qiskit/advanced/aqua/generating_random_variates.ipynb

{
 "cells": [
  {
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"
    }
   },
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "![qiskit_header.png](attachment:qiskit_header.png)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# _*Qiskit Aqua: Generating Random Variates*_ \n",
    "\n",
    "The latest version of this notebook is available on https://github.com/Qiskit/qiskit-iqx-tutorials.\n",
    "\n",
    "***\n",
    "### Contributors\n",
    "Albert Akhriev<sup>[1]</sup>, Jakub Marecek<sup>[1]</sup>\n",
    "\n",
    "### Affiliation\n",
    "- <sup>[1]</sup>IBMQ"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Introduction\n",
    "\n",
    "While classical computers use only pseudo-random routines, quantum computers\n",
    "can generate true random variates.\n",
    "For example, the measurement of a quantum superposition is intrinsically random,\n",
    "as suggested by Born's rule.\n",
    "Consequently, some of the\n",
    "best random-number generators are based on such quantum-mechanical effects.\n",
    "Further, with a logarithmic amount of random bits, quantum computers can produce\n",
    "linearly many more bits, which is known as \n",
    "randomness expansion protocols. \n",
    "\n",
    "In practical applications, one wishes to use random variates of well-known\n",
    "distributions, rather than random bits.\n",
    "In this notebook, we illustrate ways of generating random variates of several popular\n",
    "distributions on IBM Q.\n",
    "\n",
    "## Random Bits and the Bernoulli distribution\n",
    "\n",
    "It is clear that there are many options for generating random bits (i.e., Bernoulli-distributed scalars, taking values either 0 or 1). Starting from a simple circuit such as a Hadamard gate followed by measurement, one can progress to vectors of Bernoulli-distributed elements. By addition of such random variates, we could get binomial distributions. By multiplication we could get geometric distributions, although perhaps leading to a circuit depth that may be impractical at the moment, though.\n",
    "\n",
    "Let us start by importing the basic modules and creating a quantum circuit for generating random bits:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2019-08-22T01:39:05.937968Z",
     "start_time": "2019-08-22T01:39:03.521337Z"
    }
   },
   "outputs": [],
   "source": [
    "import matplotlib.pyplot as plt\n",
    "%matplotlib inline\n",
    "import numpy as np\n",
    "import sys, math, time\n",
    "import warnings\n",
    "warnings.filterwarnings(\"ignore\", category=DeprecationWarning)\n",
    "\n",
    "from qiskit import BasicAer\n",
    "from qiskit import QuantumCircuit, QuantumRegister, ClassicalRegister, execute\n",
    "\n",
    "# In this example we use 'qasm_simulator' backend.\n",
    "glo_backend = BasicAer.get_backend(\"qasm_simulator\")"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "In the next step we create a quantum circuit, which will be used for generation:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2019-08-22T01:39:06.471867Z",
     "start_time": "2019-08-22T01:39:05.946725Z"
    }
   },
   "outputs": [
    {
     "data": {
      "image/png": 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AKpVKO2usrWasZmb5yiuvAADefvvtxordqAr+k4FnJ7wJj0EzdcY/X+4NZ4nMKOmPefnll+Ho6IhTp07h0KFDBq1TX0kCwLp16xAeHo7x48ejXbt2yM/Pb7TcRGKQRVECgIuLi874gwcPkJqaitGjR2vH1Go1kpKS6swsMzMzoVQq0bNnz6YJ/RhD3vUYej/Kwt8u42HJXXTWjERLVUfd8d8L4dTAA3ka636D+vYHb9y4UXsQ1pNs2LDBmJGe2rJ12wFUf69qf9zUav9Ojxw5EgDw6aefGrTuk0oSAPLz83H48GGMGTMGzz33HD7//H+/jLwfJZkCk99H6ehYvXkxOztbZzw6Ohr5+fno3bu3diwoKAi3bt3CkSNHtGNlZWXYvXs3hg8fbhJXHSnISYfSyqbOEa/5l9Jgp3KFrb2zSMkaZtWqVWJHkLSa/fLfffed3mWFSrJGzfPUPC+RKTH5GWW3bt2g0WiwZs0aODg4oEOHDkhISEBycjIA3X/YY8eOxeDBgxESEoLo6Gi0a9cOW7ZsQW5uLnbt2iXWSzCqgpx0OHf1g5m57o8+//J3ktrs+vi+MGqYTp06AQAuXrwouJwhJVn7eTp37mzElETNg8kXpZmZGeLj4zF//nyEhYVBpVJh1qxZCA8PR1RUFDQajXZZhUKBpKQkREZGIiIiAsXFxfD19cXRo0dN5p3ykFfq3xz5XMi2Jk7yx3h4eCArK0vsGJLl4+MDa2tr/P7774LL1ZxTrO9iAikpKfD09MTdu3eNmpOoOTD5ogSA7t274/jx4zpjM2bMgIeHB1q00D1FonXr1oiJidG+i65PWVkZKioqUFlZibKyMpSWlsLKyopH+JFk/PLLLwYtN3nyZAQGBmL//v2Cy927d8/g00uIpMbk91E+SUZGxlPPEufNm4cWLVrg5MmTmDZtGlq0aIGrV68aOSGR+MrKyvSWJJGpk2VRFhcXIzs7W+dAnobYsWMHqqqqdP7r0qWLcUOSoKFDh4odgYhkQhabXh9nZ2fHOxxI3LZt0tqnSkTSJcsZJUlfWFiY2BGISCZYlCRJJ06cEDsCEckEi5KIiEgAi5KIiEgAi5IkiRcbIKKmIsujXk3R+9PFTtC04uLieBm7BmroBcprLuC+NjJU52MiueGMkiRp5cqVYkcgIplgURIREQlgURIREQlgUZIkbd26VewIRCQTLEqSJLVaLXYEIpIJFiVJUkBAgNgRiEgmWJREREQCWJQkSX5+fmJHICKZYFGSJKWnp4sdgYhkgkVJREQkgEVJREQkgEVJkpSQkCB2BCKSCRYlERGRABYlSdKkSZPEjkBEMsGiJCIiEsD7UZqI1z8X5+vK7T6Y1LQUCoUoX7eh9+4k08YZJUlSeHi42BGISCZYlCRJixYtEjsCEckEi5IkaciQIWJHICKZYFGSJN28eVPsCEQkEyxKIiIiASxKkiRPT0+xIxCRTLAoSZISExPFjkAmwsLCQuwI1MyxKB9TVFSEBQsWwMnJCTY2NvD398c333wjdix6zIoVK8SOQM2Mv78//vznP2PXrl04cuQIkpOTsXXrVsyZMweurq71rrNixQqkpKTA1ta2idOSlPCCA7VUVVUhKCgIWVlZWL9+Pdq3b4/NmzdjxIgRSEtLg6+vr9gR6b/i4+Px1ltviR2DRGZmZoZ58+bh9ddfR8+ePZ+4XEVFBQ4cOIDVq1fj1KlTAKpLctWqVaioqIC/vz+OHj3aVLFJYliUtRw4cACpqalITk7GqFGjAFSfhqBWqxEVFYXk5GSRExrPPxa1R/9Jq+E1dI52rKqqCn+fZ48R82Ph5jdexHRE+j3zzDPYsWMHBg0aBADIy8vDvn37kJGRgfz8fCiVSvTs2RMDBw7EmDFjEBQUhDFjxiA6OhqPHj3CypUrUVFRgRkzZrAkSZBsirKyshIbNmxATEwMrl27hh49emDTpk0IDQ1FQEAAtm/fjn379kGlUiEwMFC7nqWlJaZMmYK1a9eipKTEJDbRFN+5jpLCfLTt5KMzXnQjB49K78O5W1+RkhEZxsfHB1999RUcHR3x66+/IiIiAgkJCaioqNBZ7uDBg3jvvffg5OSEZcuWYfHixVi+fDkAaEty165dYrwEkhDZ7KOcPXs2Vq9ejfnz5+PQoUMIDg7G1KlTkZOTgz59+gAAMjMzoVar61xf0svLC+Xl5bhw4YIY0Y2uICcdCjNzqDqqdcZv5Z6Fjb0zWqrq35/TnKSmpoodgUTSvn17HD16FI6Ojjh06BDUajW++OKLOiVZ240bNxAREYFPP/1UO/bdd9+xJMkgsijKnTt3IjY2FklJSVi6dCmGDRuGqKgoDBgwAOXl5dqivHPnDtq0aVNnfQcHB+3jDx8+xKuvvooOHTqgdevWeO6555CVldWkr+ePKshJRxuX7lBattAZv5l7Fk5dpTGb/Pnnn8WOQCLZvn07nJyc8NVXX+Gll15CYWGhQeutWLECc+bMQUVFBUpLSzFo0CBMnDixkdOSKZDFptd33nkHgYGBCAgI0Bl3c3ODhYUFvL29AVTvo6vvbgW1x8rLy+Hm5oa3334bLi4uWLduHSZPnoxz5841Wn5D7qCw+DPD73ZQkJOOwoLLiFngqDNe9rAYfccuN3q2p7FkyRLBxzdu3GjQMs1B5NoYANXfq9ofN3fNMffYsWPx4osv4u7du5g5cyYePXpk0Hq1D9yZMWMG7O3tsW3bNmzZsgX79++v8zxiv05qGobeJcbkizIvLw+ZmZn1/lHNzc2FWq2GlZUVAEClUuHOnTt1lqsZc3BwgK2tLd544w3tY6+99hqioqJQWloKa2vrRnoVxlXwnww8O+FNeAyaqTP++XJvOEtkRknyVHMx/NWrVyM/P9+gdR4vyV27dkGhUCAsLAwajQYTJkzA7t27GzM2SZwsihIAXFxcdMYfPHiA1NRUjB49WjumVquRlJRUZ2aZmZmpPYLucWlpaejSpUujlqQh73oMvR9l4W+X8bDkLjprRqKlqqPu+O+FcGrggTyNdd8+ffuDN27ciNDQUMFlNmzYYMxIT23Zuu0Aqr9XtT9u7ppD7tr/Dl1cXPDCCy/gwYMHOvsahdRXkkD16/jwww8RExODmTNn1ilKKfx8qOmY/D5KR8fqzYvZ2dk649HR0cjPz0fv3r21Y0FBQbh16xaOHDmiHSsrK8Pu3bsxfPjwOke83r17F+Hh4Xj77bcb8RUYV0FOOpRWNnWOeM2/lAY7lSts7Z1FStYwq1atEjsCNbG+favfxH333XcG7Zd8UknWOHz4MADAz8/P+GHJpJj8jLJbt27QaDRYs2YNHBwc0KFDByQkJGjPiaw5kAeo3v8xePBghISEIDo6Gu3atcOWLVuQm5tb5x/ZgwcPMG7cOEyePBnTpk1r0tf0RxTkpMO5qx/MzHV/9PmXv5PUZtfg4GCxI1AT8/LyAgCcOXNG77L6ShKo3vVy+/ZtODo6ol27dgZvyiX5MfmiNDMzQ3x8PObPn4+wsDCoVCrMmjUL4eHhiIqKgkaj0S6rUCiQlJSEyMhIREREoLi4GL6+vjh69KhOoZaXlyM4OBju7u6Smk0CwJBX6t8c+VzItiZO8sd4eHhI7mhj+mN+/vlnfPTRR3pPDQoODtZbkjU+/vhjtGrVCuXl5caOSybE5IsSALp3747jx4/rjM2YMQMeHh5o0UL3FInWrVsjJiYGMTExT3y+uXPnorKyEtu3b2+UvERU1/79+7F//369y+3ZsweJiYlITEzUe55kZGSkseKRCZNFUdYnIyMD/fv3b/B6V69eRWxsLKytrdG6dWvt+Pnz59GpUydjRiSip1BWVoZJkyaJHYNMiCyLsri4GNnZ2Vi4cGGD1+3cuTOPiGsGhg4dKnYEIpIJWRalnZ2d4OWuqPnbtk1a+1SJSLpM/vQQMk1hYWFiRyAimWBRkiSdOHFC7AhEJBMsSiIiIgEsSiIiIgEsSpIkXmyAiJoKi5IkKS4uTuwIRCQTsjw9xBS9P13sBE1r5cqVvN6rDDT0nOWau5ysjQzV+Zjoj+CMkoiISACLkoiISACLkiRp69atYkcgIplgUZIkqdVqsSMQkUywKEmSAgICxI5ARDLBoiQiIhLAoiQiIhLAoiRJ8vPzEzsCEckEi5IkKT09XewIRCQTLEoiIiIBLEoiIiIBLEqSpISEBLEjEJFMsCiJiIgEsChJkiZNmiR2BCKSCRYlERGRAN6P0kS8/rk4X1du98Ek0kehUIjydRt6704yHGeUJEnh4eFiRyAimWBRkiQtWrRI7AhEJBMsSpKkIUOGiB2BiGSCRUmSdPPmTbEjEJFMsCiJiIgEsChJkjw9PcWOQEQywaKspaioCAsWLICTkxNsbGzg7++Pb775RuxYVI/ExESxIxAZTadOnTBs2DCMGDECfn5+aNGixROXVSgUWLx4MWxtbZswobyxKP+rqqoKQUFB2LNnD9avX4+kpCQ4OjpixIgROHPmjNjx6DErVqwQOwLRH+Ln54dPP/0UN2/exNWrV3Hs2DEcPXoUP/zwA+7fv4/Tp0/jtddeQ6tWrbTrKBQKfPjhh3j//ff5ZrEJsSj/68CBA0hNTcWOHTswc+ZMDB8+HPHx8ejYsSOioqLEjmd0/1jUHpknPtYZq6qqwra5rXA5fY9IqQwXHx8vdgSip+Li4oK9e/fihx9+wKuvvgpHR0fcunULX3/9NVJSUnDu3DlUVVXB19cXmzZtwtWrVzFr1ixtSYaFhaG0tBTvvfee2C9FNmRRlJWVlVi/fj3c3d1hbW0NHx8fpKamokePHggNDQUA7Nu3DyqVCoGBgdr1LC0tMWXKFKSkpKCkpESs+EZXfOc6Sgrz0baTj8540Y0cPCq9D+dufUVKRmTaBg8ejMzMTAQFBaGoqAjvvvsuevTogbZt2yIgIAAvvPACfHx80LJlS0ycOBGpqalo3bo1duzYgezsbG1Jjhs3DikpKWK/HNmQRVHOnj0bq1evxvz583Ho0CEEBwdj6tSpyMnJQZ8+fQAAmZmZUKvVdS4/5eXlhfLycly4cEGM6I2iICcdCjNzqDqqdcZv5Z6Fjb0zWqpcRUpGZLoGDhyIw4cPQ6VS4fDhw1Cr1fjzn/+M7OzsOsuWlpbiyy+/xNChQzFz5kw8fPgQbm5uqKiowPjx41mSTczki3Lnzp2IjY1FUlISli5dimHDhiEqKgoDBgxAeXm5tijv3LmDNm3a1FnfwcFB+zgATJ8+Hc7OzrC3t0e/fv3w3XffNd2LMZKCnHS0cekOpaXuAQM3c8/Cqas0ZpOpqaliRyAyWOvWrREXFwcbGxt8+umnePHFF3H9+nW96ykUCgwYMABWVlaoqqqCubk5evfu3QSJqTaTL8p33nkHgYGBCAgI0Bl3c3ODhYUFvL29AVTvn6vvYsaPj0VFReHatWsoKirCypUrMXHixMYL30gKctJRWHAZMQscdf47dWAdnLv5iR3PID///LPYEYgM9u6776JDhw5IS0vDvHnzUFlZqXedx/dJLlu2DACwcuVKeHh4NHZkqsWk7x6Sl5eHzMxMLFmypM5jubm5UKvVsLKyAgCoVCrtrLG2mrGamWXN+XtVVVWwsLDAb7/9htLSUlhbWzfWyzDobgSLPzP8zgEF/8nAsxPehMegmTrjny/3hnMDZ5SNdaeE+n5mtW3cuNGgZZqDyLUxAKq/V7U/bu6kmLs5Zm7fvj1effVVlJWVYfbs2aioqNC7zuMlWbNP0t3dHXPnzkVERATmzZtXZx1qGEPvuGLSM8q8vDwA1UeZ1fbgwQOkpqZqN7sCgFqtxvnz5+t84zIzM6FUKtGzZ0/t2PTp09GiRQuMHDkSCxcubNSSNLbC3y7jYclddNaMREtVR+1/FWWlePh7IZx4IA+RUc2dOxdKpRJ79uzBxYsX9S7/pJIEgOjoaADAtGnTYG9v36i56X9Mekbp6OgIAMjOzsbo0aO149HR0cjPz9fZ1h8UFIRPPvkER44c0R75WlZWht27d2P48OE6J/d+/vnn2LFjB5KTk3H//v1Gfx2GvOsx9H6UBTnpUFrZ1DniNf9SGuxUrrC1dzZ6tqeh7+CpjRs3ao9YfpINGzYYM9JTW7ZuO4Dq71Xtjzmc0dkAABX2SURBVJs7KeZuDpkfn9k9//zzAIB//vOfBq37pJIEgEuXLiEtLQ0DBw5E//79ceTIEe1jzf1nI2UmXZTdunWDRqPBmjVr4ODggA4dOiAhIQHJyckAoDOjHDt2LAYPHoyQkBBER0ejXbt22LJlC3Jzc7Fr1646z21hYYGgoCD4+PigX79+6N69e5O9rj+iICcdzl39YGau+6PPv/xdgze7imnVqlViRyDSS6FQwNfXFwDwww8/6F1WqCRr/Pvf/8bAgQPRp08fnaKkxmPSm17NzMwQHx8PtVqNsLAwhISEwNHREeHh4VAqldBoNNplFQoFkpKSMG7cOERERGDs2LG4ceMGjh49qlOoj3v06BGuXLnSBK/GOIa8sgGT3jhRZ/y5kG0Y8/qXTR/oKQUHB4sdgUivli1bomXLlrh3757gHW8MLUkAuHz5MoDqfZ/UNEx6RgkA3bt3x/Hjx3XGZsyYAQ8PjzrXU2zdujViYmIQExNT73Pdvn0bx44dw4svvgilUomPPvoIv/76Kw/XFoGHhweysrLEjkEk6N69e7Czs9MeNPgk5ubmaN++vUEXE/jkk0/w+eef4/fffzd2XHoCky/K+mRkZKB///5Pte7mzZsxd+5cmJmZwcvLCwcPHtTuCyUielxJSYneK3uVl5fj5ZdfhkajwalTpwSXLS0tRWlpqTEjkh6yK8ri4mJkZ2dj4cKFDV5XpVLh66+/boRURCR3ZWVlekuSxCG7orSzszPoPCZq3oYOHSp2BCKSCZM+mIdM17Zt28SOQEQywaIkSQoLCxM7AhHJBIuSJOnEiRNiRyAimWBREhERCWBREhERCWBRkiTxYgNE1FRkd3qIqXp/utgJmlZcXBwvY0fN0tNcnLzmAu5rI0N1PqbmgTNKkqSVK1eKHYGIZIJFSUREJIBFSUREJIBFSZK0detWsSMQkUywKEmS1Gq12BGISCZYlCRJAQEBYkcgIplgURIREQlgUZIk+fn5iR2BiGSCRUmSlJ6eLnYEIpIJFiUREZEAFiUREZEAFiVJUkJCgtgRiEgmWJREREQCWJQkSZMmTRI7AhHJBIuSiIhIAO9HaSJe/1ycryu3+2ASmSqFQtHkX/Np7t0pBs4oSZLCw8PFjkBEMsGiJElatGiR2BGISCZYlCRJQ4YMETsCEckEi5Ik6ebNm2JHICKZYFESEREJYFGSJHl6eoodgYhkgkVJkpSYmCh2BCJqAKVSCRsbG7FjPBUWZS1FRUVYsGABnJycYGNjA39/f3zzzTdix6J6rFixQuwIRLJka2uLV155BVu2bMHJkydx7tw5nDp1CnFxcYiMjISHh0eddZRKJXbv3o1Dhw7B1tZWhNR/DIvyv6qqqhAUFIQ9e/Zg/fr1SEpKgqOjI0aMGIEzZ86IHY8eEx8fL3YEIllp1aoV3nvvPVy/fh3/+te/EB4ejkGDBsHb2xu9e/fGyy+/jLVr1+L8+fM4duwYBg4cCOB/JTlx4kRoNBp07dpV5FfScLwyz38dOHAAqampSE5OxqhRowBUn4KgVqsRFRWF5ORkkRMa1z8WtUf/SavhNXSOdqyqqgp/n2ePEfNj4eY3XsR0RNScDBs2DLGxsXB1dQUAfPvtt9i3bx9Onz6NmzdvwtLSEmq1GoMHD8bkyZMxbNgwnDx5Eps3b0anTp0wfvx4FBYWYvjw4cjMzBT51TScLGaUlZWVWL9+Pdzd3WFtbQ0fHx+kpqaiR48eCA0NBQDs27cPKpUKgYGB2vUsLS0xZcoUpKSkoKSkRKz4Rld85zpKCvPRtpOPznjRjRw8Kr0P5259RUpGRM3N+PHjceTIEbi6uuKHH35Anz59MGjQILz77rv4v//7P5w7dw4ZGRmIjY3F3Llz0b59e7z99tuorKzE4sWLdUry1KlTYr+cpyKLopw9ezZWr16N+fPn49ChQwgODsbUqVORk5ODPn36AAAyMzOhVqvrXO/Qy8sL5eXluHDhghjRG0VBTjoUZuZQdVTrjN/KPQsbe2e0VLmKlMxwqampYkcgMnn9+vXD7t27YWFhgffeew8DBw7E6dOnBde5f/8+3nzzTZw8eVI7lpqaKtmSBGRQlDt37kRsbCySkpKwdOlSDBs2DFFRURgwYADKy8u1RXnnzh20adOmzvoODg7ax2v74osvoFAoJHkD4YKcdLRx6Q6lZQud8Zu5Z+HUVRqzyZ9//lnsCEQmzcrKCjt27IClpSW2bNmCpUuXoqKiQu96Nfskhw0bhvv376O0tBRBQUEYM2ZME6RuHCa/j/Kdd95BYGAgAgICdMbd3NxgYWEBb29vANX75+q7en59Y7///jvefvttqNXqOo81BkOu6r/4M8Ovwl+Qk47CgsuIWeCoM172sBh9xy43eransWTJEsHHN27caNAyzUHk2hgA1d+r2h83d1LMLcXMQPPMHR4eDg8PD2RlZWHp0qUGrVP7wJ2aza2DBw/Gxo0b8cEHHyA5ORmVlZXa5cV+jYbevcSkizIvLw+ZmZn1/kHNzc2FWq2GlZUVAEClUtWZNQL/m0nWzCyB6vJ99dVXkZSU1EjJG1fBfzLw7IQ34TFops7458u94SyRGSURNR6FQoGFCxcCACIjI/Hw4UO969RXkqdOncKZM2fw2muvoVu3bggMDJTkgZEmX5QA4OLiojP+4MEDpKamYvTo0doxtVqNpKSkOjPLzMxMKJVK9OzZEwBw5coVJCUlISMjo8mK0pB3PYbej7Lwt8t4WHIXnTUj0VLVUXf890I4NfBAnsa6n5y+fcIbN27UHoj1JBs2bDBmpKe2bN12ANXfq9ofN3dSzC3FzEDzyF37756fnx+eeeYZXL16FQcPHtS77pNKEqg+mHL79u1Yu3Ytpk6dqlOUUvjZACa+j9LRsXrTYnZ2ts54dHQ08vPz0bt3b+1YUFAQbt26hSNHjmjHysrKsHv3bgwfPlx7kmxERARWr14NCwuLJngFxleQkw6llU2dI17zL6XBTuUKW3tnkZI1zKpVq8SOQGSy+vatfsN8/PhxnU2l9REqyRopKSk6zys1Jj2j7NatGzQaDdasWQMHBwd06NABCQkJ2nc0NQfyAMDYsWMxePBghISEIDo6Gu3atcOWLVuQm5uLXbt2AQCOHTuGe/fuYdy4caK8HmMoyEmHc1c/mJnr/ujzL38nqc2uwcHBYkcgMlk1x1+cPXtWcDlDShKo3jJXUVGB7t27Q6lUory8vFFyNxaTLkozMzPEx8dj/vz5CAsLg0qlwqxZsxAeHo6oqChoNBrtsgqFAklJSYiMjERERASKi4vh6+uLo0ePagv166+/xvfff6+dqRYVFeH06dO4dOkSli9v2EEwYhnySv2bI58L2dbESf6YmoMMiMj4Dh8+jNu3b+u9hOfixYv1liQAPHr0CH/7299QUVEh+gE8T8OkixIAunfvjuPHj+uMzZgxAx4eHmjRQvf0iNatWyMmJgYxMTH1PldERATmzp2r/fzll19GSEgIpkyZYvzgREQi2b9/P/bv3693uU2bNkGj0WDTpk16z5N88803jZSu6Zl8UdYnIyMD/fv3b/B6rVq1QqtWrbSfW1lZwcHBQWeMiEguysrKMGvWLLFjNDrZFWVxcTGys7O1hz7/ESdOnPjjgeipDB06VOwIRCQTsitKOzs7g64uQc3btm3S2qdKRNJl0qeHkOkKCwsTOwIRyQSLkiSJm72JqKmwKImIiASwKImIiASwKEmSeLEBImoqLEqSpLi4OLEjEJFMyO70EFP1/nSxEzStlStX8nqvREbU0Dt51NzlZG1kqM7HpogzSiIiIgEsSiIiIgEsSpKkrVu3ih2BiGSCRUmSVHO/PCKixsaiJEkKCAgQOwIRyQSLkoiISACLkoiISADPo6RmqWfPnoKPr1y5Uu8yRETGwBklSdKbb74pdgQikgkWJRERkQAWJRERkQAWJRERkQAWJRERkQAWJRERkQAWJRERkQAWpYx06dIFarUavXr1Qq9evfDTTz+JHYkMcOLECajVari5uWHu3LmoqKgQO5Jer732Gjp27AilUjqnal+7dg3PP/88PDw84OXlheXLl4sdySAjRoyAj48PNBoNJk2ahHv37okdqUEWLlzY7H9PWJQyc+TIEfz444/48ccf4e3tLXYc0qOyshJz585FfHw8Ll++jHv37uGzzz4TO5ZekydPxqlTp8SO0SBKpRLr1q1DVlYWTp8+jbS0NOzbt0/sWHolJCTg7NmzOHfuHDp16oQNGzaIHclgJ0+eRElJidgx9GJREjVj6enpaN++PTw9PQEAc+bMQWJiosip9Bs0aBCcnZ3FjtEg7dq1Q9++fQEAlpaW0Gg0yM3NFTmVfvb29gCq31SVlJRAoVCInMgwDx8+xLJly7B+/Xqxo+jFopSZsWPHolevXnjjjTdQVlYmdhzSIy8vD66urtrPO3XqhGvXromYSB5u376NvXv3YsSIEWJHMci4cePg4uKCixcv4k9/+pPYcQzy1ltvYc6cOWjbtq3YUfRSVFVVVYkdgprGtWvX4OrqipKSEsyaNQt9+vSRzH4YqYk7eAL5N24DgPb/7ZxUOh/X8O7RFc8N7F3v8yQkJGDv3r3aza3nz5/H9OnTcebMGaNnvnz1Og4e+177uVBupbk5QoJHwcbaSvA5lUolysvLjZ61RmVlJf755VEU3S/RmxkA+vt64tleHoLP+fDhQwQGBmLMmDGNVjpns37Bie9/1H4ulNumhRVCJo2CUmku+JyVlZWIioqCo6Njo+QuKyvHJ/GHUPrwkd7MAPD8wN7w6tG13uc6d+4cIiIikJKSAoVC0ei/J38UZ5QyUjMzsbW1xdy5c5GWliZyItPVx6s78m/c1v4BAVDn4/wbt3HrTiF6e3V/4vO4urrqbP67du0aOnbs2CiZu3VqD3MzM4Nyu3Vpr7ckm4KZmRl6eboZlPlecQl8PJ4RfL6KigpMnz4dvr6+jToz83TvjIcPHxmUW+3eRW9JAtXfi1dffRU7duxojMiwsFDCw62zQZnLysrR063TE5/r22+/xfnz59G1a1d06dIFFRUV6NKlS7M9EIlF+V/p6ekYPXo0WrduDVtbW/Tv3x9xcXFixzKakpIS7S9hRUUFEhMTodFoRE5lup7p3B7q7l30LhfwbC+0bmX3xMf79u2L69ev4/z58wCAjz/+GBMmTDBWTB1mCgXGPD9A73ItbVtg6LO9GiXD0/DxeAad2jvpXe6FwX6wtrIUXCY0NBQtW7bEe++9Z6x49bJQKjFq2LN6l3NStcazvTyf+Pi9e/eQn5+v/TwxMRFqtdooGeszsI8aqjat9C43+rn+UJo/udzDwsLw66+/4sqVK7hy5QrMzc1x5coVtGql/7nFwKIEcPz4cfj7++Obb75BcHAwFixYgN9++w2TJ09u9H8wTaWgoABDhgyBRqOBRqNBVVUVoqKixI5l0kYP7Q9z8yf/E7NvaYshz/oIPoe5uTk++ugjTJo0Cc888wzs7OwwY8YMY0fV6tLRBZqe3QSXGRnQD1Z6Cmf+/Pno2LEjKioq0LFjR4SHhxszpg6FQoExzw8UXMalrQP8ND0El/n222/xySefICMjA76+vujVqxc2bdpkzKg6vLp3RVfXdoLLjHlugODvUFFREcaNGwdvb29oNBr8+OOP+OCDD4wdVUtpbo4Xh/UXXMatcwd4PPPk2aQUyX4fZXl5OXr27Im8vDx8//336NWr+p1yUVER+vXrhytXriA7OxudO3cWOWnjqaqqksyRclJz6MS/kfrvs/U+NnnMMPiq3Zs4kX53i+7jvY/iUF7P+ZodnB0RPms8zJrh78sXB47hzM+X631s7pQX4da5QxMn0u/6b7ewJfZL1PdHuEc3V4S8PKrJM+lTVVWFj79IxuWr1+s8plAosDhkIlzaOoiQrPHIYka5b98+jBkzBk5OTrCyskLnzp0xbdo0/PTTTzh27Bh++eUXTJs2TVuSQPUh13/5y1/w6NEjxMbGipi+8e3efwxHv04XO4ZJGjbAF3Y2LeqMu7Zzgo+nmwiJ9Gtj3xKD+9W/WX7M8wOaZUkCQOCQfrCwqHviuqd752ZZkgDQwcURfbzrznTNFAq8+Jz+zeBiUPx3E319b677+fQ0uZIETLwoy8vLMWXKFLz00ks4e/YsJkyYgMWLF8PX1xeJiYm4fv06Tpw4AQB44YUX6qw/cuRIAEBqampTxm5SvxbcwtmsX2BmZtK/CqKxtrLEC4P71hkf24wLBwCG9u+FlnY2OmPePbrp3VQoJvtWdgh4bFO2uZkZRuvZVCi2kUP8YGlpoTPWv7caTqrWIiXSz6WtA/r59NQZs7ayxIh6ftdNgUn/dVy0aBG++OILzJs3DxcuXMDf//53REdHY+/evbh8+TL8/f1x6dIlAIC7e91NYC4uLrCzs9MuY4r+L+00rK0s4d/XS+woJquvpofOu+xenm7o1KF5n4xvZWmBwCH9tJ8rzc0xamg/gTWahyH9fGDf0lb7uX9fLzi2sRcxkX4t7Wzw3ABf7ectrK0w3L/+04WakxGD+8KqVsE/P7B3vVtPTIHJ7qM8efIkhgwZgsDAQCQnJz9xH9wLL7yAlJQUXLp0CW5udTeFdejQAcXFxSgqKmq0rMvWbW+05yYiovqtjQw1aDmTnVG+//77AIC1a9fyQBUiInpqzfuS7X9ASkoKunTpAh8f4cPva66T+KQZ471799CmTRuj56vN0Hc1xvRrwS1s2vElhvv3wfBBfZr868vRnaL7sGthXWd/VHNXcOsunB0b99+Asd28XQhVm1aS2vdeVVWFm7cL4SSx73XBrbtwUrU26QmJSRZlYWEh7t+/jz599BdAzb7JS5cu1Vn+t99+Q3FxMfr1a9x9M2Juev3q21P46ltp3eWBiMgYZL3ptWa3640bN/QuGxAQAAA4evRonceOHDmiswwREcmPyR7M4+bmhpycHBw9ehTDhw/XeezixYvo0aP63KXy8nL06NED169ff+IFBy5evIguXbo09UtoNP/acxS/XP0VkQumokUzuF4nEVFzZrJFGRcXh8mTJ8Pc3BxBQUFwc3PDjRs3kJaWBk9PT+zZs0e77PHjxzFy5EhYW1tjypQpaNmyJRITE3H16lWsX79eMretMQT3TRIRNYzJFiVQven03XffRXp6OkpLS+Hk5IR+/frh9ddfx+DBg3WW/eGHH7By5UqkpaWhrKwM3t7eiIiIwOTJk0VK3zgyL/4HySe+x2uzJnA2SURkAJMuSqpfZWWlpI4GJCISE4uSiIhIAKcVREREAliUREREAliUREREAliUREREAliUREREAliUREREAliUREREAliUREREAliUREREAliUREREAliUREREAliUREREAliUREREAliUREREAliUREREAliUREREAliUREREAliUREREAliUREREAliUREREAliUREREAliUREREAliUREREAliUREREAliUREREAliUREREAliUREREAv4fIyW5TeXMdK4AAAAASUVORK5CYII=\n",
      "text/plain": [
       "<Figure size 588.756x385.28 with 1 Axes>"
      ]
     },
     "execution_count": 2,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# Number of qubits utilised simultaneously.\n",
    "glo_num_qubits = 5\n",
    "\n",
    "def create_circuit(num_target_qubits: int) -> QuantumCircuit:\n",
    "    \"\"\"\n",
    "    Creates and returns quantum circuit for random variate generation.\n",
    "    :param num_target_qubits: number of qubits to be used.\n",
    "    :return: quantum curcuit.\n",
    "    \"\"\"\n",
    "    assert isinstance(num_target_qubits, int) and num_target_qubits > 0\n",
    "    q = QuantumRegister(num_target_qubits)\n",
    "    c = ClassicalRegister(num_target_qubits)\n",
    "    circuit = QuantumCircuit(q, c)\n",
    "    circuit.h(q)\n",
    "    circuit.barrier()\n",
    "    circuit.measure(q, c)\n",
    "    return circuit\n",
    "\n",
    "# Create and plot generating quantum circuit.\n",
    "circuit = create_circuit(glo_num_qubits)\n",
    "#print(circuit)\n",
    "circuit.draw(output='mpl')"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Uniformly-distributed scalars and vectors\n",
    "\n",
    "It is clear that there are many options for approximating uniformly-distributed scalars by the choice of an integer from a finite range uniformly at random, e.g., by a binary-code construction from the Bernoulli-distributed vectors. In the following snippet, we generate random bits, which we then convert using the binary-code construction, up to the machine precision of a classical computer."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2019-08-22T01:39:08.718153Z",
     "start_time": "2019-08-22T01:39:08.706634Z"
    }
   },
   "outputs": [],
   "source": [
    "def uniform_rand_float64(circuit: QuantumCircuit, num_target_qubits: int,\n",
    "                         size: int, vmin: float, vmax: float) -> np.ndarray:\n",
    "    \"\"\"\n",
    "    Generates a vector of random float64 values in the range [vmin, vmax].\n",
    "    :param circuit: quantum circuit for random variate generation.\n",
    "    :param num_target_qubits: number of qubits to be used.\n",
    "    :param size: length of the vector.\n",
    "    :param vmin: lower bound.\n",
    "    :param vmax: upper bound.\n",
    "    :return: vector of random values.\n",
    "    \"\"\"\n",
    "    assert sys.maxsize == np.iinfo(np.int64).max    # sizeof(int) == 64 bits\n",
    "    assert isinstance(size, int) and size > 0\n",
    "    assert isinstance(vmin, float) and isinstance(vmax, float) and vmin <= vmax\n",
    "    nbits = 7 * 8                                   # nbits > mantissa of float64\n",
    "    bit_str_len = (nbits * size + num_target_qubits - 1) // num_target_qubits\n",
    "    job = execute(circuit, glo_backend, shots=bit_str_len, memory=True)\n",
    "    bit_str = ''.join(job.result().get_memory())\n",
    "    scale = float(vmax - vmin) / float(2**nbits - 1)\n",
    "    return np.array([vmin + scale * float(int(bit_str[i:i+nbits], 2))\n",
    "                     for i in range(0, nbits * size, nbits)], dtype=np.float64)\n",
    "\n",
    "def uniform_rand_int64(circuit: QuantumCircuit, num_target_qubits: int,\n",
    "                       size: int, vmin: int, vmax: int) -> np.ndarray:\n",
    "    \"\"\"\n",
    "    Generates a vector of random int64 values in the range [vmin, vmax].\n",
    "    :param circuit: quantum circuit for random variate generation.\n",
    "    :param num_target_qubits: number of qubits to be used.\n",
    "    :param size: length of the vector.\n",
    "    :param vmin: lower bound.\n",
    "    :param vmax: upper bound.\n",
    "    :return: vector of random values.\n",
    "    \"\"\"\n",
    "    assert sys.maxsize == np.iinfo(np.int64).max        # sizeof(int) == 64 bits\n",
    "    assert isinstance(size, int) and size > 0\n",
    "    assert isinstance(vmin, int) and isinstance(vmax, int) and vmin <= vmax\n",
    "    assert abs(vmin) <= 2**52 and abs(vmax) <= 2**52    # 52 == mantissa of float64\n",
    "    return np.rint(uniform_rand_float64(circuit, num_target_qubits,\n",
    "                                        size, float(vmin), float(vmax))).astype(np.int64)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Uniform distribution over floating point numbers.\n",
    "In this example we draw a random vector of floating-point values uniformly distributed within some arbitrary selected interval:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2019-08-22T01:39:15.167103Z",
     "start_time": "2019-08-22T01:39:09.337402Z"
    },
    "scrolled": true
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Uniform distribution over floating point numbers:\n",
      "  sample type: <class 'numpy.ndarray'> , element type: float64 , shape: (54321,)\n",
      "  sample min: -7.6685, max: 19.5196\n",
      "  sampling time: 5.48 secs\n"
     ]
    },
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "# Draw a sample from uniform distribution.\n",
    "start_time = time.time()\n",
    "sample = uniform_rand_float64(circuit, glo_num_qubits, size=54321, vmin=-7.67, vmax=19.52)\n",
    "sampling_time = time.time() - start_time\n",
    "\n",
    "# Print out some details.\n",
    "print(\"Uniform distribution over floating point numbers:\")\n",
    "print(\"  sample type:\", type(sample), \", element type:\", sample.dtype, \", shape:\", sample.shape)\n",
    "print(\"  sample min: {:.4f}, max: {:.4f}\".format(np.amin(sample), np.amax(sample)))\n",
    "print(\"  sampling time: {:.2f} secs\".format(sampling_time))\n",
    "\n",
    "# Plotting the distribution.\n",
    "plt.hist(sample.ravel(),\n",
    "         bins=min(int(np.ceil(np.sqrt(sample.size))), 100),\n",
    "         density=True, facecolor='b', alpha=0.75)\n",
    "plt.xlabel(\"value\", size=12)\n",
    "plt.ylabel(\"probability\", size=12)\n",
    "plt.title(\"Uniform distribution over float64 numbers in [{:.2f} ... {:.2f}]\".format(\n",
    "            np.amin(sample), np.amax(sample)), size=12)\n",
    "plt.grid(True)\n",
    "# plt.savefig(\"uniform_distrib_float.png\", bbox_inches=\"tight\")\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Uniform distribution over integers.\n",
    "Our next example is similar to the previous one, but here we generate a random vector of integers:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2019-08-22T01:39:21.029197Z",
     "start_time": "2019-08-22T01:39:15.181257Z"
    }
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Uniform distribution over bounded integer numbers:\n",
      "  sample type: <class 'numpy.ndarray'> , element type: int64 , shape: (54321,)\n",
      "  sample min: 37, max: 841\n",
      "  sampling time: 5.53 secs\n"
     ]
    },
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "# Draw a sample from uniform distribution.\n",
    "start_time = time.time()\n",
    "sample = uniform_rand_int64(circuit, glo_num_qubits, size=54321, vmin=37, vmax=841)\n",
    "sampling_time = time.time() - start_time\n",
    "\n",
    "# Print out some details.\n",
    "print(\"Uniform distribution over bounded integer numbers:\")\n",
    "print(\"  sample type:\", type(sample), \", element type:\", sample.dtype, \", shape:\", sample.shape)\n",
    "print(\"  sample min: {:d}, max: {:d}\".format(np.amin(sample), np.amax(sample)))\n",
    "print(\"  sampling time: {:.2f} secs\".format(sampling_time))\n",
    "\n",
    "# Plotting the distribution.\n",
    "plt.hist(sample.ravel(),\n",
    "         bins=min(int(np.ceil(np.sqrt(sample.size))), 100),\n",
    "         density=True, facecolor='g', alpha=0.75)\n",
    "plt.xlabel(\"value\", size=12)\n",
    "plt.ylabel(\"probability\", size=12)\n",
    "plt.title(\"Uniform distribution over int64 numbers in [{:d} ... {:d}]\".format(\n",
    "            np.amin(sample), np.amax(sample)), size=12)\n",
    "plt.grid(True)\n",
    "# plt.savefig(\"uniform_distrib_int.png\", bbox_inches=\"tight\")\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Normal distribution\n",
    "\n",
    "To generate random variates with a standard normal distribution using two independent \n",
    "samples $u_1, u_2$ of the uniform distribution on the unit interval [0, 1], one can\n",
    "consider the Box-Muller transform to obtain a 2-vector:\n",
    "\n",
    "\\begin{align}\n",
    "\\begin{bmatrix}\n",
    "%R\\cos(\\Theta )=\n",
    "{\\sqrt {-2\\ln u_{1}}}\\cos(2\\pi u_{2}) \\\\\n",
    "% R\\sin(\\Theta )=\n",
    "{\\sqrt {-2\\ln u_{1}}}\\sin(2\\pi u_{2})\n",
    "\\end{bmatrix},\n",
    "\\end{align}\n",
    "\n",
    "wherein we have two independent samples of the standard normal distribution.\n",
    "In IBM Q, this is implemented as follows: "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2019-08-22T01:39:21.059196Z",
     "start_time": "2019-08-22T01:39:21.040829Z"
    }
   },
   "outputs": [],
   "source": [
    "def normal_rand_float64(circuit: QuantumCircuit, num_target_qubits: int,\n",
    "                        size: int, mu: float, sigma: float) -> np.ndarray:\n",
    "    \"\"\"\n",
    "    Draws a sample vector from the normal distribution given the mean and standard\n",
    "    deviation, using the Box-Muller method. \n",
    "    \"\"\"\n",
    "    TINY = np.sqrt(np.finfo(np.float64).tiny)\n",
    "    assert isinstance(size, int) and size > 0\n",
    "    rand_vec = np.zeros((size,), dtype=np.float64)\n",
    "\n",
    "    # Generate array of uniformly distributed samples, factor 1.5 longer that\n",
    "    # actually needed.\n",
    "    n = (3 * size) // 2\n",
    "    x = np.reshape(uniform_rand_float64(circuit, num_target_qubits,\n",
    "                                        2*n, 0.0, 1.0), (-1, 2))\n",
    "\n",
    "    x1 = 0.0                # first sample in a pair\n",
    "    c = 0                   # counter\n",
    "    for d in range(size):\n",
    "        r2 = 2.0\n",
    "        while r2 >= 1.0 or r2 < TINY:\n",
    "            # Regenerate array of uniformly distributed samples upon shortage.\n",
    "            if c >= n:\n",
    "                c = 0\n",
    "                n = max(size // 10, 1)\n",
    "                x = np.reshape(uniform_rand_float64(circuit, num_target_qubits,\n",
    "                                                    2*n, 0.0, 1.0), (-1, 2))\n",
    "\n",
    "            x1 = 2.0 * x[c, 0] - 1.0        # first sample in a pair\n",
    "            x2 = 2.0 * x[c, 1] - 1.0        # second sample in a pair\n",
    "            r2 = x1 * x1 + x2 * x2\n",
    "            c += 1\n",
    "\n",
    "        f = np.sqrt(np.abs(-2.0 * np.log(r2) / r2))\n",
    "        rand_vec[d] = f * x1\n",
    "        \n",
    "    return (rand_vec * sigma + mu)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The following example demonstrates how to draw a random vector of normally distributed variates:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2019-08-22T01:39:24.088003Z",
     "start_time": "2019-08-22T01:39:22.505982Z"
    }
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Normal distribution (mu=2.400, sigma=5.100):\n",
      "  sample type: <class 'numpy.ndarray'> , element type: float64 , shape: (4321,)\n",
      "  sample min: -16.1580, max: 20.3295\n",
      "  sampling time: 1.32 secs\n"
     ]
    },
    {
     "data": {
      "image/png": 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s9dM7LkxmTyKBlnXXcm+nuzn3k6uYdcgQ6lX6GZ3beb5n4dxO+GZzE276YjD37fMXailzw/yf0vI/7NNgMXcu7pexdTiXDC8Wzu2E335+PmcVvMK+DRdlfF13dhzBXxafzpdfZnxVzpXLi4VzKfrgA3h2xaHcsOcjWVnf3rss5Zetn+Xqq7OyOufK5MXCuRSYwRVXwI3tH6ZpnZKsrffqtmN47bVgrCnnouDFwrkUTJ4cDDt+we7PZXW9+bU3cu218NvfZnW1zn3Pi4VzSTKDG24IvjBXu9b2rK//wgth1ix4882sr9q57BULSb0kfSppnqRrylheT9K4cPk7ktqH88+U9EHMbbukA7OV27lS06c3ZdUqGDAgcdtMqFcPbrzR9y5cNLJSLCTlEVzx7jigCzBQUpe4ZucDq82sI3AnMBzAzB43swPN7EBgELDQzD7IRm7nSpnBgw924Pe/h7y86HIMGgRffAFTp0aXwdVM2dqz6A7MM7MFZrYZGAv0jWvTF3g4nJ4AHCXtMNLOQGBMRpM6V4aXX4b16/MivzhR7drwm98EV+FzLpuyVSxaA7HjISwJ55XZxsy2AmuB5nFtTseLhYvA7bfDGWcsinSvotS558K0afDRR1EncTVJtob7KGsszvivvVbYRlIPYL2ZlfknImkoMBSgoKCAkpISiouLdy5thnim5OVSrs8+y2fmzJ/wf/+3gOLir4OZAwdmP0jM9jjxxHZccUUDLr00d7ZTqVz63cXKxVy5mKk82SoWS4DYC0y2AZaW02aJpNpAY2BVzPIBVLBXYWYjgZEA3bp1s/z8fAoLCyufPI2Ki4s9U5JyKdeoUXDVVdC0acP/ZbrppuwHibn40UEHwd57Q0lJC3r3/mn2s1Qgl353sXIxVy5mKk+2isU0oJOkDsCXBB/8Z8S1mQgMBt4C+gGTzcwAJNUC+gNHZCmvcwAsXgwvvgj33gvvvx9xmJjBCRsD5503haefbv2/s7PiBy/0K+u5NMpKsTCzrZKGAZOAPOABM5st6WZguplNBO4HHpU0j2CPIvYExSOAJWa2IBt5nSt1991wzjnQuHHUSXZ08cVwwAG7sW4dNGwYdRpX3WVtiHIzewF4IW7eDTHTGwn2Hsp6bDGQW/vartpbtw4efBBmzIg6Sdnat4f991/DY4+15MILo07jqjv/Brdz5Rg7Fn7+8+BDOVedfPKX3HNP8D0Q5zLJi4VzZTCDESPgl7+MOknFDjpoDVIwZpVzmeTFwrkyTJsGq1fDscdGnaQCRUXos7lcsuUO/j7gP1GncdWcFwvnyjBiBFx0EdSqAn8hZ7R6leI1B7JsU7Ooo7hqrAr8KTiXXStXwrPPwnnnxcwsKoK5c4Ofabi+djrtWnsD/Vq+zoNfHRd1FFeNebFwLs7DD0Pv3tCiRdRJknfB7s8zatnxbLeYgRBKC1sOFjhX9XixcC6GGTzwAAwZEnWS1Byy6yfsmreByasPijqKq6a8WDgXY/p02LABjqhiYwVIwdX7Ri07IeoorpryYuFcjAcfDL6xvcPg+FXAmQWv8tKq7qzY3CjqKK4a8mLhXGjjRhg3DgYPjjrJzmlap4Q+Laby6Ne/iDqKq4a8WDgXeuYZOPhgaNcu6iQ779zdXuQRLxYuA7I2NpRzue7BB4MLCwFV9uyhI5vMZOWWRswq2Yv9833cTZc+vmfhHMFQ5NOnw8knR52kcmrJOKvgVR79+pioo7hqxouFc8Ajj8Bpp8Euu0SdpPIGFbzM6K+PYpv5n7dLH383uRrPLPgi3jnnRJ0kPfZtuIjW9Vbw2uqDo47iqhEvFq7Gmz49KBjdu0edJH3OLniZR77yA90ufbJWLCT1kvSppHmSriljeT1J48Ll70hqH7Nsf0lvSZot6UNJ9bOV21V/o0fDmVseQj2rz9AYA1pN5rmVP+O7rdWgX83lhKwUC0l5wL3AcUAXYKCkLnHNzgdWm1lH4E5gePjY2sBjwEVmth9QCGzJRm5X/W3bFlzkaGBB9bogRIu633Jkk5k8ubyKfRXd5axs7Vl0B+aZ2QIz2wyMBfrGtekLPBxOTwCOkiTgF8AsM5sJYGYrzWxblnK7aq64GFq3hn0aLI46StqdvdvL/gU9lzayLFyPUVI/oJeZDQnvDwJ6mNmwmDYfhW2WhPfnAz2As4CuQCugJTDWzG4rYx1DgaEABQUFXUeNGkV+fn5mX1iKSkpKPFOSspXrttv2oX37dZx2YOI9i5JmzchftSrjmVJRUabNW/I49aqzeOjGJ2jeeAN07pydTDX8PZWKXMtUVFQ0w8y6lbUsW1/KK2uknfgqVV6b2sBhwCHAeuA1STPM7LUfNDQbCYwE6Natm+Xn51NYWFjZ3GlVXFzsmZKUjVwbN8Lbb8N990Hrsy5InGngQArHjMloplQlynTqrq35cuQaTm3zFEyZkp1MNfg9lapczFSebHVDLQHaxtxvAywtr014nKIxsCqc/7qZrTCz9cALgJ8T6CrthRfgoIOCbqjqakCryYz9puofsHfRy1axmAZ0ktRBUl1gADAxrs1EoHQIt37AZAv6yCYB+0tqEBaRI4GPs5TbVWOjR8MZZ0SdIrOOavoen21ow8INBVFHcVVcVoqFmW0FhhF88M8BnjCz2ZJultQnbHY/0FzSPOAK4JrwsauBOwgKzgfAe2b2fDZyu+pr7Vp45RU45ZSok2RWnVrbOLXFGzyx3PcuXOVkbSBBM3uBoAspdt4NMdMbgf7lPPYxgtNnnUuLp5+Gnj2hadOok2TegFaTuWL+xVwVdRBXpfk3uF2NVBO6oEod3uRDvtrcjE8/jTqJq8q8WLgaZ8UKePddOKGGXIE0T9s5rWUxY8dGncRVZV4sXI3zzDNwbN0pNDih+gzvkciAVpMZOzYYA8u5neHFwtU448dDv5avRx0jq3o0msPGjTBrVtRJXFWVdLGQ1DyTQZzLhpUrgy/iHd/8naijZJUEAwbgXVFup6WyZ7FY0rOS+oXflXCuynn2WTjmGGiYtzHqKFl3+ukwbpx3Rbmdk0qx2BN4Dbga+ErSSEmHZSaWc5kxYQL0L/ME7ervgAMgLw/efz/qJK4qSrpYmNlyM7vHzA4BfgZ8AzwqaUH45bo9M5bSuTRYvRrefBOOPz7qJNGQoF+/oGA6l6qdPcC9W3hrBMwHWgPvl3VRI+dyxcSJwRfxdt016iTR6dcvOMDvXVEuVakc4N5P0p8lLQL+AXwG7G9mx5jZ+QSD+12XoZzOVdr48TW3C6rUwQfD1q1+VpRLXSp7Fm8AuwL9zKyLmQ03sy9LF5rZQuCuNOdzLi3WroU33oDevaNOEi0pKJjjx0edxFU1qRSLk81smJm9GztT0veXuY8d68m5XDJxYvDdu0aNok4SPe+KcjsjlYEEnyM4RhHvJaBZeuI4lxkTJgQfkjVa+E31Qww2fjGG2bN348c/jjiTqzIS7llIqiUpL5iUwvult07A1szHdG7nffttcJG4Pn0St60JJOjX8g3vinIpSaYbaiuwGWgQTm+JuX0MjMhYOufS4Lnn4IgjoHHjqJPkjn4tX/dTaF1KkumG6kBwfezXgSNi5huw3Mw2ZCKYc+niZ0HtqEejOaydBx9/DF26RJ3GVQUJ9yzM7AszW2hme4bTpbdFqRQKSb0kfSppXlnfx5BUT9K4cPk7ktqH89tL2iDpg/D2z1ReoKvZvvsOJk/2Lqh4tWSceqp/Qc8lr8I9C0kjzWxoOP1Iee3M7OwEz5MH3AscAywBpkmaaGax19I+H1htZh0lDQCGA6eHy+ab2YEJX41zcZ5/Hn7+85pxRbxU9e8PF18MN/g5jC4JibqhPo+Znl+J9XQH5pnZAgBJY4G+BMc8SvUFbgynJwB/l6RKrNM5PwuqAoceGlwI6tNPYZ99ok7jcp0sCydbS+oH9DKzIeH9QUAPMxsW0+ajsM2S8P58oAeQD8wG5gLfAr81s/+UsY6hwFCAgoKCrqNGjSI/Pz+zLyxFJSUlnilJ6ci1YUMt+vc/lNG3PE6jhpsqn6lZM/JXrar086RTpTJ17szdd3eiefNNnHXWovRlqsbvqXTLtUxFRUUzzKxbWcsSdUP1TGYFZjY5QZOy9hDiq1R5bZYB7cxspaSuwDOS9jOzb+MyjARGAnTr1s3y8/MpLCxMJn7WFBcXe6YkpSPX+PFBF1SfiQ+lJ9PAgRSOGZOW50qXSmWaMgUJfvUrGDVqr/RlqsbvqXTLxUzlSdQNdX8Sz2FAonfaEqBtzP02wNJy2iyRVBtoDKyyYNdnE4CZzQj3ODoD05PI5mqw78+CejzqJLnrsMNg6VKYNw86dow6jctlFRYLM+uQpvVMAzpJ6gB8CQwAzohrMxEYDLwF9AMmm5lJaklQNLZJ2gvoBCxIUy5XHRUVsX5bPSZNncCIL88Ev1RXufLy4JRTgmM71/iY0a4CWbkGt5ltBYYBk4A5wBNmNju8DkbpSY33A80lzQOuAErfukcAsyTNJDjwfZGZ5VbHscs5L67qQfdGn9Ci7reJG9dwpWNFOVeRRMcs5pjZvuH0YnY8zgCAmbVLtCIzewF4IW7eDTHTG4EdvjplZk8CTyZ6fudiTVh+BP1avhF1jCrhyCNh0SJYsAD2St+hC1fNJDpmcUHM9FmZDOJcumzYVpcXV/bg7o5/jzpKlVC7Npx8ctAVddWLRT9cOGVKNKFczkl0zOK/MdOvZz6Oc5U3adUhHLzrXFrVXRN1lCqjf3+4/nq4qmHUSVyuSuVKeXXDYwyfSVoX/rxFUv1MBnQuVeOXF9K/pf9vk4rCwqAbauGGgqijuByVygHufwA9gUuBQ8KfR+KjzrocsnEjvLCqBye32OF7m64CderASSfBkyuOSNzY1UipFIuTgN5m9qKZfWxmL4bzTspMNOdS9/LLcEDD+exWb3XUUaqc/v1h/DeFUcdwOSqVYvEVwTUtYu1C8A1r53LChAnBtRpc6nr2hM82tGbRxlZRR3E5qMJiIaln6Q14FHhJ0gWSjgvHYnoBKHc0WueyadOm4EJHp/opszulTh3o22IqTy73rii3o50Z7uO6uPsXEgwn7lw0wmtLv7Lip/x4+wB2r+ff2UxKUdEOs/q17MEfvziLy9v6hS7cD2VruA/nMm788kL/Il4lHd10BmfNuY4lG1vQpv6KqOO4HJKV4T6cy7TN22vz75U/8y6oSqpbayt9mk/lKT8rysVJ5XsWjSTdIWmGpC8kLSq9ZTKgc8l4dXVXujT4gtb1/L/hyurX8nXGLz8y6hgux6SyZzECOBi4GWgGXAIsAu7MQC7nUjJ++ZF+FlSaHNNsBh+t68DSTc2jjuJySCrF4hfAqWb2LLAt/Hk6MCgjyZxL0ubttZm44lAvFmlSr9YWejd/i6eWHx51FJdDUikWtYC14XSJpCYE37HwS6a4SE1efRD7NFjsB2TTqH/L15ngXVEuRirFYibB8B4A/wHuJRgCZG66QzmXCh8LKv1+0XQaM9ftzVdfRZ3E5YpUisUFwMJw+lJgI9AEODuZB0vqJelTSfMk7XBNLkn1JI0Ll78jqX3c8naSSiRdmUJmV81t2QLPrvi5nwWVZvXztnB8s3d46qmok7hckXSxMLMFZjY/nF5uZueb2elm9nGix0rKI9gTOQ7oAgyU1CWu2fnAajPrSHDQPP6LfncCLyab19UMkydDx12+pF39b6KOUu30b1nMBP9ungul9D0LSedJekXS7PDn+ZKUxEO7A/PCgrMZGAv0jWvTF3g4nJ4AHFX63JJOIrju9uxU8rrqb8IE6N+qOOoY1dKxzabx3nvwjddhB8iszCul7thQuo3gA/0u4AtgT4LuqH+b2VUJHtsP6GVmQ8L7g4AeZjYsps1HYZsl4f35QA9gA/AqcAxwJVBiZn8pYx1DgaEABQUFXUeNGkV+fn5Sry1bSkpKPFOSksm1davo1+9n/PPqCezWvCTzmZo1I39Vbg0lkulMt4zrywEHrKFPn+THC63K76lsy7VMRUVFM8ysW1nLEo0NFesc4ODSD3MASc8B7wEVFgugrL2P+CpVXpubgDvNrKSinRgzGwmMBOjWrZvl5+dTWFiYIFZ2FRcXe6YkJZPrlVegc2cY8PJ92ck0cKioSRUAABzPSURBVCCFY8ZkZV3JynSmlcu/5B8P9uGO928KZiRxmdWq/J7KtlzMVJ5UuqG+C2/x875N4rFLgLYx99sAS8trI6k20BhYRbB3cZukhcCvgOskDcPVeOPHB9dgcJlzXLN3mPbdPizf3DjqKC5iiYYo36v0RtD99JSkYyTtK+kXwHiS+wb3NKCTpA6S6gIDgIlxbSYCg8PpfsBkCxxuZu3NrH2Y4U9m9vekX6GrlrZuhaefhn79ok5SvTXI20SvZtN4ZsVhUUdxEUvUDTWPoCsotv8nflzjnkCFH95mtjXcG5gE5AEPmNlsSTcD081sIsFw6I9KmkewRzEg+ZfhapriYthzT+jg4yJnXL+WrzNq2QlcsMfzOw5rnkS3lKseEg1RnrZRac3sBYKLJcXOuyFmeiNQYaeCmd2Yrjyuahs3Dk4/PeoUNcPxzd9hyKdXsnJLI5rXSabX2VVHKReD8MtxP5PUNnFr59Jv82Z46ikvFtnSMG8jxzSdwdPLvSuqJktliPLdJb1O0DX1FDBf0huS9shYOufK8HLXa+mybRbtBheVebU3l34DCyYz9pueUcdwEUplz+IfBONDNTWz3YGmwPvAPzMRzLnyjPnmKAa08r7ybDq+2dvMKOnMsk3Noo7iIpJKsTgM+LWZrQMIf14FHJqJYM6VZf16eH7lT+nfsjjqKDXKLnmb6dN8Kk8s9z25miqVYrGaYFynWPsAa9IXx7mKPfcc9Gg0h1Z1/W2XbWcUvMaYr70rqqZKpVjcBrwq6VZJv5R0K/BKON+5rBgzBga0mhx1jBrpqCYz+Hzjbszf4Icpa6JURp29j+DKeC2AE8OfA8NhNpzLuLVrg1FmT27xn6ij1Ei1a22nf8vXGfuNd0XVREkVC0l5kh4G3jSzIWZ2fPjT/8VzWfP008HJT03qrIs6So01sGAyo78+miTHH3XVSFIDCZrZtnB4j+0ZzuNcucaOhXPPxc+/i9DPGs1m3bb6fLhuL/bPX+Df6K5BUjlmcSdwk6Q6mQrjXHmWL4e334bevaNOUrPVkjGg1WRGf31U1FFclqVSLC4BfgN8J2mxpEWlPzOUzbnvTZgAxx8PDRtGncSVfkHPu6JqllSuZ3FWxlI4l8Bjj8G110adwgHs33A+DfM28ta3+3FoY794ZU2Ryp7FW8BRwCiCAQFHAUcD72Qgl3Pfmz8f5s2DY4+NOokDkGBgq9e8K6qGSXW4j54El1I9JPx5JDAiA7mc+96jj8LAgVDHj5bljIEFkxm//Ei2bM+LOorLklS6oU4C9jaz0q/OfizpHYKBBc9LezLnALOgWDzxRNRJXKy9d1lKx12W8tKq7pzY4q2o47gsSGXP4iugQdy8XYDkr+TuXIqmToX69eHgg6NO4uIN3m0SD3/lfYM1RSrF4lHgJUkXSDpO0lCCYxePSOpZeivvwZJ6SfpU0jxJ15SxvJ6kceHydyS1D+d3l/RBeJsp6eTUXqKryh55BM4+O+gnd7nltJZTeHV1V1Zt2TXqKC4LUumGujD8eV3c/IvCGwSXYN0r/oGS8oB7gWOAJcA0SRPN7OOYZucDq82so6QBwHCC4UU+ArqFl2bdHZgp6d9mtjWF7K4K2rgxOGV25syok7iyNKmzjl7N3mXsNz25uPWzUcdxGZbK2FAdkrjtUChC3YF5ZrbAzDYDY4G+cW36Ag+H0xOAoyTJzNbHFIb6BAXJVWdFRTB3Lv/ueiMHbZ9BmzZRB3Ll8a6omkOWhW/WSOoH9DKzIeH9QUAPMxsW0+ajsM2S8P78sM0KST2AB4A9gUFm9nQZ6xgKDAUoKCjoOmrUKPLz8zP90lJSUlLimZIxdy4lzZrx51sO4YiDF3DsYO2wPAolzZqRv2pVJOsuT9SZtm0Tp197Bndc8RztdlsLnTvn5nuK3Hyv51qmoqKiGWbWraxlqXRDVUZZPc7xVarcNmb2DrCfpH2BhyW9aGYbf9AwGP12JEC3bt0sPz+fwsLCSgdPp+LiYs+UjJtu4pne5zJ7dnMmNbqI/MIXd1geheKBAykcMyaSdZcnFzKd22hX5ty/jbP3GgNTpuTme4rcfK/nYqbyZKtYLAHaxtxvAywtp80SSbWBxsAP/mUyszmS1gE/BqZnLq6L2ivvdKJPi6nk197o19nOcYMLJtFr1m38ocMD+Lcuqq9UzoaqjGlAJ0kdJNUFBgAT49pMBAaH0/2AyWZm4WNqA0jak+DqfAuzE9tFwQye/+8+XLD7c1FHcUn4cf5Cdqu7itdW+/nN1VlW9izCM5mGAZOAPOABM5st6WZguplNBO4HHpU0j2CPYkD48MOAayRtIRgi/WIzW5GN3C4ab3/bhe3ba3FY4w+jjuKSNHi3STz4VS9+EXUQlzHZ6obCzF4g+F5G7LwbYqY3Av3LeNyjBN/xcDXEqGUncNzPP0W5dSzZVeDMglf53efnsXJl1ElcpmSrG8q5pHz3HTy14nCO/Wk0Zzy5ndOsznec2GIqjzwSdRKXKV4sXE4ZNw6KmrxPs8Yboo7iUjR09+cYORK/zkU15cXC5ZRRo+D83V9M3NDlnNJjTB9+2DjiJC4TvFi4nPHhh7BkCRzb9N2oo7idIMEFF8Bzz+0edRSXAV4sXM4YNQrOOQdq19oedRS3k84+G6ZObUGOfdHdpYEXC5cT1q0LLp06dGjUSVxltGgBPXqs5LHHok7i0s2LhcsJjz8Ohx8O7dpFncRV1oknLvMD3dWQFwsXOSss4t4r5vN/n1/pQ3tUAwccsIZt2+CNN6JO4tLJi4WL3Jtrf8zG7XU4qul7UUdxlVVUhD6byyXb7uKefl4tqhMvFi5y9y49iYtbT6SWvN+iuji7YBLFaw7giy+iTuLSxYuFi9RXX8FLq7ozuOClqKO4NMqvvZHBu73MP/4RdRKXLl4sXKTuuw/6t3ydJnXWRR3Fpdn/tX6G+++H9eujTuLSwYuFi8ymTTBiBFzS+qmoo7gM2HuXpfz0pzB6dNRJXDp4sXCRGT0a9t8ffpL/edRRXIZceincc4+fRlsdeLFwkTCDO+6AK6+MOonLpKP/WMTW+QspPuhyPy26istasZDUS9KnkuZJuqaM5fUkjQuXvyOpfTj/GEkzJH0Y/uyZrcwuQ4qKePnAq6j1+XyO/qN/gFRnElzeZgK3Lz496iiukrJSLCTlAfcCxwFdgIGSusQ1Ox9YbWYdgTuB4eH8FcCJZvYTgsuu+oWQqoG/Lu7PFW3HI0WdxGXaoIKX+aCkI7NK9oo6iquEbO1ZdAfmmdkCM9sMjAX6xrXpCzwcTk8AjpIkM3vfzJaG82cD9SXVy0pqlxEzS/Zm9rr2DGz1WtRRXBbUz9vCZW2eZPiigVFHcZWQrWLRGlgcc39JOK/MNma2FVgLNI9rcyrwvpltylBOlwV/XXwaw1o/Q91aW6OO4rLkoj3+zUurDmHhwqiTuJ0ly8JpCpL6A8ea2ZDw/iCgu5ldEtNmdthmSXh/fthmZXh/P2Ai8Aszm1/GOoYCQwEKCgq6jho1ivz8/Ay/stSUlJTU+ExLl9bnl0MP5PE/jCW/webyczVrRn6OjXPtmZJTXqZ/PdmdTQ1bcOml8yJI5X9/ySgqKpphZt3KWlY7SxmWAG1j7rcBlpbTZomk2kBjYBWApDbA08DZZRUKADMbCYwE6Natm+Xn51NYWJjO11BpxcXFNT7T0KFwaYtH6P3swxW2Kx44kMIxY7KUKjmeKTnlZdpn0yS6vPsk//pXG1q2jCCX//1VSra6oaYBnSR1kFQXGECwlxBrIsEBbIB+wGQzM0lNgOeBa83szSzldRmwaBFMmACXtX4y6iguArvXW0X//nDnnVEncTsjK8UiPAYxDJgEzAGeMLPZkm6W1Cdsdj/QXNI84Aqg9PTaYUBH4HeSPghvrbKR26XXbbfBkCHQou63UUdxEbnuOvjXv2DFiqiTuFRlqxsKM3sBeCFu3g0x0xuB/mU87g/AHzIe0GXUsmXBN7bnzCHYr3Q1Uvv2cNppcPvtMHx4wuYuh/g3uF1W3HprcH3mgoKok7ioXXddMIDk119HncSlwouFy6yiIhb+dACPjVjLddNO9iEfHG3bwplnBt2SrurwYuEy7saF5/B/rZ+lVd01UUdxOeLaa+HBB4PuSVc1ZO2YhauZZq9rz4urujO3x9lRR3G5INyz3AMY3PBibr65v18gqYrwPQuXUb/9/DyuajeWxrX94kbuh36752NMmBCe9OBynhcLlzFTp8L07/bh4j2ejTqKy0HN63zLNdfA1VdHncQlw4uFy4jt24ML39y6133sklf+sB6uZhs2DD76CKZMiTqJS8SLhcuIhx6CevXgjFavRh3F5bB6vYq4dZeb+HWfuWwv7Olny+UwLxYu7dauheuvh7vvxq9X4RLq37KYXWpt5oFlx0UdxVXAi4VLuz/8AY47DrqVOXalcz8kwYjOd3H95+ezYnOjqOO4cvipsy6tPvww6IL68MOok7iq5ID8+ZxR8BpXL7iQ+6MO48rkxcKlzbZtcMEFwZ7FbrtFncZVNTe1f4gu7z7Emwdfws8bf/S/BX70Oyd4N5RLmxEjoO4nM7lgTHig0g9WuhQ0qr2eOzqO4KK5l7N5u/8fm2u8WLi0WLwYbroJRnb+K7WU+asvuuqpf8tiOtRfxi1fDIo6iovjxcJVmllwBbzLLoMfNVyc+AHOlUMK/uEYubQ3077dJ+o4LoYXC1dpI0bAypVwzTWJ2zqXyG71VnNXx3s555Nr2LitTtRxXChrxUJSL0mfSponaYePFUn1JI0Ll78jqX04v7mkKZJKJP09W3ldcj7+GG68ER57DOr437VLkwGtJrNvgy/47efnRx3FhbJSLCTlAfcCxwFdgIGSusQ1Ox9YbWYdgTuB0utobQR+B1yZjawueZs2Bdcl+NOfoHPnqNO46kSCf3a+gyeWF/L881GncZC9PYvuwDwzW2Bmm4GxQN+4Nn2Bh8PpCcBRkmRm68zsvwRFw+WQq66C9kv+w5DHi/zsJ5d2Lep+y+h9/8B55wUnULhoySzzZ65I6gf0MrMh4f1BQA8zGxbT5qOwzZLw/vywzYrw/jlAt9jHxK1jKDAUoKCgoOuoUaPIz8/P4KtKXUlJSbXJ9MorrXjooQ786zfjyG+Q/oECS5o1I3/VqrQ/b2V4puSkO9Po6UczdWpz7rrrA2rX3vnPq+r095cpRUVFM8yszLEXsnUyc1kjBMX/1pNpUy4zGwmMBOjWrZvl5+dTWFiYdMBsKC4urvqZioqYWbI3I2f+hdcOGMb+zy7ITK6BAykcMyYjz72zPFNy0p3piNeG0rs3PPvskfztb5XIVR3+/iKUrW6oJUDbmPttgKXltZFUG2gM5Na/TI6VWxpx6kc3cU/Hv7F/fmYKhXOxatWC0aPh1Vfxq+pFKFt7FtOATpI6AF8CA4Az4tpMBAYDbwH9gMmWjT4yl7QNG6DPh3/k1JZvMLBgctRxXE1RVEQT4N+N9uCwy/5Gp07NOProqEPVPFnZszCzrcAwYBIwB3jCzGZLullSn7DZ/UBzSfOAK4DvT6+VtBC4AzhH0pIyzqRyGbZtW3Dm0571v+bPe90XdRxXA3VssJRx+93MmWf6QJVRyNoALGb2AvBC3LwbYqY3Av3LeWz7jIZzFTKDX/0K1qyBF3803IfzcJE5sslM7r4aevWC4mLo1CnqRDWHf4PbVcgsOEX2zTfhqaegXq0tUUdyNdyAfxVxU/7tHPOTr1j8s9OijlNj+NCOrlylheK114KDi02aRJ3IucCQPV7gu20N6DnzDl5bBO3aRZ2o+vNi4cq0fTtccQW88UZQKJo1izqRcz90edsJ1JJxeOd+TNr/qv8NYunXv8gILxZuBxs3wtlnw7JlXihcbruszZM0qV1C0cw7ee4n19F117lRR6q2vFi4H1i5Ek46CfbYA155BerXjzqRcxUbvNskGueV0GvWcO7tdDd+FCMzvFi4702fDv37w2mnwZ//HHwZysd7clXBSS3fZM/6X3PSR7cw67dw883h+9eljW9Ohxn8859w3HHwl7/A8OH+h+aqnoN2nce0rr/kjTfg2GNhafwYEa5SfM+ihluxoi59+sCiRcHpsZ0vLAK/aoirolrVXcPkV+CPf4SDDw6GBzn55KhTVQ9eLGqasFvJDB79+hdc9tWvueQSePJJqFs34mzOpUHtY4r4PfCLPbow6IzrGNunNXfeGXWqqs87G2qg6d925rD3/8Y9S05h+PBZ3HyzFwpX/fys8cd8eMh5dO4MBxwAEya0YdOmqFNVXV4sapBPP4VBc66lz0d/ZMjuz/Nu11/SuXNJ1LGcy5hd8jZzyy3wn//A9OlN+dGP4OGHg7HOXGq8WNQAs2bBwIFw+OHwowaL+KT7YM7d/SUf48nVDEVF/OiXRdx63pM80uRSRl02i/12XcTIff7KhiOOjTpdleHFopratAkefxwOOwyOPx4OPBDmz4fr93ycRrXXRx3PuUgc3uRD3jjwMkZ0vot/r/wZe749luuvh88+izpZ7vMD3NXI1q0weTKMGwfPPBOcDXLFFXDiiVCnTjkPmjsXbropqzmdi5IEPZu+T8+m7zN3fRtGrHuUww6DvfeGQYPglFOgoCDqlLnH9yyquCVL4MEH4YwzYPfd4Xe/g/32gw8+CL6BfcopFRQK52q4zg2WcNddwd/R9dcHY6H96EfQowfccgu8+y5s8YGWAd+zqFI2bgyOP7z7bnB7Z8IiVm5pzFFN3+OYptO5dcZvfjj6pn/72rmk1KkDJ5wQ3DZvhv/+F557DoYMgc8/h+7dgy7dbt2CM6vatg32UGqSrBULSb2Au4E8YJSZ3Rq3vB7wCNAVWAmcbmYLw2XXAucD24BLzWxStnJn2/r1wX85ixfDwoXwySf/uy1eHPzX0707HHEEXDnnJvZruJA8bQ8ePPiFCp/bOZdY3brQs2dwA1i9GqYeeS3/fejH3Pu3TsxctzcbGzRn//1h332hQwfYa6/gZ4cOwcCb1bGQZKVYSMoD7gWOAZYA0yRNNLOPY5qdD6w2s46SBgDDgdPDS6gOAPYD9gBeldTZzHLu5Lft24P//tevh/Unns6G7fVYv60e67fXZ/22erzTfzALF8KqVT+8rVwJ33wTFIP166FNm+A/l3btguJw3nnBm3KvC4+hbq2t8BnBLT/iF+xcdZBgD7wpcEJzOKH529/P+2bcFGbNCk5H//xzeOed4OeCBcGeSUHBD28tW0LjxtCoUfjzz9fSqPZ65vU+gtZXn0X9CY9Rr14wcGe9ekHByrWCk609i+7APDNbACBpLNAXiC0WfYEbw+kJwN8lKZw/1sw2AZ+H1+juDryViaBHHw0lJcHB4q1bg/Oxy5reYdmGzWy22tSrtYUGtTbSIO9v4c9N7FJrEw1qbWLDi7uz997QvHnw38d++wU/mzcP3kxt2wbT379JiopgEfByeN+PMDmXE1qdXsTRwNGxM5sAB8O6bfX5enPT4LagKd982pRvNjdh5baGfL61Ad9ua8i3W/vw7bYGLH2oNbeu+CmbDg3+0dy0Kbht3hwUjdICUqcO5OUld7vqqmDk6HSTWebPtZfUD+hlZkPC+4OAHmY2LKbNR2GbJeH9+UAPggLytpk9Fs6/H3jRzCbErWMoMDS8uw9BV9aKTL6undACz5SsXMzlmZKTi5kgN3PlWqY9zaxlWQuytWdR1g5VfJUqr00yj8XMRgIjv38yabqZdUslZKZ5puTlYi7PlJxczAS5mSsXM5UnWx0bS4C2MffbAPEDCH/fRlJtoDGwKsnHOuecy6BsFYtpQCdJHSTVJThgPTGuzURgcDjdD5hsQR/ZRGCApHqSOgCdgHezlNs55xxZ6oYys62ShgGTCE6dfcDMZku6GZhuZhOB+4FHwwPYqwgKCmG7JwgOhm8F/i/JM6FGJm6SdZ4pebmYyzMlJxczQW7mysVMZcrKAW7nnHNVm5+M6ZxzLiEvFs455xKqVsVC0u2SPpE0S9LTkprELLtW0jxJn0rK2iD2kvpLmi1pu6RuMfPbS9og6YPw9s9sZaooV7gskm0Vl+FGSV/GbJ/jo8gRZukVbot5kq6JKkc8SQslfRhun+kRZXhA0jfh96RK5zWT9Iqkz8KfTXMgU6TvJ0ltJU2RNCf8u7ssnB/ptkqJmVWbG/ALoHY4PRwYHk53AWYC9YAOwHwgL0uZ9iX4kmAx0C1mfnvgowi3VXm5IttWcfluBK7MgfdUXrgN9gLqhtumS9S5wmwLgRYRZzgCODj2vQzcBlwTTl9T+ncYcaZI30/A7sDB4fSuwNzwby3SbZXKrVrtWZjZy2a2Nbz7NsF3MiBmyBAz+xwoHTIkG5nmmNmn2VhXKirIFdm2ylHfD1VjZpuB0qFqHGBmbxCcvRirL/BwOP0wkIHBJ1LOFCkzW2Zm74XT3wFzgNZEvK1SUa2KRZzzgBfD6dbA4phlS8J5Uesg6X1Jr0s6POowoVzaVsPCLsUHItw9z6XtEc+AlyXNCIe7yRUFZrYMgg9JoFXEeUrlwvsJSe2Bg4B3yN1ttYMqdz0LSa8Cu5Wx6HozezZscz3BdzIeL31YGe3Tds5wMpnKsAxoZ2YrJXUFnpG0n5l9G3GujG6rH6yognzAP4BbwnXfAvyV4B+AbMva9tgJPzezpZJaAa9I+iT8r9rtKCfeT5LygSeBX5nZt8q1oWUrUOWKhZkdXdFySYOB3sBRFnYEkuEhQxJlKucxm4BN4fSMcODEzkDaDlTuTC6yOLxKsvkk3Qc8l4kMScjZ4WbMbGn48xtJTxN0meVCsfha0u5mtkzS7sA3UQcys69Lp6N6P0mqQ1AoHjezp8LZObetylOtuqEUXGDpaqCPma2PWZRzQ4ZIaqngOh9I2ivMtCDKTKGc2FbhH06pk4GPymubYckMVZN1khpK2rV0muDkjqi2UbzYoXsGA+XtxWZN1O8nBbsQ9wNzzOyOmEU5t63KFfUR9nTeCA7GLgY+CG//jFl2PcFZLZ8Cx2Ux08kE/51uAr4GJoXzTwVmE5xd8x5wYpa3VZm5otxWcfkeBT4EZhH8Qe0e4fvqeIKzV+YTdOFFkiMu017he2dm+D6KJBcwhqBLdUv4fjofaA68RnCJrteAZjmQKdL3E3AYQRfYrJjPp+Oj3lap3Hy4D+eccwlVq24o55xzmeHFwjnnXEJeLJxzziXkxcI551xCXiycc84l5MXCuSyRVChpSdQ5nNsZXiycc84l5MXCOedcQl4snEuRpGskTYibd7ekeySdG17g5jtJCyRdWMHzmKSOMfcfkvSHmPu9wwv1rJE0VdL+mXlFziXmxcK51I0BjpfUCCAc4+s0YDTBQHC9gUbAucCdkg5OdQXhYx4ALiQYEuJfwERJ9dLyCpxLkRcL51JkZl8QjOdVeqGansB6M3vbzJ43s/kWeB14GdiZa5VcAPzLzN4xs21m9jDBOF4/TcdrcC5VXiyc2zmjgYHh9BnhfSQdJ+ltSaskrSEYLK7FTjz/nsCvwy6oNeFztQX2SEN251LmxcK5nTMeKJTUhmAE39FhF9GTwF8IroDWBHiBsi+gBLAeaBBzP/ZCUIuBP5pZk5hbAzMbk/ZX4lwSvFg4txPMbDlQDDwIfG5mc4C6QD1gObBV0nEE15kozwfAGZLywmuxHBmz7D7gIkk9FGgo6YTSa1g4l21eLJzbeaOBo8OfmNl3wKXAE8Bqgu6pii6UdBlwIrAGOBN4pnSBmU0nOG7x9/C55gHnpPsFOJcsv56Fc865hHzPwjnnXEJeLJxzziXkxcI551xCXiycc84l5MXCOedcQl4snHPOJeTFwjnnXEJeLJxzziX0/zaVsiv5E5JXAAAAAElFTkSuQmCC\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "# Mean and standard deviation.\n",
    "mu = 2.4\n",
    "sigma = 5.1\n",
    "\n",
    "# Draw a sample from the normal distribution.\n",
    "start_time = time.time()\n",
    "sample = normal_rand_float64(circuit, glo_num_qubits, size=4321, mu=mu, sigma=sigma)\n",
    "sampling_time = time.time() - start_time\n",
    "\n",
    "# Print out some details.\n",
    "print(\"Normal distribution (mu={:.3f}, sigma={:.3f}):\".format(mu, sigma))\n",
    "print(\"  sample type:\", type(sample), \", element type:\", sample.dtype, \", shape:\", sample.shape)\n",
    "print(\"  sample min: {:.4f}, max: {:.4f}\".format(np.amin(sample), np.amax(sample)))\n",
    "print(\"  sampling time: {:.2f} secs\".format(sampling_time))\n",
    "\n",
    "# Plotting the distribution.\n",
    "x = np.linspace(mu - 4.0 * sigma, mu + 4.0 * sigma, 1000)\n",
    "analyt = np.exp(-0.5 * ((x - mu) / sigma)**2) / (sigma * math.sqrt(2.0 * math.pi))\n",
    "plt.hist(sample.ravel(),\n",
    "         bins=min(int(np.ceil(np.sqrt(sample.size))), 100),\n",
    "         density=True, facecolor='r', alpha=0.75)\n",
    "plt.plot(x, analyt, '-b', lw=1)\n",
    "plt.xlabel(\"value\", size=12)\n",
    "plt.ylabel(\"probability\", size=12)\n",
    "plt.title(\"Normal distribution: empirical vs analytic\", size=12)\n",
    "plt.grid(True)\n",
    "# plt.savefig(\"normal_distrib.png\", bbox_inches=\"tight\")\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "There is a substantial amount of further work needed to either certify the quality of the source of random numbers (cf. NIST SP 800-90B, Recommendation for the Entropy Sources Used for Random Bit Generation) or to use random variates within quantum algorithms (cf. <a href=\"https://github.com/Qiskit/qiskit-aqua/tree/master/qiskit/aqua/components/uncertainty_models\">uncertainty_models</a> within Qiskit Aqua)."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2019-08-22T01:39:36.760562Z",
     "start_time": "2019-08-22T01:39:36.752562Z"
    }
   },
   "outputs": [
    {
     "data": {
      "text/html": [
       "<h3>Version Information</h3><table><tr><th>Qiskit Software</th><th>Version</th></tr><tr><td>Qiskit</td><td>None</td></tr><tr><td>Terra</td><td>0.9.0</td></tr><tr><td>Aer</td><td>0.3.0</td></tr><tr><td>Ignis</td><td>0.2.0</td></tr><tr><td>Aqua</td><td>0.5.6</td></tr><tr><td>IBM Q Provider</td><td>0.3.2rc1</td></tr><tr><th>System information</th></tr><tr><td>Python</td><td>3.7.4 (default, Aug 13 2019, 15:17:50) \n",
       "[Clang 4.0.1 (tags/RELEASE_401/final)]</td></tr><tr><td>OS</td><td>Darwin</td></tr><tr><td>CPUs</td><td>4</td></tr><tr><td>Memory (Gb)</td><td>16.0</td></tr><tr><td colspan='2'>Wed Aug 21 21:39:36 2019 EDT</td></tr></table>"
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      "text/plain": [
       "<IPython.core.display.HTML object>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "text/html": [
       "<div style='width: 100%; background-color:#d5d9e0;padding-left: 10px; padding-bottom: 10px; padding-right: 10px; padding-top: 5px'><h3>This code is a part of Qiskit</h3><p>&copy; Copyright IBM 2017, 2019.</p><p>This code is licensed under the Apache License, Version 2.0. You may<br>obtain a copy of this license in the LICENSE.txt file in the root directory<br> of this source tree or at http://www.apache.org/licenses/LICENSE-2.0.<p>Any modifications or derivative works of this code must retain this<br>copyright notice, and modified files need to carry a notice indicating<br>that they have been altered from the originals.</p></div>"
      ],
      "text/plain": [
       "<IPython.core.display.HTML object>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "import qiskit.tools.jupyter\n",
    "%qiskit_version_table\n",
    "%qiskit_copyright"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
  }
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