLOA84DGDp0KDNnzgRgzJgx9OvXj0WLFgEwcOBAJkyYwOzZswHo3bs3U6dOZeHChTz55JMA1NfXs379euCQXShfXVVDQwMbN24EYPLkyaxevZo1a9YAMG7cOHr16sWSJUsAGDJkCKNHj2bu3LkA9OnTh8mTJzNv3jw2bdoEwJQpU1ixYgXr1hUPQsaPH8/WrVtZunQpAMOGDWP48OHMm1e0RPTt25f6+nrmzp1LY2MjAFOnTuXBBx/kkUceAeCII46gsbGRZcuK/9RHjBjB4MGDaWhoAKB///5MmjSJOXPmsGXLFgCmTZvGfffdx2OPPQbAxIkT2bBhA8uXLwdg1KhRDBgwgIULFwJwwAEHMHHiRGbNmkVmEhFMnz6dRYsW8cQTTwAwadIkHn/8cVauXAns2udp1apVABx22GHU1dWxePFiAAYNGsTYsWOZM2cOAHV1dUyZMmWn/p2Klh91N2vXru2Qz9P2RGbuxrfQ7C+KGAqsAaZn5uxm2y8F3p6Z43Zw/k+ARzPzXTs4rhdwN/CbzLxwe8fW19dn0/9cdsW5l+/yJdQFXXNRrSvQ7uJntnvqqM9sRCzIzPrW9nVmm+ajwFZgcIvtg4EOa4PMzK1AA3BYR11TkiToxNDMzM3AAuDkFrtOpuhF2yEiIoCjKDoYSZLUYTqzTRPgS8D1ETEfuAN4HzAUuBogIq4DyMyzmk6IiKPLl/2B58rfN2fmknL/vwJ3AcvKYy6kCM1We+RKkrSzOjU0M/OmiBgIfIJiPOVi4HWZ+VB5SGvjNX/f4vc3Ag8Bo8rf9we+AQwB/loePy0z53do8ZKkHq+z7zTJzK8CX21j34mtbIsdXO/DwIc7pDhJkrbDuWclSarI0JQkqSJDU5KkigxNSZIqMjQlSarI0JQkqSJDU5KkigxNSZIqMjQlSarI0JQkqSJDU5KkigxNSZIqMjQlSarI0JQkqSJDU5KkigxNSZIqMjQlSarI0JQkqSJDU5KkigxNSZIqMjQlSarI0JQkqSJDU5KkigxNSZIqMjQlSarI0JQkqSJDU5KkigxNSZIqMjQlSarI0JQkqSJDU5KkigxNSZIqMjQlSarI0JQkqSJDU5KkigxNSZIqMjQlSarI0JQkqSJDU5KkigxNSZIqMjQlSarI0JQkqSJDU5KkigxNSZIqMjQlSarI0JQkqSJDU5KkigxNSZIqMjQlSarI0JQkqSJDU5KkigxNSZIqMjQlSarI0JQkqSJDU5Kkijo9NCPi/IhYERHPRMSCiDhhB8dPL497JiKWR8T7dvWakiTtjE4NzYg4A7gC+AxwDHAncGtEjGzj+NHAz8rjjgE+C3wlIt68s9eUJGlndfad5sXAtZl5TWben5kXAGuB97dx/PuAhzPzgvL4a4DvAJfswjUlSdopnRaaEbEP8DLgtha7bgNe0cZpU1o5/hdAfUTsvZPXlCRpp/TuxL/rQKAXsL7F9vXASW2cMwT4ZSvH9y6vF+29ZkScB5xX/roxIpZWKV7bHAg8WusiOsM3P1zrCqQO4We2/V7S1o7ODM0uITO/AXyj1nXsqSKiITPra12HpGr8zHaszgzNR4GtwOAW2wcD69o4Z10bx28prxc7cU1JknZKp7VpZuZmYAFwcotdJ1P0eG3N3DaOb8jMZ3fympIk7ZTOfjz7JeD6iJgP3EHRO3YocDVARFwHkJlnlcdfDXwwIi4Hvg4cD7wLOLPqNdXhfLQt7Vn8zHagyMzO/Qsjzgc+AhwMLAY+nJmzy30zATLzxGbHTwe+DEwAHgY+n5lXV72mJEkdpdNDU5KkPZVzz0qSVJGhKUlSRYamJEkVGZrqEBERTX82vZak7sbQVIfIzIyIvbLUtN0Albq2iDAH2sHes9plEbE/8BqKyfMPAX4F/CwzH6plXZLa1vSFNg2BdjE0tcsi4nvANOCPwOPAcRSTRM8CPp2ZM2tXnaSWIuI44B8oVpJ6Bvg5xQpSCzPzuVrW1tUZmtol5eQTtwAnAEuBvuXPccA5wBHApZl5ba1qlPQ3EXEs8P9TzOH9Y2A4xZOig4FfA5/IzN/VrsKuzdDULomIzwHHZOZrW9k3GPgY8HZgamY+2Nn1SXq+iPgB8CTwnqZHs2W75quAS4Ajgfdm5k9qV2XXZQOwdtUs4PCImNhyR2auB/4VeAA4sZPrktS6EcDSsvNeRESvzHwuM38JvA2YD3wkIvrXtsyuydDUrpoHLANuiIj3RsQhLXrjbQVGUbR1Sqq9nwBnR8R+ZWf3rVDcbWbmX4BPUiyvOKp2JXZdPp7VLouIccBlwCRgBXAXxeT6G4BTgFdm5siaFShpm4gYC/wPxXrE3wRuy8z7mu0/BbglM/erUYldmqGpDhERvYDXAv8IjC83DwN+C/xXZt5Rq9okPV9EHA1cBBwNPEHxZXct0I+ibXNmZn6wRuV1aYamdlr5jfW1wJ8pHtEuzszGsi3kcIretJsys7GGZUpqRUTsQ9Hr/Y3AaGAIRWh+FpiRmX+tYXldlqGpnRIR/05xV9lI0VX9RcB9wI3AVzJzYw3Lk9RCOQnJOyiGgS0H/gDMz8zVEdEP2CczH6thiXsEQ1PtVrZhzgM+AMwG1lC0Z74feAtFW+b7M3NGRIQzjki1FREjgO9R9JxdAYylaNN8iGJigyvKTkDaAUNT7RYRnwRelZmvLH/fq2kWkYh4MfBl4Cjg1T7ikWovIq4CDgM+lJkPlNumUDwtejtFe+bZmTm3dlXuGRxyop2xGjgoIg4GyMznImKfiKgrQ/IKilmBTqtlkZK2eTnw48x8ICJ6A2Tm3Mw8n2JGoFXAh8FFFnbE0NTO+AVwAPDd8tsqmbm5qcNPZi6imM+yd+1KlATberbPBc6MiL0zc0u5va78orsBuBI4OiKOsTll+wxNtUvZRvkw8DqgF/CtiPh+RHwgIg6LiIMi4lKKgdE31rJWSVBOXvBDig5ANzfN3pWZjc16tt9PccfpVJc7YJumdlpEjAZOp5gibygwBugD3A18OzO/VrPiJAHbvuhmREwD/pNieMn9wM+AWyke3f4jsCEz31i7SvcMhqbaJSIGAAMpwnEjRU+8/SgmNOgHPAc8mJlralakpFZFxEHAqcCrgWMpvuj+GbgB+JqLKuyYoanKIuJM4L0Ua2c+SjGhwR+A24Cf2mVd6noi4njgxRT9EJYCC4GkWPO2F9DfsKzO0FQl5cDoZcB/A/8XeAnwSmAyxeQGvwcuzsxHalWjpL+JiBcBX6RYuaSO4gvuFuARinU0f5iZ68pjtw0b0/YZmqokIi4E3pGZL2+xvQ/w98C/U0xqMD0zn+z8CiU1FxH/DJwNXJSZt0XECcDLKBaIH0txx3l+Zm6uYZl7HIcEqKo6ij4FIzJzVUTsC2zJzE0Uy4LdDcwAplJ0MJBUW6cDV2fmbQCZ+Vvgt+VTo9cBV1H0RzizZhXugRxyoqq+T9EB6ByAzHwmM7c0Gyh9P/AYxUTtkmooIuooho+8MiL2K7f1KnvS/iUzv0fxWX5pRBxWy1r3NIamqloFXA78n4j4Q0RcEhEDy+AcGBGvAY4EflTLIqWergzGRoq5ZuuBc8oFp7e2mLjg98AhwKZa1Lmnsk1T7RIRhwIfoei2fiDFkJO/UKz0fktmfrh21UlqUt5hfgz4KMXE7NdSLD79BDAOeDNwYmYeVasa90SGpiop1957SWYuK38fRtGp4GiKdpGbgXvsVCB1LRExhmJe2TdR9HT/E39byu9TmTmzdtXteQxNbVdEHAn8E8WsIX+l6LI+G/heZt5Xy9okta5cCH4skJm5oNzWi6IJpR54GPhN2ZFP7WBoarsiYhnFlFtzKXpbD6YI0BdTrMP3+XIuWkldQES8F7iM4jHsiygmMPgh8NWmZcGaHet6t+3kkBO1KSL+gaKz2Duaxl6WK7yPA06hGAPWJyI+6GNZqfbKWbs+RrE832KK6S6PAd4InBERNwKXNq1za2C2n3eaalNEfAw4AXhDax+uiPg74BvAGzNzfmfXJ+n5IuI3wPzM/GizbXUUM3j9HXAexWIKn6lRiXs8h5xoe2YCrwI+Ut5htjSDYizYSZ1ZlKQXKtss17fcXi4B9mBmfgG4Dnhz2ZFPO8HQVJsy8y7g08D7gf+MiMkR0b/85gowkqJjwcwalSipVK6b+RvgnyLiPRFxYCuHfQcYQdEnQTvBx7NqVdMEzhGxN8UjnYspQnIxcAfFB28s8MfMfEPtKpXUpJyh63PAG4BZ/O1p0FrgWeCDwD9l5tCaFbmHMzTVpojYNzOfafb7NOAMii7rDwDzKSY0WFejEiWVmn3RfTFwFkVAHkqxKPw6ihWJVlLMR/vNWtW5pzM09QIRMYRisudJFHPJzgduysw7mx3zvECVVFvlU6G+mflEs21HAW8F9qf4ojsLuM9lwHaeoakXiIifUnxDvZ9iEPTxwARgOfAFit53W1yDT+oaIqJpCNhEihWJZlE8BfrfmhbWDTlOU88TEa+imLzgmMxcXS4BVkcRmmcDHwf6AV8yMKXai4ipFOMyHwKupFiN6A3AjyLiQYqp8m6oYYndineaep6IuAyYlpmvamVff+ACiuCclpkLO7k8SS1ExA+AJzLz3BbbX0rRge/1wMcz87pa1NfdOORELf0SOLJ83PM85axAnwfuBKZ3dmGSWjWAoqMPUPSgLZtOHgAuBG4DLoqIQbUqsDsxNNXSAuBW4KqI+FhEvKxpEdtSX2A8xaMgSbX3I+DciBgLkJlbyl60UXbWuxToDwyvYY3dho9n9QIRMYLiEexJFJM+/47im+xWYAowLjPH1a5CSU0i4iCKBafHADcCvwB+n5kbyv2nAd/JzNZm9VI7GZpqU9ld/R3AKyg6jR1A8Wj2S5l5by1rk/Q35eLw51M0m2wBVgNPU6x1+1LgJ5n5kdpV2H0YmgK2Tep8FHAasAG4B2homrggIsZRDIze7MoIUtcUEeMpVjQZT9HW2Qf4MsXamU/XsrbuwtAUABFxJUVgrqX4sI2ieCT7Y4o1M23DlLqQshnl3RRDxFYA9wF3ZuaicvL2Ppm5sZY1dkeGppq+nd5FMQvQgsx8rGwneQ/wXmAIcBHwDe8ypdqLiNHALRR3kvMpFk4YDPwFmAN8NjNX1qq+7szQFBHxf4BTMnNa+XvvzNzSbP9nKO5CX5WZD9eoTEmliLgaGAac26wJZSRwJkXb5n7AezLzx7WrsntyyImgmC7v4LIzAeUUeb3L2YAArqHoVHB6rQqU9DwTgFmZuS4i9i6/6P4pMz+fmS8BbgfeFxF7RUTUuNZuxdAUwGyKHnc/iYi3RkRdOdbrGYDMXAH0AhprWaSkbX4FvCMi+mXms+UX3b0jok+5//9SLN33cptUOpaPZwVARAyl6GV3JEV39fnAr8vXHwD+ARiVmU/VrEhJAETEy4D/peisd1lmzmix/6UUS4INsNdsxzI0tU1ENE30fDLFQOlxFGMzZwFfz8wba1ieJKCc6SfL5pT/BI4DHgN+SzGb13iKppQVmfnW2lXaPRmaPVxEDKdYBgzgKWAJsIkiNPtStGU+mpmP16ZCSW0p+x2cRPFF9+UUbZ2PU/RD+K5DxTqeodmDRcT7KcZ5TaQIx+UUj2N/A/wgM1fVsDxJLUTEXsDfAQdRDDf5I/DbzPxrGaAJ9MvMR2tYZrdmaPZQ5aPYPwBfBL5G8SE8CTiR4vHOw8CFmbmk6XFQrWqVBBHRD/gW8ErgOWANEBRfeH9JcWe5rDzWBeJ3E0Ozh4qIC4B3ZObkVvZNBT5LMQ7s5X5rlWovIv4FeCtwTmb+ruzsUw8cT/Fo9mHg3Zn55xqW2e055KTn2gz0i4gjoJh7NiL2AcjMOcDbgWeA19SuREnNnEKxWsnvADLzgcz8LvBB4MMUE7NfX8P6egRDs+f6AcUjnovKsV6Nmbm5bDMhM/9EMSWXa/BJNRYRvYHFwJvLKS6JiF7lY9itmTkbeB8wPCIm1rLW7s7Q7IHKGUIeBz5B0evu4Yj4Vjn2i4gYGRHvoBiz+f3aVSoJilm6gO9QzAN9SUQMLsOyebvlgxQLLdicshvZptmDRcT+wEiK9TLfRNE2AsWA6QCuz8zLalKcpG3KJ0B7AWcDn6FY3/YW4CbgTxTL+r0BGJ+Zx9aqzp7A0OxhImIQ8I/AP1F8I93E31ZGuAvYm2Lc5s8z88EalSmpDeWX3XdRzNJ1NMX6t88Av6NY3WRerWrrCQzNHiYirqUYAP2/FI9oB1A8hh0LPAJ8wg+d1HVERH9gQ/NhX+Wd574UE5AcATzl57ZzGJo9SNmWuQF4XdlxoGnbSGAycA7FTEBvzcyFNStU0jYR8XWKuaDnAw9l5pOtHHNAZj7hmOrdz45APct4ihXeNzdtyMJDmfl94I0Uj2rfUpvyJDUXEWcC51JMQvJj4AsRcVpEHNq0oklE9AW+HRFHGpi7n3eaPUj5IfsJ8CLgLOCPLWcNKSc9eE9mHt35FUpqLiKuAbZSTMx+GvBO4BBgKfAziiXCxgFXZOY+taqzJ/FOswfJzE3Av1DMWXkdcFZEjCi/qRIRLwKmU4wHk1RD5djMFcBfMnN5Zv5XZh4JHEux8tA7KYaEfQUnNeg03mn2QOUsQJ8ETqVY2WQu8GeKuWfXUkzTdW/tKpQERVslMDgzHyhn7Hq2RYegM4AbgEmZeXeNyuxRDM0erBx+8nrg7ym6rC8Gbs7MB2pZl6S2lT1nIzO3RsS5FI9mX1TrunoKQ1OAqyJIe6KIuBjolZlfqHUtPYWhKUl7qIjYG2g5nZ52I0NTkqSK7D0rSVJFhqYkSRUZmpIkVWRoSpJUkaEpSVJF/w+8JzQjARVlkgAAAABJRU5ErkJggg==\n",
      "text/plain": [
       "<Figure size 504x360 with 1 Axes>"
      ]
     },
     "execution_count": 8,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "aer_sim = Aer.get_backend('qasm_simulator')\n",
    "t_qc = transpile(qc, aer_sim)\n",
    "qobj = assemble(t_qc)\n",
    "results = aer_sim.run(qobj).result()\n",
    "counts = results.get_counts()\n",
    "plot_histogram(counts)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Since we have 8 qubits, these results correspond to measured phases of:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "            Register Output           Phase\n",
      "0  10000000(bin) = 128(dec)  128/256 = 0.50\n",
      "1  00000000(bin) =   0(dec)    0/256 = 0.00\n",
      "[0.5, 0.0]\n"
     ]
    }
   ],
   "source": [
    "rows, measured_phases = [], []\n",
    "for output in counts:\n",
    "    decimal = int(output, 2)  # Convert (base 2) string to decimal\n",
    "    phase = decimal/(2**n_count)  # Find corresponding eigenvalue\n",
    "    measured_phases.append(phase)\n",
    "    # Add these values to the rows in our table:\n",
    "    rows.append([f\"{output}(bin) = {decimal:>3}(dec)\", \n",
    "                 f\"{decimal}/{2**n_count} = {phase:.2f}\"])\n",
    "# Print the rows in a table\n",
    "headers=[\"Register Output\", \"Phase\"]\n",
    "df = pd.DataFrame(rows, columns=headers)\n",
    "print(df)\n",
    "print(measured_phases)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We can now use the continued fractions algorithm to attempt to find $s$ and $r$. Python has this functionality built in: We can use the `fractions` module to turn a float into a `Fraction` object, for example:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "Fraction(5998794703657501, 9007199254740992)"
      ]
     },
     "execution_count": 10,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "Fraction(0.666)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Because this gives fractions that return the result exactly (in this case, `0.6660000...`), this can give gnarly results like the one above. We can use the `.limit_denominator()` method to get the fraction that most closely resembles our float, with denominator below a certain value:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "Fraction(2, 3)"
      ]
     },
     "execution_count": 11,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# Get fraction that most closely resembles 0.666\n",
    "# with denominator < 15\n",
    "Fraction(0.666).limit_denominator(15)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Much nicer! The order (r) must be less than N, so we will set the maximum denominator to be `15`:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "   Phase Fraction  Guess for r\n",
      "0    0.5      1/2            2\n",
      "1    0.0      0/1            1\n"
     ]
    }
   ],
   "source": [
    "rows = []\n",
    "for phase in measured_phases:\n",
    "    frac = Fraction(phase).limit_denominator(15)\n",
    "    rows.append([phase, f\"{frac.numerator}/{frac.denominator}\", frac.denominator])\n",
    "# Print as a table\n",
    "headers=[\"Phase\", \"Fraction\", \"Guess for r\"]\n",
    "df = pd.DataFrame(rows, columns=headers)\n",
    "print(df)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We can see that two of the measured eigenvalues provided us with the correct result: $r=4$, and we can see that Shor’s algorithm has a chance of failing. These bad results are because $s = 0$, or because $s$ and $r$ are not coprime and instead of $r$ we are given a factor of $r$. The easiest solution to this is to simply repeat the experiment until we get a satisfying result for $r$.\n",
    "\n",
    "### Quick Exercise\n",
    "\n",
    "- Modify the circuit above for values of $a = 2, 8, 11$ and $13$. What results do you get and why?"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## 4. Modular Exponentiation\n",
    "\n",
    "You may have noticed that the method of creating the $U^{2^j}$ gates by repeating $U$ grows exponentially with $j$ and will not result in a polynomial time algorithm. We want a way to create the operator:\n",
    "\n",
    "$$ U^{2^j}|y\\rangle = |a^{2^j}y \\bmod N \\rangle $$\n",
    "\n",
    "that grows polynomially with $j$. Fortunately, calculating:\n",
    "\n",
    "$$ a^{2^j} \\bmod N$$\n",
    "\n",
    "efficiently is possible. Classical computers can use an algorithm known as _repeated squaring_ to calculate an exponential. In our case, since we are only dealing with exponentials of the form $2^j$, the repeated squaring algorithm becomes very simple:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "metadata": {
    "tags": [
     "thebelab-init"
    ]
   },
   "outputs": [],
   "source": [
    "def a2jmodN(a, j, N):\n",
    "    \"\"\"Compute a^{2^j} (mod N) by repeated squaring\"\"\"\n",
    "    for i in range(j):\n",
    "        a = np.mod(a**2, N)\n",
    "    return a"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "47"
      ]
     },
     "execution_count": 14,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "a2jmodN(7, 2049, 53)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "If an efficient algorithm is possible in Python, then we can use the same algorithm on a quantum computer. Unfortunately, despite scaling polynomially with $j$, modular exponentiation circuits are not straightforward and are the bottleneck in Shor’s algorithm. A beginner-friendly implementation can be found in reference [1].\n",
    "\n",
    "## 5. Factoring from Period Finding\n",
    "\n",
    "Not all factoring problems are difficult; we can spot an even number instantly and know that one of its factors is 2. In fact, there are [specific criteria](https://nvlpubs.nist.gov/nistpubs/FIPS/NIST.FIPS.186-4.pdf#%5B%7B%22num%22%3A127%2C%22gen%22%3A0%7D%2C%7B%22name%22%3A%22XYZ%22%7D%2C70%2C223%2C0%5D) for choosing numbers that are difficult to factor, but the basic idea is to choose the product of two large prime numbers.\n",
    "\n",
    "A general factoring algorithm will first check to see if there is a shortcut to factoring the integer (is the number even? Is the number of the form $N = a^b$?), before using Shor’s period finding for the worst-case scenario. Since we aim to focus on the quantum part of the algorithm, we will jump straight to the case in which N is the product of two primes.\n",
    "\n",
    "### Example: Factoring 15\n",
    "\n",
    "To see an example of factoring on a small number of qubits, we will factor 15, which we all know is the product of the not-so-large prime numbers 3 and 5."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 15,
   "metadata": {
    "tags": [
     "thebelab-init"
    ]
   },
   "outputs": [],
   "source": [
    "N = 15"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The first step is to choose a random number, $a$, between $1$ and $N-1$:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 16,
   "metadata": {
    "tags": [
     "thebelab-init"
    ]
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "7\n"
     ]
    }
   ],
   "source": [
    "np.random.seed(1) # This is to make sure we get reproduceable results\n",
    "a = randint(2, 15)\n",
    "print(a)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Next we quickly check it isn't already a non-trivial factor of $N$:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 17,
   "metadata": {
    "scrolled": true
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "1"
      ]
     },
     "execution_count": 17,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "from math import gcd # greatest common divisor\n",
    "gcd(a, N)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Great. Next, we do Shor's order finding algorithm for `a = 7` and `N = 15`. Remember that the phase we measure will be $s/r$ where:\n",
    "\n",
    "$$ a^r \\bmod N = 1 $$\n",
    "\n",
    "and $s$ is a random integer between 0 and $r-1$."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 18,
   "metadata": {
    "tags": [
     "thebelab-init"
    ]
   },
   "outputs": [],
   "source": [
    "def qpe_amod15(a):\n",
    "    n_count = 8\n",
    "    qc = QuantumCircuit(4+n_count, n_count)\n",
    "    for q in range(n_count):\n",
    "        qc.h(q)     # Initialize counting qubits in state |+>\n",
    "    qc.x(3+n_count) # And auxiliary register in state |1>\n",
    "    for q in range(n_count): # Do controlled-U operations\n",
    "        qc.append(c_amod15(a, 2**q), \n",
    "                 [q] + [i+n_count for i in range(4)])\n",
    "    qc.append(qft_dagger(n_count), range(n_count)) # Do inverse-QFT\n",
    "    qc.measure(range(n_count), range(n_count))\n",
    "    # Simulate Results\n",
    "    aer_sim = Aer.get_backend('aer_simulator')\n",
    "    # Setting memory=True below allows us to see a list of each sequential reading\n",
    "    t_qc = transpile(qc, aer_sim)\n",
    "    qobj = assemble(t_qc, shots=1)\n",
    "    result = aer_sim.run(qobj, memory=True).result()\n",
    "    readings = result.get_memory()\n",
    "    print(\"Register Reading: \" + readings[0])\n",
    "    phase = int(readings[0],2)/(2**n_count)\n",
    "    print(\"Corresponding Phase: %f\" % phase)\n",
    "    return phase"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "From this phase, we can easily find a guess for $r$:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 19,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Register Reading: 00000000\n",
      "Corresponding Phase: 0.000000\n"
     ]
    },
    {
     "name": "stderr",
     "output_type": "stream",
     "text": [
      "/home/ceb/miniconda3/lib/python3.8/site-packages/qiskit/utils/deprecation.py:62: DeprecationWarning: Using a qobj for run() is deprecated as of qiskit-aer 0.9.0 and will be removed no sooner than 3 months from that release date. Transpiled circuits should now be passed directly using `backend.run(circuits, **run_options).\n",
      "  return func(*args, **kwargs)\n"
     ]
    },
    {
     "data": {
      "text/plain": [
       "Fraction(0, 1)"
      ]
     },
     "execution_count": 19,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "phase = qpe_amod15(a) # Phase = s/r\n",
    "Fraction(phase).limit_denominator(15) # Denominator should (hopefully!) tell us r"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 20,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "1\n"
     ]
    }
   ],
   "source": [
    "frac = Fraction(phase).limit_denominator(15)\n",
    "s, r = frac.numerator, frac.denominator\n",
    "print(r)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Now we have $r$, we might be able to use this to find a factor of $N$. Since:\n",
    "\n",
    "$$a^r \\bmod N = 1 $$\n",
    "\n",
    "then:\n",
    "\n",
    "$$(a^r - 1) \\bmod N = 0 $$\n",
    "\n",
    "which means $N$ must divide $a^r-1$. And if $r$ is also even, then we can write:\n",
    "\n",
    "$$a^r -1 = (a^{r/2}-1)(a^{r/2}+1)$$\n",
    "\n",
    "(if $r$ is not even, we cannot go further and must try again with a different value for $a$). There is then a high probability that the greatest common divisor of $N$ and either $a^{r/2}-1$, or $a^{r/2}+1$ is a proper factor of $N$ [2]:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 21,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[15, 1]\n"
     ]
    }
   ],
   "source": [
    "guesses = [gcd(a**(r//2)-1, N), gcd(a**(r//2)+1, N)]\n",
    "print(guesses)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The cell below repeats the algorithm until at least one factor of 15 is found. You should try re-running the cell a few times to see how it behaves."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 22,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "\n",
      "Attempt 1:\n",
      "Register Reading: 00000000\n",
      "Corresponding Phase: 0.000000\n",
      "Result: r = 1\n",
      "\n",
      "Attempt 2:\n",
      "Register Reading: 01000000\n",
      "Corresponding Phase: 0.250000\n",
      "Result: r = 4\n",
      "Guessed Factors: 3 and 5\n",
      "*** Non-trivial factor found: 3 ***\n",
      "*** Non-trivial factor found: 5 ***\n"
     ]
    }
   ],
   "source": [
    "a = 7\n",
    "factor_found = False\n",
    "attempt = 0\n",
    "while not factor_found:\n",
    "    attempt += 1\n",
    "    print(\"\\nAttempt %i:\" % attempt)\n",
    "    phase = qpe_amod15(a) # Phase = s/r\n",
    "    frac = Fraction(phase).limit_denominator(N) # Denominator should (hopefully!) tell us r\n",
    "    r = frac.denominator\n",
    "    print(\"Result: r = %i\" % r)\n",
    "    if phase != 0:\n",
    "        # Guesses for factors are gcd(x^{r/2} ±1 , 15)\n",
    "        guesses = [gcd(a**(r//2)-1, N), gcd(a**(r//2)+1, N)]\n",
    "        print(\"Guessed Factors: %i and %i\" % (guesses[0], guesses[1]))\n",
    "        for guess in guesses:\n",
    "            if guess not in [1,N] and (N % guess) == 0: # Check to see if guess is a factor\n",
    "                print(\"*** Non-trivial factor found: %i ***\" % guess)\n",
    "                factor_found = True"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## 6. References\n",
    "\n",
    "1. Stephane Beauregard, _Circuit for Shor's algorithm using 2n+3 qubits,_ [arXiv:quant-ph/0205095](https://arxiv.org/abs/quant-ph/0205095)\n",
    "\n",
    "2. M. Nielsen and I. Chuang, _Quantum Computation and Quantum Information,_ Cambridge Series on Information and the Natural Sciences (Cambridge University Press, Cambridge, 2000). (Page 633)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 127,
   "metadata": {},
   "outputs": [
    {
     "name": "stderr",
     "output_type": "stream",
     "text": [
      "/home/ceb/miniconda3/lib/python3.8/site-packages/qiskit/aqua/__init__.py:86: DeprecationWarning: The package qiskit.aqua is deprecated. It was moved/refactored to qiskit-terra For more information see <https://github.com/Qiskit/qiskit-aqua/blob/main/README.md#migration-guide>\n",
      "  warn_package('aqua', 'qiskit-terra')\n"
     ]
    },
    {
     "data": {
      "text/html": [
       "<h3>Version Information</h3><table><tr><th>Qiskit Software</th><th>Version</th></tr><tr><td><code>qiskit-terra</code></td><td>0.18.3</td></tr><tr><td><code>qiskit-aer</code></td><td>0.9.0</td></tr><tr><td><code>qiskit-ignis</code></td><td>0.6.0</td></tr><tr><td><code>qiskit-ibmq-provider</code></td><td>0.16.0</td></tr><tr><td><code>qiskit-aqua</code></td><td>0.9.5</td></tr><tr><td><code>qiskit</code></td><td>0.30.1</td></tr><tr><th>System information</th></tr><tr><td>Python</td><td>3.8.5 (default, Sep  4 2020, 07:30:14) \n",
       "[GCC 7.3.0]</td></tr><tr><td>OS</td><td>Linux</td></tr><tr><td>CPUs</td><td>6</td></tr><tr><td>Memory (Gb)</td><td>15.401931762695312</td></tr><tr><td colspan='2'>Tue Oct 12 10:24:36 2021 -03</td></tr></table>"
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      ]
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     "metadata": {},
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    }
   ],
   "source": [
    "import qiskit.tools.jupyter\n",
    "%qiskit_version_table"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "\n",
      "Here is the list of cloud backends that are available to you:\n",
      "ibmq_qasm_simulator\n",
      "ibmq_armonk\n",
      "ibmq_santiago\n",
      "ibmq_bogota\n",
      "ibmq_lima\n",
      "ibmq_belem\n",
      "ibmq_quito\n",
      "simulator_statevector\n",
      "simulator_mps\n",
      "simulator_extended_stabilizer\n",
      "simulator_stabilizer\n",
      "ibmq_manila\n",
      "\n",
      "Here is the list of local backends that are available to you: \n",
      "aer_simulator\n",
      "aer_simulator_statevector\n",
      "aer_simulator_density_matrix\n",
      "aer_simulator_stabilizer\n",
      "aer_simulator_matrix_product_state\n",
      "aer_simulator_extended_stabilizer\n",
      "aer_simulator_unitary\n",
      "aer_simulator_superop\n",
      "qasm_simulator\n",
      "statevector_simulator\n",
      "unitary_simulator\n",
      "pulse_simulator\n"
     ]
    }
   ],
   "source": [
    "from qiskit import IBMQ, Aer\n",
    "provider = IBMQ.load_account()\n",
    "available_cloud_backends = provider.backends() \n",
    "print('\\nHere is the list of cloud backends that are available to you:')\n",
    "for i in available_cloud_backends: print(i)\n",
    "\n",
    "available_local_backends = Aer.backends() \n",
    "print('\\nHere is the list of local backends that are available to you: ')\n",
    "for i in available_local_backends: print(i)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {},
   "outputs": [],
   "source": [
    "aer_sim = Aer.get_backend('qasm_simulator')"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
  }
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