algorithms/old_iterative_phase_estimation_algorithm.ipynb

{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "<img src=\"https://raw.githubusercontent.com/Qiskit/qiskit-tutorials/master/images/qiskit-heading.png\" alt=\"Note: In order for images to show up in this jupyter notebook you need to select File => Trusted Notebook\" width=\"500 px\" align=\"left\">"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Iterative Phase Estimation Algorithm\n",
    "\n",
    "\n",
    "The latest version of this notebook is available on https://github.com/qiskit/qiskit-tutorial.\n",
    "\n",
    "For more information about how to use the IBM Q Experience (QX), consult the [tutorials](https://quantumexperience.ng.bluemix.net/qstage/#/tutorial?sectionId=c59b3710b928891a1420190148a72cce&pageIndex=0), or check out the [community](https://quantumexperience.ng.bluemix.net/qstage/#/community).\n",
    "\n",
    "***\n",
    "### Contributors\n",
    "Antonio Córcoles, Jay Gambetta, Rudy Raymond\n",
    "\n",
    "### Qiskit Package Versions"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "{'qiskit': '0.10.3',\n",
       " 'qiskit-terra': '0.8.1',\n",
       " 'qiskit-ignis': '0.1.1',\n",
       " 'qiskit-aer': '0.2.1',\n",
       " 'qiskit-ibmq-provider': '0.2.2',\n",
       " 'qiskit-aqua': '0.5.1'}"
      ]
     },
     "execution_count": 1,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "import qiskit\n",
    "qiskit.__qiskit_version__"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Quantum Phase Estimation (QPE)\n",
    "\n",
    "The Quantum Phase Estimation (QPE) algorithm solves the problem of finding unknown eigenvalues of a unitary operator. The attractiveness of the QPE algorithm is due to the fact that it is a key ingredient of some other very powerful algorithms, like order-finding and Shor's. \n",
    "\n",
    "In a standard textbook, such as Nielsen & Chuang <i>Quantum Computation and Quantum Information</i>, in the QPE, each bit of the phase is encoded in a different qubit on a register using the phase kickback property of controlled-unitary operations. This is followed by an inverse Quantum Fourier Transform operation, which yields an n-bit approximation to the phase by reading the n-qubit register."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Iterative Phase Estimation Algorithm (IPEA)\n",
    "The QPE algorithm can, however, be realized in a much smaller qubit system, by iterating the steps on a system of just two qubits. This is called the Iterative Phase Estimation Algorithm (IPEA).\n",
    "\n",
    "Consider the problem of finding $\\varphi$ given $|\\Psi\\rangle$ and $U$ in $U |\\Psi\\rangle = e^{i \\phi} | \\Psi \\rangle$, with $\\phi = 2 \\pi \\varphi$. Let's assume for now that $\\varphi$ can be written as $\\varphi = \\varphi_1/2 + \\varphi_2/4 + ... + \\varphi_m/2^m = 0.\\varphi_1 \\varphi_2 ... \\varphi_m$, where we have defined the notation $0.\\varphi_1 \\varphi_2 ... \\varphi_m$. Now, if we have two qubits, $q_0$ and $q_1$, and we initialize them as $q_0 \\rightarrow |+\\rangle$ and $q_1 \\rightarrow |\\Psi \\rangle$, then, after applying a control-U between $q_0$ and $q_1$ $2^t$ times, the state of $q_0$ can be written as $|0\\rangle + e^{i 2 \\pi 2^{t} \\varphi} | 1 \\rangle$. That is, the phase of $U$ has been kicked back into $q_0$ as many times as the control operation has been performed.\n",
    "\n",
    "For $t=0$, we have a total phase in $q_0$ of $e^{i 2 \\pi 2^{0} \\varphi} = e^{i 2 \\pi \\varphi} = e^{i 2 \\pi 0.\\varphi_1 \\varphi_2 ... \\varphi_m}$\n",
    "\n",
    "For $t=1$, the phase would be $e^{i 2 \\pi 2^{1} \\varphi} = e^{i 2 \\pi \\varphi_1} e^{i 2 \\pi 0.\\varphi_2 \\varphi_3 ... \\varphi_m}$\n",
    "\n",
    "For $t=2$, $e^{i 2 \\pi 2^{2} \\varphi} = e^{i 2 \\pi 2 \\varphi_1} e^{i 2 \\pi \\varphi_2} e^{i 2 \\pi 0.\\varphi_3 \\varphi_4 ... \\varphi_m}$\n",
    "\n",
    "And for $t=m-1$, $e^{i 2 \\pi 2^{m-1} \\varphi} = e^{i 2 \\pi 2^{m-2} \\varphi_1} e^{i 2 \\pi 2^{m-3} \\varphi_2} ... e^{i 2 \\pi 2^{-1} \\varphi_m} = e^{i 2 \\pi 0.\\varphi_m}$. Note that if we perform a Hadamard operation on the state $|0\\rangle + e^{i 2 \\pi 0.\\varphi_m}|1\\rangle$ and perform a measurement in the standard basis, we obtain $|0\\rangle$ if $\\varphi_m = 0$ and $|1\\rangle$ if $\\varphi_m = 1$. \n",
    "\n",
    "In the first step of the IPEA, we directly measure the least significant bit of the phase $\\varphi$, $\\varphi_m$, by initializing the 2-qubit register as described above, performing $2^{m-1}$ control-$U$ operations between the qubits, and measuring $q_0$ in the diagonal basis.\n",
    "\n",
    "For the second step, we initialize the register in the same way and apply $2^{m-2}$ control-$U$ operations. The phase in $q_0$ after these operations is now $e^{i 2 \\pi 0.\\varphi_{m-1}\\varphi_{m}}= e^{i 2 \\pi 0.\\varphi_{m-1}} e^{i 2 \\pi \\varphi_m/4}$. We see that prior to extracting the phase bit $\\varphi_{m-1}$, we must perform a phase correction of $\\varphi_m /2$. This is equivalent to a rotation around the $Z-$axis of angle $-\\varphi_m /4$.\n",
    "\n",
    "Therefore, the $k$th step of the IPEA, giving $\\varphi_{m-k+1}$, consists of the register initialization ($q_0$ in $|+\\rangle$, $q_1$ in $|\\Psi\\rangle$), the application of control-$U$ $2^{m-k}$ times, a rotation around $Z$ of angle $\\omega_k = -2 \\pi 0.0\\varphi_{k+1} ... \\varphi_m$, a Hadamard transform to $q_0$, and a measurement of $q_0$ in the standard basis. Note that $q_1$ remains in the state $|\\Psi\\rangle$ throughout the algorithm."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## IPEA circuit\n",
    "\n",
    "Let's first initialize the API and import the necessary packages"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2018-09-26T14:28:50.747843Z",
     "start_time": "2018-09-26T14:28:48.229088Z"
    }
   },
   "outputs": [],
   "source": [
    "from math import pi\n",
    "import numpy as np\n",
    "import scipy as sp\n",
    "import matplotlib.pyplot as plt\n",
    "%matplotlib inline\n",
    "\n",
    "# importing Qiskit\n",
    "from qiskit import BasicAer, IBMQ\n",
    "from qiskit import QuantumCircuit, ClassicalRegister, QuantumRegister\n",
    "from qiskit import execute\n",
    "from qiskit.tools.visualization import plot_histogram\n",
    "from qiskit.tools.monitor import job_monitor"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2018-09-26T14:28:51.950686Z",
     "start_time": "2018-09-26T14:28:50.750412Z"
    }
   },
   "outputs": [],
   "source": [
    "# Load saved IBMQ accounts\n",
    "IBMQ.load_account()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Now you can try the following circuit in the quantum simulator for a phase of $-5\\pi/8 = 2 \\pi \\varphi$ and $m=4$. Note that the IPEA cannot be run in the real device in this form, due to the current lack of feedback capability."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2018-09-26T14:28:53.401913Z",
     "start_time": "2018-09-26T14:28:51.952881Z"
    }
   },
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 504x360 with 1 Axes>"
      ]
     },
     "execution_count": 4,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# We first define controlled gates used in the IPEA\n",
    "def cu1fixed(qProg, c, t, a):\n",
    "    qProg.u1(-a, t)\n",
    "    qProg.cx(c, t)\n",
    "    qProg.u1(a, t)\n",
    "    qProg.cx(c, t)\n",
    "\n",
    "def cu5pi8(qProg, c, t):\n",
    "    cu1fixed(qProg, c, t, -5.0*pi/8.0)\n",
    "\n",
    "# We then prepare quantum and classical registers and the circuit\n",
    "qr = QuantumRegister(2)\n",
    "cr = ClassicalRegister(4)\n",
    "circuitName=\"IPEAonSimulator\"\n",
    "ipeaCircuit = QuantumCircuit(qr, cr)\n",
    "\n",
    "# Apply IPEA\n",
    "ipeaCircuit.h(qr[0])\n",
    "for i in range(8):\n",
    "    cu5pi8(ipeaCircuit, qr[0], qr[1])\n",
    "ipeaCircuit.h(qr[0])\n",
    "ipeaCircuit.measure(qr[0], cr[0])\n",
    "\n",
    "ipeaCircuit.reset(qr[0])\n",
    "\n",
    "ipeaCircuit.h(qr[0])\n",
    "for i in range(4):\n",
    "    cu5pi8(ipeaCircuit, qr[0], qr[1])\n",
    "ipeaCircuit.u1(-pi/2, qr[0]).c_if(cr, 1)\n",
    "ipeaCircuit.h(qr[0])\n",
    "ipeaCircuit.measure(qr[0], cr[1])\n",
    "\n",
    "ipeaCircuit.reset(qr[0])\n",
    "\n",
    "ipeaCircuit.h(qr[0])\n",
    "for i in range(2):\n",
    "    cu5pi8(ipeaCircuit, qr[0], qr[1])\n",
    "ipeaCircuit.u1(-pi/4, qr[0]).c_if(cr, 1)\n",
    "ipeaCircuit.u1(-pi/2, qr[0]).c_if(cr, 2)\n",
    "ipeaCircuit.u1(-3*pi/4, qr[0]).c_if(cr, 3)\n",
    "ipeaCircuit.h(qr[0])\n",
    "ipeaCircuit.measure(qr[0], cr[2])\n",
    "\n",
    "ipeaCircuit.reset(qr[0])\n",
    "\n",
    "ipeaCircuit.h(qr[0])\n",
    "cu5pi8(ipeaCircuit, qr[0], qr[1])\n",
    "ipeaCircuit.u1(-pi/8, qr[0]).c_if(cr, 1)\n",
    "ipeaCircuit.u1(-2*pi/8, qr[0]).c_if(cr, 2)\n",
    "ipeaCircuit.u1(-3*pi/8, qr[0]).c_if(cr, 3)\n",
    "ipeaCircuit.u1(-4*pi/8, qr[0]).c_if(cr, 4)\n",
    "ipeaCircuit.u1(-5*pi/8, qr[0]).c_if(cr, 5)\n",
    "ipeaCircuit.u1(-6*pi/8, qr[0]).c_if(cr, 6)\n",
    "ipeaCircuit.u1(-7*pi/8, qr[0]).c_if(cr, 7)\n",
    "ipeaCircuit.h(qr[0])\n",
    "ipeaCircuit.measure(qr[0], cr[3])\n",
    "\n",
    "backend = BasicAer.get_backend('qasm_simulator')\n",
    "shots = 1000\n",
    "\n",
    "results = execute(ipeaCircuit, backend=backend, shots=shots).result()\n",
    "plot_histogram(results.get_counts())"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The results are given in terms of $\\varphi = 0.\\varphi_1 \\varphi_2 \\varphi_3 \\varphi_4$, with the least significant digit ($\\varphi_4$) as the leftmost bit in the classical register. The result is $\\varphi = 11/16$, from which $\\phi = 2\\pi \\varphi = 11 \\pi/8 = 2 \\pi - 5\\pi/8$, as encoded in the circuit. "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## IPEA in the real device\n",
    "\n",
    "As we have mentioned before, we currently lack the ability to use measurement feedback or feedforward, along with qubit resetting, on the real device in the Quantum Experience. However, we still can implement a segmentized version of the IPEA by extracting the information about the phase one bit at a time.\n",
    "\n",
    "Try the following four circuits in the real device. They estimate the same phase as in the previous example (-5$\\pi/8$), one bit at a time, from least ($\\varphi_4$) to most ($\\varphi_1$) significant bit."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2018-09-26T14:28:55.118404Z",
     "start_time": "2018-09-26T14:28:53.403783Z"
    },
    "scrolled": true
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Available backends: [[<QasmSimulatorPy('qasm_simulator') from BasicAer()>, <StatevectorSimulatorPy('statevector_simulator') from BasicAer()>, <UnitarySimulatorPy('unitary_simulator') from BasicAer()>], [<IBMQBackend('ibmqx4') from IBMQ()>, <IBMQBackend('ibmqx2') from IBMQ()>, <IBMQBackend('ibmq_16_melbourne') from IBMQ()>, <IBMQSimulator('ibmq_qasm_simulator') from IBMQ()>]]\n",
      "Job Status: job has successfully run\n"
     ]
    }
   ],
   "source": [
    "# We then prepare quantum and classical registers and the circuit\n",
    "qr = QuantumRegister(5)\n",
    "cr = ClassicalRegister(5)\n",
    "realStep1Circuit = QuantumCircuit(qr, cr)\n",
    "\n",
    "# Apply IPEA\n",
    "realStep1Circuit.h(qr[0])\n",
    "for i in range(8):\n",
    "    cu5pi8(realStep1Circuit, qr[0], qr[1])\n",
    "realStep1Circuit.h(qr[0])\n",
    "realStep1Circuit.measure(qr[0], cr[0])\n",
    "\n",
    "#connect to remote API to be able to use remote simulators and real devices\n",
    "print(\"Available backends:\", [BasicAer.backends(), IBMQ.backends()])\n",
    "\n",
    "backend = IBMQ.get_backend(\"ibmq_5_yorktown\")\n",
    "shots = 1000\n",
    "   \n",
    "job_exp1 = execute(realStep1Circuit, backend=backend, shots=shots)\n",
    "job_monitor(job_exp1)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2018-09-26T14:46:51.283780Z",
     "start_time": "2018-09-26T14:28:55.120518Z"
    }
   },
   "outputs": [
    {
     "data": {
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\n",
      "text/plain": [
       "<Figure size 504x360 with 1 Axes>"
      ]
     },
     "execution_count": 6,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "results1 = job_exp1.result()\n",
    "plot_histogram(results1.get_counts())"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "In the first step of IPEA as above, we obtain the bit \"1\" with probability close to one. We then proceed to the second step of IPEA, assuming that we have identified the result of the first step correctly, as below. "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2018-09-26T14:46:51.501883Z",
     "start_time": "2018-09-26T14:46:51.285861Z"
    },
    "scrolled": true
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "\r",
      "Job Status: job has successfully run\n"
     ]
    }
   ],
   "source": [
    "realStep2Circuit = QuantumCircuit(qr, cr)\n",
    "\n",
    "# Apply IPEA\n",
    "realStep2Circuit.h(qr[0])\n",
    "for i in range(4):\n",
    "    cu5pi8(realStep2Circuit, qr[0], qr[1])\n",
    "realStep2Circuit.u1(-pi/2, qr[0]) # Assuming the value of the measurement on Step 1\n",
    "realStep2Circuit.h(qr[0])\n",
    "realStep2Circuit.measure(qr[0], cr[0])\n",
    "\n",
    "job_exp2 = execute(realStep2Circuit, backend=backend, shots=shots)\n",
    "job_monitor(job_exp1)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2018-09-26T14:47:15.561184Z",
     "start_time": "2018-09-26T14:46:51.503838Z"
    }
   },
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 504x360 with 1 Axes>"
      ]
     },
     "execution_count": 8,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "results2 = job_exp2.result()\n",
    "plot_histogram(results2.get_counts())"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "In the second step of IPEA as above, we obtain the bit \"1\" with probability close to one. We then proceed to the third step of IPEA, assuming that we have identified the result of the first and second steps correctly, as below."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2018-09-26T14:47:15.760111Z",
     "start_time": "2018-09-26T14:47:15.563852Z"
    }
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Job Status: job has successfully run\n"
     ]
    }
   ],
   "source": [
    "realStep3Circuit = QuantumCircuit(qr, cr)\n",
    "\n",
    "# Apply IPEA\n",
    "realStep3Circuit.h(qr[0])\n",
    "for i in range(2):\n",
    "    cu5pi8(realStep3Circuit, qr[0], qr[1])\n",
    "realStep3Circuit.u1(-3*pi/4, qr[0]) # Assuming the value of the measurement on Step 1 and Step 2\n",
    "realStep3Circuit.h(qr[0])\n",
    "realStep3Circuit.measure(qr[0], cr[0])\n",
    "\n",
    "job_exp3 = execute(realStep3Circuit, backend=backend, shots=shots)\n",
    "job_monitor(job_exp3)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2018-09-26T14:47:40.065810Z",
     "start_time": "2018-09-26T14:47:15.761971Z"
    }
   },
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 504x360 with 1 Axes>"
      ]
     },
     "execution_count": 10,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "results3 = job_exp3.result()\n",
    "plot_histogram(results3.get_counts())"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "In the third step of IPEA as above, we obtain the bit \"0\" with probability close to one. We then proceed to the fourth step of IPEA, assuming that we have identified the result of the first, second, and third steps correctly, as below."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2018-09-26T14:47:40.199847Z",
     "start_time": "2018-09-26T14:47:40.069259Z"
    }
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Job Status: job has successfully run\n"
     ]
    }
   ],
   "source": [
    "realStep4Circuit = QuantumCircuit(qr, cr)\n",
    "\n",
    "# Apply IPEA\n",
    "realStep4Circuit.h(qr[0])\n",
    "cu5pi8(realStep4Circuit, qr[0], qr[1])\n",
    "realStep4Circuit.u1(-3*pi/8, qr[0]) # Assuming the value of the measurement on Step 1, 2, and 3\n",
    "realStep4Circuit.h(qr[0])\n",
    "realStep4Circuit.measure(qr[0], cr[0])\n",
    " \n",
    "job_exp4 = execute(realStep4Circuit, backend=backend, shots=shots)\n",
    "job_monitor(job_exp4)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2018-09-26T14:48:04.209177Z",
     "start_time": "2018-09-26T14:47:40.202791Z"
    }
   },
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 504x360 with 1 Axes>"
      ]
     },
     "execution_count": 12,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "results4 = job_exp4.result()\n",
    "plot_histogram(results4.get_counts())"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "In the fourth step of the IPEA, we identify the bit \"1\" with high probability. In summary, we can conclude with high probability that the binary string of the phase is \"1011\"; that is, eleven in the decimal.  \n",
    "\n",
    "We have left aside the case when $\\varphi$ does not accept a decomposition of the form $\\varphi = \\varphi_1/2 + \\varphi_2/4 + ... + \\varphi_m/2^m$. In that case, it can be shown that we can still use the IPEA to obtain $\\varphi$ to an accuracy of $2^{-m}$ with greater than a constant probability independent of $m$ (around $81\\%$ [1])."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### References\n",
    "\n",
    "[1] M. Dobsicek *et al. Phys. Rev. A* **76**, 030306 (2007)"
   ]
  }
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