{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Manipulating and measuring observables\n",
    "This notebook introduces the Observable class that allows to describe, manipulate and sample observables over quantum states produced by circuits.\n",
    "\n",
    "## Defining a new observable\n",
    "\n",
    "We will take as example a simple observable that counts the number of ones in a quantum state over 5 qubits.\n",
    "\n",
    "This observable can be written as:\n",
    "\n",
    "$$ O = \\Sigma_i (1 - \\sigma_z^i)/2 $$\n",
    "\n",
    "An observable is initialized with the number of qubits it acts on:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "from qat.core import Observable, Term\n",
    "nbqbits = 5\n",
    "one_count = Observable(nbqbits)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "New Pauli terms can be added to the observable.\n",
    "\n",
    "First, we need to write our observable $O$ as a sum of weighted Pauli operators:\n",
    "\n",
    "$$ O = N/2 - \\Sigma_i \\frac{1}{2}\\sigma_z^i $$\n",
    "\n",
    "\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "# The sigma Z terms:\n",
    "for i in range(nbqbits):\n",
    "    one_count.add_term(Term(-0.5, \"Z\", [i]))\n",
    "# And the constant term:\n",
    "one_count.constant_coeff += nbqbits/2"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We can print our observable to check if it is correct"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "print(one_count)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Sampling an observable over the final state of a circuit\n",
    "Lets build a simple circuit and approximate the expectation of our observable over its final state.\n",
    "\n",
    "Because PyLinalg does not natively supports observable sampling, we will use an intermediate plugin `ObservableSplitter` in order to split the observable carrying job into a collection of basic sampling jobs."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "from qat.lang.AQASM import Program, X, CNOT, RX\n",
    "\n",
    "prog_2_ones = Program()\n",
    "qbits = prog_2_ones.qalloc(nbqbits)\n",
    "prog_2_ones.apply(X, qbits[0])\n",
    "prog_2_ones.apply(CNOT, qbits[0], qbits[2])\n",
    "circ_2_ones = prog_2_ones.to_circ()\n",
    "\n",
    "from qat.qpus import PyLinalg\n",
    "from qat.plugins import ObservableSplitter\n",
    "\n",
    "qpu = ObservableSplitter() | PyLinalg()\n",
    "job = circ_2_ones.to_job(\"OBS\", observable=one_count, nbshots=30)\n",
    "print(\"Number of ones:\", qpu.submit(job).value)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Now with a less obvious circuit:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "prog = Program()\n",
    "qbits = prog.qalloc(5)\n",
    "for i, qb in enumerate(qbits):\n",
    "    prog.apply(RX(0.324 * i), qb)\n",
    "circ = prog.to_circ()\n",
    "job = circ.to_job(\"OBS\", observable=one_count, nbshots=30)\n",
    "print(\"Number of ones:\", qpu.submit(job).value)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Of course, we can reduce the deviation of this result by increasing the number of samples:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "job = circ.to_job(\"OBS\", observable=one_count, nbshots=1000)\n",
    "print(\"Number of ones:\", qpu.submit(job).value)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Or even compute the exact value of the observable using an \"infinite\" number of shots"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "job = circ.to_job(\"OBS\", observable=one_count)\n",
    "print(\"Exact number of ones:\", qpu.submit(job).value)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
  }
 ],
 "metadata": {
  "authors": [
   "Simon Martiel"
  ],
  "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": "3.6.8"
  }
 },
 "nbformat": 4,
 "nbformat_minor": 2
}