nine_qubits_QEC_code.ipynb

{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Quantum Error Correction Project\n",
    "## Philippe Noël, Alaisha Sharma, Nicolas Weninger\n",
    "### File #2: 9-qubits code"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {},
   "outputs": [],
   "source": [
    "import numpy as np\n",
    "from qiskit import QuantumProgram\n",
    "import Qconfig\n",
    "from qiskit.tools.visualization import plot_histogram, plot_state"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {},
   "outputs": [
    {
     "name": "stderr",
     "output_type": "stream",
     "text": [
      "/anaconda/lib/python3.6/site-packages/qiskit/_quantumprogram.py:707: DeprecationWarning: set_api() will be deprecated in upcoming versions (>0.5.0). Using the API object instead is recommended.\n",
      "  \"Using the API object instead is recommended.\", DeprecationWarning)\n"
     ]
    }
   ],
   "source": [
    "Q_program = QuantumProgram()\n",
    "Q_program.set_api(Qconfig.APItoken, Qconfig.config['url']) # set the APIToken and API url"
   ]
  },
  {
   "attachments": {},
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### Overview of the 9-qubit code\n",
    "Source: https://pdfs.semanticscholar.org/f456/dfade192d442fce02e394fc01402adba7fc4.pdf\n",
    "\n",
    "<img src=\"img/9_qubit_detailed.png\">"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### 1. Run the theoretical simulation with a local quantum simulator as a control experiment."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {},
   "outputs": [],
   "source": [
    "# define program\n",
    "qp1 = QuantumProgram()\n",
    "n = 9  # number of qubits\n",
    "q = qp1.create_quantum_register(\"q\", n) # quantum register\n",
    "c = qp1.create_classical_register(\"c\", n) # classical register"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {},
   "outputs": [],
   "source": [
    "# create circuit\n",
    "SimControl9 = qp1.create_circuit(\"SimControl9\", [q], [c])"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### Create an arbitrary state $|\\psi\\rangle$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "<qiskit.extensions.standard.h.HGate at 0x10fe55080>"
      ]
     },
     "execution_count": 5,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "SimControl9.h(q[0])"
   ]
  },
  {
   "attachments": {},
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### The following cell encodes the arbitrary state $|\\psi\\rangle$ to 9 qubits as follows:\n",
    "\n",
    "<img src=\"img/9_qubits_single_qubit_encoding_circuit.png\">"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "<qiskit.extensions.standard.cx.CnotGate at 0x10ff0dcf8>"
      ]
     },
     "execution_count": 6,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# Encoding Stage\n",
    "SimControl9.cx(q[0],q[3])\n",
    "SimControl9.cx(q[0],q[6])\n",
    "\n",
    "SimControl9.h(q[0])\n",
    "SimControl9.h(q[3])\n",
    "SimControl9.h(q[6])\n",
    "\n",
    "SimControl9.cx(q[0],q[1])\n",
    "SimControl9.cx(q[0],q[2])\n",
    "\n",
    "SimControl9.cx(q[3],q[4])\n",
    "SimControl9.cx(q[3],q[5])\n",
    "\n",
    "SimControl9.cx(q[6],q[7])\n",
    "SimControl9.cx(q[6],q[8])"
   ]
  },
  {
   "attachments": {},
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "<img src=\"img/9_qubits_total.jpg\">"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### This cell performs X-error correction for the 9-qubit code using the ancilla qubits n+1 and n+2, where n is the first qubit in the block of three."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "<qiskit.extensions.standard.ccx.ToffoliGate at 0x10ff0df28>"
      ]
     },
     "execution_count": 7,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "SimControl9.cx(q[0],q[1])\n",
    "SimControl9.cx(q[0],q[2])\n",
    "SimControl9.ccx(q[1],q[2],q[0])\n",
    "\n",
    "SimControl9.cx(q[3],q[4])\n",
    "SimControl9.cx(q[3],q[5])\n",
    "SimControl9.ccx(q[4],q[5],q[3])\n",
    "\n",
    "SimControl9.cx(q[6],q[7])\n",
    "SimControl9.cx(q[6],q[8])\n",
    "SimControl9.ccx(q[7],q[8],q[6])"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### This cell performs Z-error correction for the 9-qubit code using the same principle as in the 3-qubit scheme."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "<qiskit.extensions.standard.ccx.ToffoliGate at 0x10fea6e48>"
      ]
     },
     "execution_count": 8,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "SimControl9.h(q[0])\n",
    "SimControl9.h(q[3])\n",
    "SimControl9.h(q[6])\n",
    "\n",
    "SimControl9.cx(q[0],q[3])\n",
    "SimControl9.cx(q[0],q[6])\n",
    "\n",
    "SimControl9.ccx(q[3],q[6],q[0]) # correct"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### Measure logical qubit"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {
    "scrolled": true
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "<qiskit._measure.Measure at 0x10fea6e80>"
      ]
     },
     "execution_count": 9,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "SimControl9.measure(q[0], c[0])"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### Execute and plot the measurement results  \n",
    "The shots paremeter allowed us to simulate the circuit quickly without taking too much time, as there are $2^{9}$ states to compute with."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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nFGDVzrbgETEfmAE8tXNAZm6rGn8gNbT4kCQ1Tr2tNnr7wKAjKZ4+t9MGYGrXQRFxOfBJ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\nRK5CYII=\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x10fea6400>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "results = qp1.execute(\"SimControl9\", shots = 50)\n",
    "plot_histogram(results.get_counts(\"SimControl9\"))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### as $|\\psi\\rangle = \\frac{1}{\\sqrt{2}}|0\\rangle + \\frac{1}{\\sqrt{2}}|1\\rangle$ this makes sense. There were no introduced errors. "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### 2. Run the theoretical simulation with a local quantum simulator with X and Z flips introduced to one qubit."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "metadata": {},
   "outputs": [],
   "source": [
    "# define program\n",
    "qp2 = QuantumProgram()\n",
    "n = 9  # number of qubits\n",
    "q = qp2.create_quantum_register(\"q\", n) # quantum register\n",
    "c = qp2.create_classical_register(\"c\", n) # classical register"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "metadata": {},
   "outputs": [],
   "source": [
    "# create circuit\n",
    "SimError9 = qp2.create_circuit(\"SimError9\", [q], [c])"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### Create an arbitrary state $|\\psi\\rangle$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "<qiskit.extensions.standard.h.HGate at 0x10ffb4208>"
      ]
     },
     "execution_count": 13,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "SimError9.h(q[0])"
   ]
  },
  {
   "attachments": {},
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### The following cell encodes the arbitrary state $|\\psi\\rangle$ to 9 qubits as follows, starting with the first two CNOTs:\n",
    "\n",
    "<img src=\"img/9_qubits_single_qubit_encoding_circuit.png\">"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "<qiskit.extensions.standard.cx.CnotGate at 0x111049ba8>"
      ]
     },
     "execution_count": 14,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# First Encoding Stage\n",
    "SimError9.cx(q[0],q[3])\n",
    "SimError9.cx(q[0],q[6])\n",
    "\n",
    "SimError9.h(q[0])\n",
    "SimError9.h(q[3])\n",
    "SimError9.h(q[6])\n",
    "\n",
    "SimError9.cx(q[0],q[1])\n",
    "SimError9.cx(q[0],q[2])\n",
    "\n",
    "SimError9.cx(q[3],q[4])\n",
    "SimError9.cx(q[3],q[5])\n",
    "\n",
    "SimError9.cx(q[6],q[7])\n",
    "SimError9.cx(q[6],q[8])"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### Introduce Z-flip and X-flip error to q0"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 15,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "<qiskit.extensions.standard.x.XGate at 0x10ffb0fd0>"
      ]
     },
     "execution_count": 15,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "SimError9.z(q[0])\n",
    "SimError9.x(q[0])"
   ]
  },
  {
   "attachments": {},
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "<img src=\"img/9_qubits_total.jpg\">"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### This cell performs X-error correction for the 9-qubit code using the ancilla qubits n+1 and n+2, where n is the first qubit in the block of three."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 16,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "<qiskit.extensions.standard.ccx.ToffoliGate at 0x11106a240>"
      ]
     },
     "execution_count": 16,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "SimError9.cx(q[0],q[1])\n",
    "SimError9.cx(q[0],q[2])\n",
    "SimError9.ccx(q[1],q[2],q[0]) # correct\n",
    "\n",
    "SimError9.cx(q[3],q[4])\n",
    "SimError9.cx(q[3],q[5])\n",
    "SimError9.ccx(q[4],q[5],q[3]) # correct\n",
    "\n",
    "SimError9.cx(q[6],q[7])\n",
    "SimError9.cx(q[6],q[8])\n",
    "SimError9.ccx(q[7],q[8],q[6]) # correct\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### This cell performs Z-error correction for the 9-qubit code using the same principle as in the 3-qubit scheme."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 17,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "<qiskit.extensions.standard.ccx.ToffoliGate at 0x11100c208>"
      ]
     },
     "execution_count": 17,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "SimError9.h(q[0])\n",
    "SimError9.h(q[3])\n",
    "SimError9.h(q[6])\n",
    "\n",
    "SimError9.cx(q[0],q[3])\n",
    "SimError9.cx(q[0],q[6])\n",
    "SimError9.ccx(q[3],q[6],q[0]) # correct"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### Measure logical qubit"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 18,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "<qiskit._measure.Measure at 0x10a70c2b0>"
      ]
     },
     "execution_count": 18,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "SimError9.measure(q[0], c[0])"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### Execute and plot the measurement results"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 19,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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      "text/plain": [
       "<matplotlib.figure.Figure at 0x10ff845f8>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "results2 = qp2.execute(\"SimError9\", shots=100)\n",
    "plot_histogram(results2.get_counts(\"SimError9\"))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### as $|\\psi\\rangle = \\frac{1}{\\sqrt{2}}|0\\rangle + \\frac{1}{\\sqrt{2}}|1\\rangle$ this makes sense. There was an induced error that was corrected."
   ]
  },
  {
   "attachments": {},
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### Neither was run on an IBMQ computer as the series of swaps that would have been required to map it to the 16-qubit unit would have been very complicated.\n",
    "\n",
    "<img src=\"img/16_qubits_architecture.png\">"
   ]
  }
 ],
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