Faster Amplitude Estimation.ipynb

{
 "cells": [
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {},
   "outputs": [],
   "source": [
    "import numpy as np\n",
    "from qiskit import Aer, QuantumCircuit\n",
    "from qiskit.aqua.utils import CircuitFactory\n",
    "from qiskit.aqua.algorithms import AmplitudeEstimation, FasterAmplitudeEstimation\n",
    "from qiskit.aqua.components.uncertainty_models import NormalDistribution\n",
    "from qiskit.quantum_info import Statevector\n",
    "\n",
    "\n",
    "import matplotlib.pyplot as plt\n",
    "%matplotlib inline\n",
    "\n",
    "from scipy.stats import norm"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {},
   "outputs": [],
   "source": [
    "class BernoulliAFactory(CircuitFactory):\n",
    "    r\"\"\"Circuit Factory representing the operator A in a Bernoulli problem.\n",
    "\n",
    "    Given a probability $p$, the operator A prepares the state $\\sqrt{1 - p}|0> + \\sqrt{p}|1>$.\n",
    "    \"\"\"\n",
    "\n",
    "    def __init__(self, probability=0.5):\n",
    "        #\n",
    "        super().__init__(1)\n",
    "        self._probability = probability\n",
    "        self.i_state = 0\n",
    "        self._theta_p = 2 * np.arcsin(np.sqrt(probability))\n",
    "\n",
    "    def build(self, qc, q, q_ancillas=None, params=None):\n",
    "        # A is a rotation of angle theta_p around the Y-axis\n",
    "        qc.ry(self._theta_p, q[self.i_state])\n",
    "\n",
    "    def value_to_estimation(self, value):\n",
    "        return value"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {},
   "outputs": [],
   "source": [
    "class BernoulliQFactory(CircuitFactory):\n",
    "    r\"\"\"Circuit Factory representing the operator A in a Bernoulli problem.\n",
    "\n",
    "    Given a probability $p$, the operator A prepares the state $\\sqrt{1 - p}|0> + \\sqrt{p}|1>$.\n",
    "    \"\"\"\n",
    "\n",
    "    def __init__(self, probability=0.5):\n",
    "        #\n",
    "        super().__init__(2)\n",
    "        self._probability = probability\n",
    "        self.i_state = 0\n",
    "        self._theta_p = 2 * np.arcsin(np.sqrt(probability))\n",
    "\n",
    "    def build(self, qc, q, q_ancillas=None, params=None):\n",
    "        # A is a rotation of angle theta_p around the Y-axis\n",
    "        qc.ry(2 * self._theta_p, q[self.i_state])\n",
    "        \n",
    "    def build_power(self, qc, q, power, q_ancillas=None):\n",
    "        qc.ry(2 * power * self._theta_p, q[self.i_state])        \n",
    "\n",
    "    def value_to_estimation(self, value):\n",
    "        return value"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {},
   "outputs": [],
   "source": [
    "from qiskit.circuit.library import RYGate\n",
    "\n",
    "class BernoulliS11QFactory(CircuitFactory):\n",
    "    r\"\"\"Circuit Factory representing the operator A in a Bernoulli problem.\n",
    "\n",
    "    Given a probability $p$, the operator A prepares the state $\\sqrt{1 - p}|0> + \\sqrt{p}|1>$.\n",
    "    \"\"\"\n",
    "\n",
    "    def __init__(self, probability=0.5):\n",
    "        #\n",
    "        super().__init__(2)\n",
    "        self._probability = probability\n",
    "        self._theta_p = 2 * np.arcsin(np.sqrt(probability))\n",
    "        \n",
    "    def required_ancillas(self):\n",
    "        return 1\n",
    "\n",
    "    def build(self, qc, q, q_ancillas=None, params=None):\n",
    "        # A is a rotation of angle theta_p around the Y-axis\n",
    "        self.build_power(qc, q, 1, q_ancillas)\n",
    "        \n",
    "    def build_power(self, qc, q, power, q_ancillas=None):\n",
    "        qc.ccx(0, 1, 2)\n",
    "        #qc.cry(2 * power * self._theta_p, 2, 0)\n",
    "        qc.cry(2 * power * self._theta_p, 2, 1)\n",
    "        qc.ccx(0, 1, 2)\n",
    "#         qc.append(RYGate(2 * power * self._theta_p).control(2), [0, 1, 2])\n",
    "        \n",
    "\n",
    "    def value_to_estimation(self, value):\n",
    "        return value"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 85,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "0.030975999999999997\n"
     ]
    }
   ],
   "source": [
    "n = 2\n",
    "#X = NormalDistribution(n, mu=1, sigma=1, low=0, high=2)\n",
    "y = 0\n",
    "# a_factory = SingleRV(X, y, 1)\n",
    "p = (0.176)**2\n",
    "a_factory = BernoulliAFactory(p)\n",
    "q_factory = BernoulliQFactory(p)\n",
    "backend = Aer.get_backend('qasm_simulator')\n",
    "\n",
    "N = [1000, 1000]\n",
    "delta = 1 - 0.01\n",
    "maxiter = 3\n",
    "\n",
    "print(p)\n",
    "a = np.sqrt(p)\n",
    "\n",
    "faster = FasterAmplitudeEstimation(N, delta, maxiter, a_factory, q_factory, i_objective=[0])\n",
    "#faster._x_factory = a_factory\n",
    "# faster = FasterAmplitudeEstimation(N, delta, maxiter, a_factory, q_factory)\n",
    "#faster._x_factory = None\n",
    "\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 91,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "0.0308891684730608"
      ]
     },
     "execution_count": 91,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "result = faster.run(backend)\n",
    "result['estimation']"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 71,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "0.03224264518583157"
      ]
     },
     "execution_count": 71,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": []
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "0.0576"
      ]
     },
     "execution_count": 8,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "p"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "0.05763189814265775"
      ]
     },
     "execution_count": 9,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "np.arcsin(p)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "reps = 1000\n",
    "results = []\n",
    "maxiters = [2]\n",
    "for maxiter in maxiters:\n",
    "    faster = FasterAmplitudeEstimation(N, delta, maxiter, a_factory)\n",
    "    inside = 0\n",
    "    for _ in range(reps):\n",
    "        result = faster.run(backend)\n",
    "        if result['value_ci'][0] <= p and p <= result['value_ci'][1]:\n",
    "            inside += 1\n",
    "    print(N, delta, maxiter, 'confidence:', inside/reps)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "scrolled": true
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "1 -1.9153754392602926\n",
      "[0.38323556853974877, 1.6793026100265211] True\n",
      "1 -1.9457796655360402\n",
      "1 1.8983435390644192\n",
      "3 -3.070674878630545\n",
      "5 -0.1305617798610741\n",
      "10 -0.545211727683385\n",
      "[0.8698663142448932, 0.9603221347121461] True\n",
      "1 -1.9656214028218646\n",
      "1 1.9478723981115798\n",
      "3 -2.891106491815511\n",
      "5 0.09990989706871115\n",
      "10 -0.39667283080928845\n",
      "20 -1.0973323672040916\n",
      "40 -2.684324050745464\n",
      "79 0.44034012848574555\n",
      "158 0.45192656221409666\n",
      "[0.9187078852956052, 0.9244233120614351] True\n",
      "1 -1.9238026095357599\n",
      "1 1.948718623201206\n",
      "3 -2.9235098479769226\n",
      "5 -0.02042991137445622\n",
      "10 -0.47152712307554084\n",
      "20 -1.078074129969064\n",
      "40 -2.6933611551201544\n",
      "79 0.4125836506259257\n",
      "158 0.37548068648157323\n",
      "316 0.39884177383933533\n",
      "632 0.3960235154840463\n",
      "1264 0.32913768458642023\n",
      "2528 0.2215186869294156\n",
      "[0.9214145854385846, 0.9217719685768764] True\n",
      "1 -1.9129365048968887\n",
      "1 1.9495315943515537\n",
      "3 -2.866566049168211\n",
      "5 0.04304300967375341\n",
      "10 -0.3263590248237056\n",
      "20 -1.0937015806828392\n",
      "40 -2.6547967721233765\n",
      "79 0.46000874276787235\n",
      "158 0.4485584554455757\n",
      "316 0.451125120848994\n",
      "632 0.4233131158523193\n",
      "1264 0.34212438320550365\n",
      "2528 0.21919574776989334\n"
     ]
    }
   ],
   "source": [
    "maxiters = list(range(3, 20, 4))\n",
    "results = []\n",
    "for maxiter in maxiters:\n",
    "    faster = FasterAmplitudeEstimation(N, delta, maxiter, a_factory)\n",
    "    faster._x_factory = None\n",
    "    result = faster.run(backend)\n",
    "    results.append(result)\n",
    "    print(result['value_ci'], result['value_ci'][0] <= p and p <= result['value_ci'][1])"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 19,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "theta 0.05632978476122883 sin(theta) 0.0563 a/4 0.05\n"
     ]
    }
   ],
   "source": [
    "print('theta', np.arcsin(0.0563), 'sin(theta)', 0.0563, 'a/4', 0.2 / 4)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 20,
   "metadata": {},
   "outputs": [],
   "source": [
    "a = np.sqrt(0.2)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 21,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "(0.8660254037844386,\n",
       " 0.22360679774997896,\n",
       " 0.43301270189221935,\n",
       " 0.11180339887498948)"
      ]
     },
     "execution_count": 21,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "np.sqrt(15 * (1 - a**2)) / 4, np.sqrt(1 - a**2) / 4, np.sqrt(15) * a /4, a / 4"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 67,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "0.11180339887498948\n",
      "1 0.32982002668121896\n",
      "2 0.5313456531533876\n",
      "3 0.7063039969678867\n"
     ]
    }
   ],
   "source": [
    "p = 0.2\n",
    "a = np.sqrt(p)\n",
    "theta = np.arcsin(a / 4)\n",
    "print(np.sin(theta))\n",
    "for m in range(1, 4):\n",
    "    print(m, np.sin((2*m + 1) * theta))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 62,
   "metadata": {},
   "outputs": [],
   "source": [
    "zero = Statevector.from_int(0, dims=(2,2))\n",
    "three = Statevector.from_int(3, dims=(2,2))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 64,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "Statevector([0.       +0.j, 0.       +0.j, 0.       +0.j, 0.1118034+0.j],\n",
       "            dims=(2, 2))"
      ]
     },
     "execution_count": 64,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "circuit = QuantumCircuit(2)\n",
    "circuit.ry(2 * np.arcsin(np.sqrt(0.2)), 0)\n",
    "circuit.ry(2 * np.arcsin(0.25), 1)\n",
    "Statevector.from_int(0, dims=(2,2)).evolve(circuit) @ three"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 70,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "Statevector([0.8660254+0.j, 0.4330127+0.j, 0.2236068+0.j, 0.1118034+0.j,\n",
       "             0.       +0.j, 0.       +0.j, 0.       +0.j, 0.       +0.j],\n",
       "            dims=(2, 2, 2))"
      ]
     },
     "execution_count": 70,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "circuit = QuantumCircuit(3)\n",
    "circuit.ry(2 * np.arcsin(np.sqrt(0.2)), 0)\n",
    "circuit.ry(2 * np.arcsin(0.25), 1)\n",
    "Statevector.from_int(0, dims=(2,2,2)).evolve(circuit)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 82,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "Statevector([0.        +0.j, 0.        +0.j, 0.        +0.j,\n",
       "             0.06708204+0.j, 0.        +0.j, 0.        +0.j,\n",
       "             0.        +0.j, 0.        +0.j],\n",
       "            dims=(2, 2, 2))"
      ]
     },
     "execution_count": 82,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "zero = Statevector.from_int(0, dims=(2,2,2))\n",
    "three = Statevector.from_int(3, dims=(2,2,2))\n",
    "\n",
    "\n",
    "q = QuantumCircuit(3)\n",
    "q.ccx(0, 1, 2)\n",
    "q.ry(2 * 2 * np.arcsin(np.sqrt(p)), 2)\n",
    "# q.cry(2 * 2 * np.arcsin(np.sqrt(0.2)), 2, 1)\n",
    "q.ccx(0, 1, 2)\n",
    "zero.evolve(circuit).evolve(q) @ three"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 23,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "Statevector([0.6846532+0.j, 0.6846532+0.j, 0.1767767+0.j, 0.1767767+0.j,\n",
       "             0.       +0.j, 0.       +0.j, 0.       +0.j, 0.       +0.j],\n",
       "            dims=(2, 2, 2))"
      ]
     },
     "execution_count": 23,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "circuit = faster.construct_circuit(0)\n",
    "circuit.draw()\n",
    "zero = Statevector.from_int(0, dims=(2,2,2))\n",
    "three = Statevector.from_int(3, dims=(2,2,2))\n",
    "zero.evolve(circuit) \n",
    "#faster.run(backend)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 24,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<pre style=\"word-wrap: normal;white-space: pre;background: #fff0;line-height: 1.1;font-family: &quot;Courier New&quot;,Courier,monospace\">         ┌──────────┐       ┌───────────┐  ┌───┐     ┌───┐  ┌──────────┐      »\n",
       "  q_0: ──┤ RY(pi/2) ├───■───┤ RY(-pi/2) ├──┤ X ├──■──┤ X ├──┤ RY(pi/2) ├───■──»\n",
       "       ┌─┴──────────┴┐  │  ┌┴───────────┴─┐├───┤  │  ├───┤┌─┴──────────┴┐  │  »\n",
       "  q_1: ┤ RY(0.50536) ├──■──┤ RY(-0.50536) ├┤ X ├──■──┤ X ├┤ RY(0.50536) ├──■──»\n",
       "       └────┬───┬────┘┌───┐└──────────────┘└───┘┌─┴─┐├───┤└────┬───┬────┘┌───┐»\n",
       "aux_0: ─────┤ X ├─────┤ H ├─────────────────────┤ X ├┤ H ├─────┤ X ├─────┤ X ├»\n",
       "            └───┘     └───┘                     └───┘└───┘     └───┘     └───┘»\n",
       "«        ┌───────────┐  ┌───┐     ┌───┐  ┌──────────┐       ┌───────────┐  »\n",
       "«  q_0: ─┤ RY(-pi/2) ├──┤ X ├──■──┤ X ├──┤ RY(pi/2) ├───■───┤ RY(-pi/2) ├──»\n",
       "«       ┌┴───────────┴─┐├───┤  │  ├───┤┌─┴──────────┴┐  │  ┌┴───────────┴─┐»\n",
       "«  q_1: ┤ RY(-0.50536) ├┤ X ├──■──┤ X ├┤ RY(0.50536) ├──■──┤ RY(-0.50536) ├»\n",
       "«       └────┬───┬─────┘└───┘┌─┴─┐├───┤└────┬───┬────┘┌───┐└────┬───┬─────┘»\n",
       "«aux_0: ─────┤ H ├───────────┤ X ├┤ H ├─────┤ X ├─────┤ X ├─────┤ H ├──────»\n",
       "«            └───┘           └───┘└───┘     └───┘     └───┘     └───┘      »\n",
       "«       ┌───┐     ┌───┐  ┌──────────┐       ┌───────────┐  ┌───┐     ┌───┐»\n",
       "«  q_0: ┤ X ├──■──┤ X ├──┤ RY(pi/2) ├───■───┤ RY(-pi/2) ├──┤ X ├──■──┤ X ├»\n",
       "«       ├───┤  │  ├───┤┌─┴──────────┴┐  │  ┌┴───────────┴─┐├───┤  │  ├───┤»\n",
       "«  q_1: ┤ X ├──■──┤ X ├┤ RY(0.50536) ├──■──┤ RY(-0.50536) ├┤ X ├──■──┤ X ├»\n",
       "«       └───┘┌─┴─┐├───┤└────┬───┬────┘┌───┐└────┬───┬─────┘└───┘┌─┴─┐├───┤»\n",
       "«aux_0: ─────┤ X ├┤ H ├─────┤ X ├─────┤ X ├─────┤ H ├───────────┤ X ├┤ H ├»\n",
       "«            └───┘└───┘     └───┘     └───┘     └───┘           └───┘└───┘»\n",
       "«         ┌──────────┐ \n",
       "«  q_0: ──┤ RY(pi/2) ├─\n",
       "«       ┌─┴──────────┴┐\n",
       "«  q_1: ┤ RY(0.50536) ├\n",
       "«       └────┬───┬────┘\n",
       "«aux_0: ─────┤ X ├─────\n",
       "«            └───┘     </pre>"
      ],
      "text/plain": [
       "         ┌──────────┐       ┌───────────┐  ┌───┐     ┌───┐  ┌──────────┐      »\n",
       "  q_0: ──┤ RY(pi/2) ├───■───┤ RY(-pi/2) ├──┤ X ├──■──┤ X ├──┤ RY(pi/2) ├───■──»\n",
       "       ┌─┴──────────┴┐  │  ┌┴───────────┴─┐├───┤  │  ├───┤┌─┴──────────┴┐  │  »\n",
       "  q_1: ┤ RY(0.50536) ├──■──┤ RY(-0.50536) ├┤ X ├──■──┤ X ├┤ RY(0.50536) ├──■──»\n",
       "       └────┬───┬────┘┌───┐└──────────────┘└───┘┌─┴─┐├───┤└────┬───┬────┘┌───┐»\n",
       "aux_0: ─────┤ X ├─────┤ H ├─────────────────────┤ X ├┤ H ├─────┤ X ├─────┤ X ├»\n",
       "            └───┘     └───┘                     └───┘└───┘     └───┘     └───┘»\n",
       "«        ┌───────────┐  ┌───┐     ┌───┐  ┌──────────┐       ┌───────────┐  »\n",
       "«  q_0: ─┤ RY(-pi/2) ├──┤ X ├──■──┤ X ├──┤ RY(pi/2) ├───■───┤ RY(-pi/2) ├──»\n",
       "«       ┌┴───────────┴─┐├───┤  │  ├───┤┌─┴──────────┴┐  │  ┌┴───────────┴─┐»\n",
       "«  q_1: ┤ RY(-0.50536) ├┤ X ├──■──┤ X ├┤ RY(0.50536) ├──■──┤ RY(-0.50536) ├»\n",
       "«       └────┬───┬─────┘└───┘┌─┴─┐├───┤└────┬───┬────┘┌───┐└────┬───┬─────┘»\n",
       "«aux_0: ─────┤ H ├───────────┤ X ├┤ H ├─────┤ X ├─────┤ X ├─────┤ H ├──────»\n",
       "«            └───┘           └───┘└───┘     └───┘     └───┘     └───┘      »\n",
       "«       ┌───┐     ┌───┐  ┌──────────┐       ┌───────────┐  ┌───┐     ┌───┐»\n",
       "«  q_0: ┤ X ├──■──┤ X ├──┤ RY(pi/2) ├───■───┤ RY(-pi/2) ├──┤ X ├──■──┤ X ├»\n",
       "«       ├───┤  │  ├───┤┌─┴──────────┴┐  │  ┌┴───────────┴─┐├───┤  │  ├───┤»\n",
       "«  q_1: ┤ X ├──■──┤ X ├┤ RY(0.50536) ├──■──┤ RY(-0.50536) ├┤ X ├──■──┤ X ├»\n",
       "«       └───┘┌─┴─┐├───┤└────┬───┬────┘┌───┐└────┬───┬─────┘└───┘┌─┴─┐├───┤»\n",
       "«aux_0: ─────┤ X ├┤ H ├─────┤ X ├─────┤ X ├─────┤ H ├───────────┤ X ├┤ H ├»\n",
       "«            └───┘└───┘     └───┘     └───┘     └───┘           └───┘└───┘»\n",
       "«         ┌──────────┐ \n",
       "«  q_0: ──┤ RY(pi/2) ├─\n",
       "«       ┌─┴──────────┴┐\n",
       "«  q_1: ┤ RY(0.50536) ├\n",
       "«       └────┬───┬────┘\n",
       "«aux_0: ─────┤ X ├─────\n",
       "«            └───┘     "
      ]
     },
     "execution_count": 24,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "m = 4\n",
    "circuit = faster.construct_circuit(m)\n",
    "circuit.draw()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 25,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "Statevector([0.        +0.j, 0.        +0.j, 0.        +0.j,\n",
       "             0.99959107+0.j, 0.        +0.j, 0.        +0.j,\n",
       "             0.        +0.j, 0.        +0.j],\n",
       "            dims=(2, 2, 2))"
      ]
     },
     "execution_count": 25,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "zero.evolve(circuit) @ three "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 26,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "sin(2m + 1)theta 0.8459471409339914\n"
     ]
    }
   ],
   "source": [
    "print('sin(2m + 1)theta', np.sin((2 * m + 1) * np.arcsin(a / 4)))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 41,
   "metadata": {},
   "outputs": [],
   "source": [
    "widths = [np.diff(result['value_ci'])[0] / 2 for result in results]\n",
    "oracle_calls = [result['num_oracle_calls'] for result in results]"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 42,
   "metadata": {},
   "outputs": [
    {
     "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": [
    "plt.loglog(oracle_calls, widths)\n",
    "plt.grid()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 84,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "<qiskit.circuit.instructionset.InstructionSet at 0x134521150>"
      ]
     },
     "execution_count": 84,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "from qiskit import transpile\n",
    "from qiskit.circuit import QuantumCircuit, Parameter\n",
    "circuit = QuantumCircuit(3)\n",
    "a, t = Parameter('alpha'), Parameter('theta')\n",
    "\n",
    "circuit.ry(2 * t, 0)\n",
    "circuit.ry(2 * a, 1)\n",
    "circuit.x([0, 1, 2])\n",
    "circuit.h(2)\n",
    "circuit.ccx(0, 1, 2)\n",
    "circuit.h(2)\n",
    "circuit.x([0, 1, 2])\n",
    "circuit.ry(-2 * t, 0)\n",
    "circuit.ry(-2 * a, 1)\n",
    "circuit.cz(0, 1)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 85,
   "metadata": {},
   "outputs": [],
   "source": [
    "transpiled = transpile(circuit, optimization_level=3)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 86,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<pre style=\"word-wrap: normal;white-space: pre;background: #fff0;line-height: 1.1;font-family: &quot;Courier New&quot;,Courier,monospace\">     ┌─────────────┐┌───┐     ┌───┐┌──────────────┐   \n",
       "q_0: ┤ RY(2*theta) ├┤ X ├──■──┤ X ├┤ RY(-2*theta) ├─■─\n",
       "     ├─────────────┤├───┤  │  ├───┤├──────────────┤ │ \n",
       "q_1: ┤ RY(2*alpha) ├┤ X ├──■──┤ X ├┤ RY(-2*alpha) ├─■─\n",
       "     └────┬───┬────┘├───┤┌─┴─┐├───┤└────┬───┬─────┘   \n",
       "q_2: ─────┤ X ├─────┤ H ├┤ X ├┤ H ├─────┤ X ├─────────\n",
       "          └───┘     └───┘└───┘└───┘     └───┘         </pre>"
      ],
      "text/plain": [
       "     ┌─────────────┐┌───┐     ┌───┐┌──────────────┐   \n",
       "q_0: ┤ RY(2*theta) ├┤ X ├──■──┤ X ├┤ RY(-2*theta) ├─■─\n",
       "     ├─────────────┤├───┤  │  ├───┤├──────────────┤ │ \n",
       "q_1: ┤ RY(2*alpha) ├┤ X ├──■──┤ X ├┤ RY(-2*alpha) ├─■─\n",
       "     └────┬───┬────┘├───┤┌─┴─┐├───┤└────┬───┬─────┘   \n",
       "q_2: ─────┤ X ├─────┤ H ├┤ X ├┤ H ├─────┤ X ├─────────\n",
       "          └───┘     └───┘└───┘└───┘     └───┘         "
      ]
     },
     "execution_count": 86,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "transpiled.draw()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
  }
 ],
 "metadata": {
  "kernelspec": {
   "display_name": "Python 3.7.5 64-bit ('latest': virtualenv)",
   "language": "python",
   "name": "python37564bitlatestvirtualenv76cfe9d7d3f749489cf2ff69593234af"
  },
  "language_info": {
   "codemirror_mode": {
    "name": "ipython",
    "version": 3
   },
   "file_extension": ".py",
   "mimetype": "text/x-python",
   "name": "python",
   "nbconvert_exporter": "python",
   "pygments_lexer": "ipython3",
   "version": "3.7.5"
  }
 },
 "nbformat": 4,
 "nbformat_minor": 2
}