Measurement of Qubits using Qiskit

Measurements.ipynb

{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Measurements\n",
    "\n",
    "Qubits are a superposition of $|0\\rangle$ and $|1\\rangle$. We write an arbitrary qubit $|\\psi\\rangle$ as:\n",
    "\n",
    "$|\\psi\\rangle = \\alpha|0\\rangle + \\beta|1\\rangle$\n",
    "\n",
    "When a measurement is made, however, the output will be either 0 or 1. There is a $|\\alpha^{2}|$ probability that the measurement will be 0 and a $|\\beta^{2}|$ probability that the measurement will be 1.\n",
    "\n",
    "Because the measurement is either 0 or 1, $|\\alpha^{2}| + |\\beta^{2}| = 1$\n",
    "\n",
    "To try to guess $|\\alpha^{2}|$ and $|\\beta^{2}|$, a quantum computer (or simulator) can count each measurement and calculate the count divided by number of tries to discover the approximate probabilities $|\\alpha^{2}|$ and $|\\beta^{2}|$.\n",
    "\n",
    "We may extract the counts of each measurement from a circuit using the **qasm_simulator** in Qiskit, or running on a real quantum computer on IBM Q Experience."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 24,
   "metadata": {},
   "outputs": [],
   "source": [
    "from qiskit import *\n",
    "from qiskit.visualization import plot_histogram"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The Qiskit simulators are available in the _Aer_ module. For this notebook, we will explore mostly the **qasm_simulator** because we are interested in simulating the measurements that a quantum computer would give."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 25,
   "metadata": {},
   "outputs": [],
   "source": [
    "simulator = Aer.get_backend(\"qasm_simulator\")"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "By default, all qubits start as $|0\\rangle$. We can verify this by using an empty circuit. In the following code, we use a system with no gates and one measurement device to ensure that 0 is always measured."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 26,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/plain": [
       "<Figure size 173.376x144.48 with 1 Axes>"
      ]
     },
     "execution_count": 26,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "circuit = QuantumCircuit(1, 1)\n",
    "circuit.measure(0, 0)\n",
    "circuit.draw(output=\"mpl\")"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 27,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "{'0': 32768}\n"
     ]
    }
   ],
   "source": [
    "job = execute(circuit, backend=simulator, shots=2**15)\n",
    "result = job.result()\n",
    "counts = result.get_counts()\n",
    "print(counts)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The simulator returns a dictionary where each key is a measurement and each value is the count of that measurement. Often, it is useful to have a histogram of the counts: that is possible with the **plot_histogram** from *qiskit.visualization*"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 28,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 504x360 with 1 Axes>"
      ]
     },
     "execution_count": 28,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "plot_histogram(counts)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "For a single qubit, there are two possible measurements: 0 and 1. To achieve a $|1\\rangle$ component in our qubit, we can use a $X$ gate"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 29,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/plain": [
       "<Figure size 233.576x144.48 with 1 Axes>"
      ]
     },
     "execution_count": 29,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "circuit = QuantumCircuit(1, 1)\n",
    "circuit.x(0)\n",
    "circuit.measure(0, 0)\n",
    "circuit.draw(output=\"mpl\")"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 30,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/plain": [
       "<Figure size 504x360 with 1 Axes>"
      ]
     },
     "execution_count": 30,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "counts = execute(circuit, backend=simulator, shots=2**15).result().get_counts()\n",
    "plot_histogram(counts)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "If there is superposition, then it is possible that some of the times the measurement will be 0 and others it will be 1.\n",
    "\n",
    "An example of gate that generates superposition is $H$. Let's see how the measurements perform when that happens"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 31,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/plain": [
       "<Figure size 233.576x144.48 with 1 Axes>"
      ]
     },
     "execution_count": 31,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "circuit = QuantumCircuit(1, 1)\n",
    "circuit.h(0)\n",
    "circuit.measure(0, 0)\n",
    "circuit.draw(output=\"mpl\")"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "metadata": {},
   "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": [
    "counts = execute(circuit, backend=simulator, shots=2**15).result().get_counts()\n",
    "plot_histogram(counts)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Global Phase\n",
    "\n",
    "One possible thought is: is it possible to differentiate between $|0\\rangle$, $-|0\\rangle$ or $i|0\\rangle$?\n",
    "\n",
    "We can build circuits to achieve those states and compare!"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 32,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/plain": [
       "<Figure size 353.976x144.48 with 1 Axes>"
      ]
     },
     "execution_count": 32,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# Circuit to generate -|0>\n",
    "circuit = QuantumCircuit(1, 1)\n",
    "circuit.x(0)\n",
    "circuit.z(0)\n",
    "circuit.x(0)\n",
    "circuit.measure(0, 0)\n",
    "circuit.draw(output=\"mpl\")"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 33,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/plain": [
       "<Figure size 504x360 with 1 Axes>"
      ]
     },
     "execution_count": 33,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "counts = execute(circuit, backend=simulator, shots=2**15).result().get_counts()\n",
    "plot_histogram(counts)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 34,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/plain": [
       "<Figure size 293.776x144.48 with 1 Axes>"
      ]
     },
     "execution_count": 34,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# Circuit to generate i|0>\n",
    "circuit = QuantumCircuit(1, 1)\n",
    "circuit.x(0)\n",
    "circuit.y(0)\n",
    "circuit.measure(0, 0)\n",
    "circuit.draw(output=\"mpl\")"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 35,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/plain": [
       "<Figure size 504x360 with 1 Axes>"
      ]
     },
     "execution_count": 35,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "counts = execute(circuit, backend=simulator, shots=2**15).result().get_counts()\n",
    "plot_histogram(counts)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "It turns out that no! In general, if we multiply a qubit by a number $z \\in \\mathbb{C}$ such that $|z|^{2} = 1$, then it has no effects in the measurements!\n",
    "\n",
    "Notice that $z$ can be written using Euler's identity:\n",
    "\n",
    "$$\n",
    "z = e^{i\\theta}\n",
    "$$\n",
    "\n",
    "Where, $\\theta \\in [0, 2\\pi]$ and $e$ is Euler's number.\n",
    "\n",
    "Hence we call that global phase: if we multiply by $z$, we are adding a global phase that can be ignored. Thus, in practice, $|\\psi\\rangle = e^{i\\theta}|\\psi\\rangle$"
   ]
  }
 ],
 "metadata": {
  "kernelspec": {
   "display_name": "Python 3",
   "language": "python",
   "name": "python3"
  },
  "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"
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 },
 "nbformat": 4,
 "nbformat_minor": 2
}