{
 "cells": [
  {
   "cell_type": "markdown",
   "id": "5c38153e",
   "metadata": {},
   "source": [
    "# Introduction to quantum photonics"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "id": "e20f1af2",
   "metadata": {},
   "outputs": [],
   "source": [
    "import qoptkit                                   # QOPTKIT module\n",
    "import numpy as np                               # Numpy\n",
    "from math  import acos,sqrt,pi                   # Mathematical module\n",
    "from qiskit import QuantumCircuit                # QISKIT Circuits \n",
    "from qiskit_aer import Aer                       # QISKIT Aer"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "400ec6a7",
   "metadata": {},
   "source": [
    "## 1 Basic concepts"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "567c03d3",
   "metadata": {},
   "source": [
    "### 1.1 Circuit and state"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "d5f6d837",
   "metadata": {},
   "source": [
    "<p style='text-align: justify'> An optical quantum computer is based on an ensemble of indistinguishable photons traveling through a set of optical gates. The kind of optical gates used and their interconnections define the circuit. The ensemble of photons define the state of the quantum operation. In a superconductor qubit based hardware, the active part of the computation is performed by the qubits. However, in an optical quantum computer the active part of the computation is performed by photons traveling trough the circuit. Therefore a quantum optical computer will consist of a set of emitters, an optical circuit and a set of detectors. Those define the input state, the transformation rules of the circuit and the output state respectively.  </p>\n",
    "    \n",
    "<p style='text-align: justifty'> Photons may arrive to the optical circuit by different channels or modes. For example, we can see next a simple circuit made of a single balanced beamsplitter with two channels labeled as <i>\"0\"</i> and <i>\"1\"</i>. </p>"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "id": "d344f067",
   "metadata": {},
   "outputs": [
    {
     "data": {
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+LyVJHv7dD1JXX9fQt3DJrHTo0CljR57daL1xx5yb+tTnlaWzD3zxAAAAALRrAnc4yC1b/VJWb1iUvr2G5ZhhE/boH33U6enZvW9WbViUZWsWHNBa6uvrc8fDf50dO2tzxfl/n5I2+AqpWPFcNletSaeOXTL+uAv36D959AXp1LFzNm1dlcUr5je0b63emJ7d+jbaZiZJepUOeKt/w4EtHAAAAIB2z5YycJCrWLkrNB4xuKzJ/pKSkhw1cFxeXvJUFq+cn6MGjj1gtTz5/B15ZensXHTGDRnS77is27Ss2XN/8/sfZs3GxSlJSfr2GpoTjn5fhg8a1+Ialq5+KUkytP/odO7UdY/+Lp26ZUi/0Vm25qUsXfNSwz9S9Oh2RNZvXp66+rpGofuWqrVv9fdtcS0AAAAAsDuBOxzk1m5anCRZse61/OShLzc5Zt3mpUmSNRsXN2r/7owr8tqbc1t8zJuufTT9+xzVqG3NxsX5xVPfyvBBZfnAe/9bi9e854m/b/T+3qe/nbFHn50rz/+HRtvkvJt1m3eF/H17DXvHMX17Dc2yNS9l/Vs/lyQ5fsQZqVg5Pwsqnsy4Y85paC9/49GUpCRjhk9qdg0AAAAA0BSBOxzkqmsqkySrNyzK6g2L3mXslkbvx448O/16H9niY3btUtrofV19XW6feWN27qzNled/Ix07NP+r473HX5QJY6blqIFj06t0QDZsWZGXlzyV+2d/LwsWP5F/vuea/OXldzd5t3pTarZvTZJ06dz9nevvvKv+t392STJlwrWZu+A/87PHvpa+vf41Q/ofl/JFv8mT8+/I+97zpxnU95hmnxMAAAAANEXgDoeIiSd8JFdf+O0m+2598IY88/K9e7SfP/HThRz7sed+nEXLn8m0Sddn2IDjWzT3E1O/0+j9oL4jM6jvyJw48px8846PZPnahXnq+X/PlFOuLaTWd9Kje9988U/uyi9nfSffv/vKbN9RkyN6Ds60yddnyimfPKDHBgAAAODwIHCHg1y3rj2TJNt3bHvHMTXbq94a26vw469avyj3Pf2dHDnghJw/8TOFrTugz/CcPu6jeey5H6f8jUebHbh37dIjSbK9di8/j9q3fx49G7X37TU0V1/wrX2sGAAAAAD2TuAOB7kBfY5OkmyqXPWOYzZsWZ4kGXjE0Y3aZ/7uB1m1/vUWH/Pis29Mz+79kiQvVTye2p012b5jW/7p59c0GrdjZ02SpK5+Z74744okyQcnfjonjjy7WccZ0u/YJMnGypXNrq3/W1vkvH3OTdmwZUWSpF/vP+xD/+zC+zLj0a+945y/uuIX6dtraLPrAAAAAID/SuAOB7mRQ8cnSZavezU763bssX/69tptWbn+1V1jh4xv1Leg4ol9emjqtEnXNwTub1uzcfEeD2Xd3dvHOf3ES5p9nK3bNiZJunbu0ew5wweNS5KsWPdqanfU7LH3+/Yd1Q0/j+EDxzW0n3r8h3Pq8R9u9nEAAAAAoKUE7nCQO2rg2AzuOyqrNizK8689nAljpjbqf3bhfdmxszaD+47KkQNPaNT3+Uvv2O/jTznl2nfc7mXdpmW56UdT0qGkY777+ZdbtG5dfV2ee/X+JMnRQ97T7Hkjh05I79KB2Vy1JvNeezATT/hIo/75rz6UHTtr06fH4Bw99OQW1QQAAAAA+6NDWxcAvLsPTf5CkuTux/93Vm+oaGhfuf613Pv0rj3JP3TGF1q/sHfxu5d/kVXrF+3RvqVqXW594C+ybM2CdOzQOeeMv2qPMbc99KX87a0X5PF5P2nU3qGkQ85773VJknuf+lbWblra0Ld209KGn8cHJ346HUp8xQEAAADQetzhDoeACWOm5pzlV+fxebflG7d/OMcffWZSX5+XlzydHTu359wJ12TC6KnvvlAre+6VB3Pbg3+ZgUeMzJB+x6Vr5+5Zv2VF3lyzIDW1W9OlU/dcef4/ZEi/4/aYu2Hz8qzesChbt23Yo++c8VfntTd/l+dffzh//5MP5fgRZyRJFi6Zle07tmX86Atz1slXHPDzAwAAAIDdCdzhEPGxc7+SY4ZOyOPzfpLXlv02STJi8Ek5++SrcurxH2rj6pp2+okXp1uXHlm2ZkHeWPFsqmq2pEunrhl4xIiMGX5Gzjr5igzoM7zF63bo0DGf+vA/ZdYLd2X2izOycOnsJMmQ/qMzedzHcuZJl6ekpKTo0wEAAACAvSqpr6+vb+si4FAwefLkzJkzp1HbyCHjc8PlM9qoIg42N995aSpWzmvUNmnSpMyePbttCgIAAACgVdngGAAAAAAACiBwBwAAAACAAgjcAQAAAACgAAJ3AAAAAAAogMAdAAAAAAAKIHAHAAAAAIACdGrrAoBDy+ata/Pg3H/Oi288ls1bV6V719459siJOX/iZzJ80Lh9WnPHzu159Pc/yjML78vajYvTsWPnHDnghJw9/spMGD21yTkr17+e+2Z9J68u/W3q6nekf+/hef+pn8ppY/94P84OAAAAAPadwB1ottUb3sgtMz6eLVXrMqDP8Jw06oNZt3lZ5r36YJ5//df55LTv5uTjzm/Rmttrt+Wffn5N3ljx+3Tv2jtjR56Vmu1VeWXZnLz25ty8/5RP5eKzb2w0Z83GxfnOXX+SAX1G5K+u+EX69BiUOS/9PHfMvDGV29bn/ad8ssjTBgAAAIBmEbgDzVJfX58f3f+FbKlal4lj/zhXfvAb6dChY5Lk6RfuzJ2PfCU/eejLOWbohPTuMbDZ6/7y6ZvzxorfZ9iA4/O5j96Wnt37JUmWrCrP9352ZX7z+/+X0UedlrJR72+Yc/+c76W6ZkuuuuCb6df7yCTJmSddloVLZuVXs7+byeMuTfeuvQo8ewAAAAB4d/ZwB5rlpYrHs2zNS+netXcum3JTQ9ieJGeedHnGDD8jNbVb89hztzZ7zarqTXnqhX9Pklz2/r9pCNuTZMTgspz33uuSJA/N/ddG8xYumZV+vY/M0P6jG7WPO+bcbK+tSsWKeS08OwAAAADYfwJ3OERsqlyVOx/5Sqb/8s+TJE/Mvz1f/X/n5IZ/ek/+9T//W5avfeWAHn/+6w8nSU4a9YF07dJjj/73nvDht8bNbPaaL1Y8lh07a9O317CMGnZqE2telCSpWDkvmypXNbRvrd6YXqUD9hjf+622yur1za4BAAAAAIoicIdDwOKVz+fvb78oT79wZ+rr6/Lmmpfzs0e/luqaLRk+eFxeqng8N991ad5Y/twBq2HZ6peS7LrzvCkjBp2UJFmzoSI1tVXNXHPBXtcc0GdESrsdsWvsmgUN7T26HZHKbev2GL+5am2SpGe3fnv0AQAAAMCBZg93OMjV1e3Mjx/4YrZWb8j40Rfm2qm35KkXfpr61Oe0Ey/Ox879Su5+7Ot5bN6t+fGDf5GvXvNwOnb4w6/2d2dckdfenNvi49507aPp3+eohvfrNi9LkvTtNazJ8X17DU2S1Kc+6ze/ucd2L01Zt3npXtdMkiN6DklV9caG4yfJ8SPOyLMv/zIr1r3a6DgvvvFounTqnpFDx7/rsQEAAACgaAJ3OMgtWPxU1m5aki6duudP3to7vXLbhiRJadc+SZJpkz+fuQv+M+s3L8vCJbNy4sizG+aPHXl2w4NFW6Jrl9JG72u2b02SdOncvenxnf8wvrqmslnHeHvNru+w5u7r7r7m1NM/lxcXPZrbZ96YT33o++nTY2DmvPTzzHvtofzRmX/pgakAAAAAtAmBOxzkKlbu2iZm7Miz0qu0/67G+vpGY7p37ZXRw0/P/NdmpmLl/EaB+/kTP91qtbaWQX1H5ouX3ZX7Zv1jvnHHRamv35l+vYfnT8/7u0wa99G2Lg8AAACAw5TAHQ5yW7bu2pd8QJ8Rex3Xr9euu9grtx2YB4Z27dIjVdUbs712W5P9u+/b3q1rz2avuWtu02vuvu5/XXNo/9G57qJ/adZxAAAAAKA1CNzhIFfSoWOSpK6+bq/jOnbsnCTpUNL4Wcgzf/eDrFr/eouPe/HZN6Zn9z88fLR/7yNTVb0xG7Ysb3L8hi0rdtWbkvTby57su+vX+6i35ja9ZpJsrFz51tg/bIvz62f+b379zL+945xvfKble9YDAAAAwP4SuMNB7oieQ5Ik63d7aGhT3g6m33546dsWVDyxTw9NnTbp+kaB+1GDxmXp6hezZFV5k+OXrH4hSTKw78iGO9ffzfBBJ+6a+w5rrt20JFXVG3eNHXhiQ/t57/3vOe+9/71ZxwAAAACA1iJwh4PcmOGTkux6eGrltvWNQvC31e6oycuLn3pr/ORGfZ+/9I5C6jj52A9mdvl/5IVFj6SmtqrRQ1KT5JmX73tr3PnNXnPcyHPTqWPnbNiyPIuWP5tRw079L2v+Mkkycsj49Ok5eD/PAAAAAAAOrA7vPgRoS8cMnZBRw07N9tqq3PPEN1JXt7NRf+2Omvz7w3+dym3rc/yIMzJ80LgDUseJI8/JUQNPzLaazfmP39zUqI6nX7gzryydla6de+TcCZ/YY+737746f3vrBZn/2sxG7aXd+uR9J/1pkuSu39yUrds2NPQtXf1ifv3M9CTJBaf92YE4JQAAAAAolDvc4RBw9QXfznd/dkXmLrgnFSvnpWOHXfu1z3vtwcx56e5s2LI8/XsflSs++I0DVkNJSUmumfqPuWXGxzN3wT1ZtPyZjBj8nqzbtDSLVz2fDh065aoLvpnePQbuMXftxiVZv+XNbKvZskffRWfekMWrns8bK57L1378wYwZPik1tdvyytLZ2VlXmymnfDJlo95/wM4LAAAAAIoicIdDQP8+R+VLH/95Hpr7L3n+9YezpnJxkmTl+tfTp8egnDP+qlx4+meb3G6mSIP7jcpfX3lfHpz7L3lx0aN5/vWZ6dalV04+7vxccNqf79Pd9V06d8/1H7s9v/n9j/LMy/fmxYrH06lD5xwzdHzOPvmqTBgz9QCcCQAAAAAUr6S+vr6+rYuAQ8HkyZMzZ86cRm0jh4zPDZfPaPVa7p/9vTzw2+9n6umfy7TJ17f68WnazXdemoqV8xq1TZo0KbNnz26bggAAAABoVfZwBwAAAACAAgjcAQAAAACgAAJ3AAAAAAAogIemwiFo2uTr7d0OAAAAAAcZd7gDAAAAAEABBO4AAAAAAFAAgTsAAAAAABRgnwL3qqqqPPDAA/n617+eSy65JEcffXRKSkpSUlKSm266qeASAYD2zrUFAAAA7cE+PTR17ty5mTZtWtG1AACHKdcWAAAAtAf7FLgnSd++fXPKKac0/PniF7+YlStXFlkbAHAYcW0BAADAoW6fAvezzjor69evb9R24403FlIQAHD4cW0BAABAe7BPe7h37Nix6DoAgMOYawsAAADag30K3AEAAAAAgMYE7gAAAAAAUACBOwAAAAAAFGCfHpp6IF133XUpLy9v6zJohkWLFqWqqqqty2iW6urq1NXVNWrr0qVLxo8f3+w1mvpcrt20ZH9Lox1p6vNQXl6eyZMnN3uNpn6vSktLM2rUqP2uD1pDWVlZpk+f3tZlAAAAQJs46AL38vLyzJkzp63L4DBQXV2935+12h3VBVVDe9DU56GysnK/P2eVlZVZvXr1fq0BAAAAwIFnSxkAAAAAACiAwB0AAAAAAAogcAcAAAAAgAIcdHu4l5WVtXUJNNPh+NDUysrKRm2dO3Urojzaic6duqWmtvHvRM+ePVv0veahqRzq/D0OAADA4eygC9ynT5/e1iVAkyZPnrzHwy8H9BnRRtVwMBrQZ0Qqt61v1FZWVpbZs2e3UUUAAAAAtKZ9Dtw3bNiQnTt3Nrx/++7hqqqqrF27tqG9W7du6dmz536UCAAcDlxbAAAAcKjb5z3cJ0yYkIEDBzb8Wbp0aZLkW9/6VqP2z372s4UVCwC0X64tAAAAONR5aCoAAAAAABRgn7eUqaioKLAMAOBw59oCAACAQ5073AEAAAAAoAACdwAAAAAAKIDAHQAAAAAACiBwBwAAAACAAuzzQ1OBZPm6V3LznZe2dRkcJJave6WtSwAAAACgDQncYT9sr61Kxcp5bV0GAAAAAHAQsKUMAAAAAAAUQOAOAAAAAAAFELgDAAAAAEAB7OEOzVRWVtbWJXAI8rkBAAAAOHyU1NfX17d1EQAAAAAAcKizpQwAAAAAABRA4A4AAAAAAAUQuAMAAAAAQAEE7gAAAAAAUACBOwAAAAAAFEDgDgAAAAAABRC4AwAAAABAAQTuAAAAAABQAIE7AAAAAAAUQOAOAAAAAAAFELgDAAAAAEABBO4AAAAAAFAAgTsAAAAAABRA4A4AAAAAAAUQuAMAAAAAQAEE7gAAAAAAUACBOwAAAAAAFEDgDgAAAAAABRC4AwAAAABAAQTuAAAAAABQAIE7AAAAAAAUQOAOAAAAAAAFELgDAAAAAEABBO4AAAAAAFAAgTsAAAAAABRA4A4AAAAAAAUQuAMAAAAAQAEE7gAAAAAAUACBOwAAAAAAFEDgDgAAAAAABRC4AwAAAABAAQTuAAAAAABQAIE7AAAAAAAUQOAOAAAAAAAFELgDAAAAAEABBO4AAAAAAFAAgTsAAAAAABRA4A4AAAAAAAUQuAMAAAAAQAEE7gAAAAAAUACBOwAAAAAAFEDgDgAAAAAABRC4AwAAAABAAQTuAAAAAABQAIE7AAAAAAAUQOAOAAAAAAAFELgDAAAAAEABBO4AAAAAAFAAgTsAAAAAABRA4A4AAAAAAAUQuAMAAAAAQAEE7gAAAAAAUACBOwAAAAAAFEDgDgAAAAAABRC4AwAAAABAAQTuAAAAAABQAIE7AAAAAAAUQOAOAAAAAAAFELgDAAAAAEABBO4AAAAAAFAAgTsAAAAAABRA4A4AAAAAAAUQuAMAAAAAQAEE7gAAAAAAUACBOwAAAAAAFEDgDgAAAAAABRC4AwAAAABAAQTuAAAAAABQAIE7AAAAAAAUQOAOAAAAAAAF+P/TeVQ7UjH8UQAAAABJRU5ErkJggg==",
      "text/plain": [
       "<Figure size 1900x320 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "# Create circuit\n",
    "hom = qoptkit.qocircuit(2);\n",
    "# Build circuit\n",
    "hom.bs(0,1,45.0,0.0)\n",
    "# Plot circuit\n",
    "hom.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "ff086211",
   "metadata": {},
   "source": [
    "A beamsplitter can be represented by the transformation between input and output bosonic operators:\n",
    "\n",
    "$$ \\begin{equation}\n",
    "\t\\begin{matrix}\n",
    "\t\t\\hat{a}_0^\\dagger \\rightarrow & \\cos(\\theta)\\hat{a}_0^\\dagger + e^{-i \\phi}\\sin(\\theta) \\hat{a}_1^\\dagger  \\\\\n",
    "\t\t\\hat{a}_1^\\dagger \\rightarrow & -e^{i \\phi} \\sin(\\theta)\\hat{a}_0^\\dagger + \\cos(\\theta) \\hat{a}_1^\\dagger.  \n",
    "\t\\end{matrix}\n",
    "\\end{equation} \n",
    "$$"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "7420b823",
   "metadata": {},
   "source": [
    "<p style='text-align: justify'> For the moment, we can think of the channels in an optical circuit as being equivalent to qubits in a computer based on superconductor qubits. We will refine this idea later on. Note that photons are bosonic states therefore states of an optical quantum machine may have more than one photon in each channel. For example we can define a state with two photons in the first channel and zero photons in the second one as follows: </p>"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "id": "d93f8a64",
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "\u001b[36m | 2, 0 >\u001b[0m:  1.00000000 + 0.00000000 j\n",
      "\n"
     ]
    }
   ],
   "source": [
    "# Define a |2,0> state\n",
    "inputst = qoptkit.state(2,st=[[1.0,[2,0]]])\n",
    "# Print the state\n",
    "inputst.prnt_state()"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "a3d60fd1",
   "metadata": {},
   "source": [
    "### 1.2 Simulation"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "4b4ce7bc",
   "metadata": {},
   "source": [
    "<p style='text-align: justify'> By convention we consider that photons travel trough a circuit from the left to the right direction. We can use the previously defined state as an input state of the beamsplitter circuit and calculate the corresponding output state. </p>"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "id": "48852a10",
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "\u001b[36m | 2, 0 >\u001b[0m:  0.50000000 + 0.00000000 j\n",
      "\u001b[36m | 1, 1 >\u001b[0m:  0.70710678 + 0.00000000 j\n",
      "\u001b[36m | 0, 2 >\u001b[0m:  0.50000000 + 0.00000000 j\n",
      "\n"
     ]
    }
   ],
   "source": [
    "# Create a simulator\n",
    "sim=qoptkit.simulator()\n",
    "# Calculate the output\n",
    "outputst=sim.run(inputst,hom)\n",
    "# Print output state\n",
    "outputst.prnt_state()"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "985934a9",
   "metadata": {},
   "source": [
    "<p style='text-align: justify'> In this particular case we can see how each photon has a 1/2 of probability to be detected in each of the channels because the beamsplitter is balanced. Therefore, this circuit implements the transformation, </p>\n",
    "\n",
    "$$ \n",
    "\\begin{equation}\n",
    "\\begin{split}\n",
    "|2, 0 \\rangle &=\\frac{\\bigl(\\hat{a}_{0}^{\\dagger}\\bigr)^{2}} {\\sqrt{2!}}\\frac{\\bigl(\\hat{a}_{1}^{\\dagger}\\bigr)^{0}}{\\sqrt{0!}}|0, 0 \\rangle \n",
    "\\rightarrow \\frac{1}{\\sqrt{2}} \\Bigl( \\hat{a}_{0}^{\\dagger}\\frac{1}{\\sqrt{2}}+\\hat{a}_{1}^{\\dagger}\\frac{1}{\\sqrt{2}}\\Bigr)^{2} |0, 0 \\rangle\n",
    "= \\frac{1}{2} |2, 0 \\rangle + \\sqrt2  |1, 1 \\rangle + \\frac{1}{2} |0, 2 \\rangle\n",
    "\\end{split}\n",
    "\\end{equation}\n",
    "$$"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "a464aa36",
   "metadata": {},
   "source": [
    "### 1.3 HOM Effect"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "ed0bf84e",
   "metadata": {},
   "source": [
    "<p style='text-align: justify'>  Of course, optical circuits may give rise to some interesting quantum effects. For example, the correlation at the output of indistinguishable photons that enter the beamsplitter by different channels. This effect is the well known  Hong-Ou-Mandel effect [1]. </p>"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "id": "c827b1d2",
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Input state:\n",
      "\u001b[36m | 1, 1 >\u001b[0m:  1.00000000 + 0.00000000 j\n",
      "\n",
      "Output state:\n",
      "\u001b[36m | 2, 0 >\u001b[0m: -0.70710678 + 0.00000000 j\n",
      "\u001b[36m | 0, 2 >\u001b[0m:  0.70710678 + 0.00000000 j\n",
      "\n"
     ]
    }
   ],
   "source": [
    "inputst = qoptkit.state(2,st=[[1.0,[1,1]]])\n",
    "outputst=sim.run(inputst,hom)\n",
    "print(\"Input state:\")\n",
    "inputst.prnt_state()\n",
    "print(\"Output state:\")\n",
    "outputst.prnt_state()"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "84bb117d",
   "metadata": {},
   "source": [
    "Note how in this case the outcome $|1,1\\rangle$ does not appear. The Hong-Ou-Mandel effect results in bunching both photons in one or the other output channel of the beamsplitter."
   ]
  },
  {
   "cell_type": "markdown",
   "id": "ce493012",
   "metadata": {},
   "source": [
    "## 2 Basic gates"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "a04019a6",
   "metadata": {},
   "source": [
    "### 2.1 Qubit encoding"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "0c0e41eb",
   "metadata": {},
   "source": [
    "<p style='text-align: justify'>  But how a qubit is defined in a quantum optical circuit? And how I can define some of the logical gates that I know? For example qubit rotations. There are various ways to encode a photonic state into a logical qubit state. In this tutorial we will use path encoding. But what is path encoding? </p>\n",
    "\n",
    "<p style='text-align: justify'>  We will use a pair of channels to define a qubit. For example, channels <i>\"0\"</i> and <i>\"1\"</i> may define qubit <i>\"0\"</i>. There will be only one photon by each pair of channels defining a qubit. The position of the photon will define the qubit value. If the photon is on the first channel this represents a logical <i>\"1\"</i>. However, if the photon is in the second channel this means a logical <i>\"0\"</i>.</p>\n",
    "\n",
    "Photonic state $\\rightarrow$ qubit:\n",
    "\n",
    "$|0,1\\rangle\\rightarrow |0 \\rangle$\n",
    "\n",
    "$|1,0\\rangle\\rightarrow |1 \\rangle$"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "af1f367e",
   "metadata": {},
   "source": [
    "<p style='text-align: justify'> We can define a map specifying the channels for each qubit. Each pair of channels define a qubit and differentt qubits are defined in order from 0 to n.</p>"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "id": "4219dd12",
   "metadata": {},
   "outputs": [],
   "source": [
    "qmap=[[0, 1], [2, 3]]"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "9cef4189",
   "metadata": {},
   "source": [
    "<p style='text-align: justify'> In this particular case, the map specifies that channels <i>\"0\"</i> and <i>\"1\"</i> define qubit <i>\"0\"</i> while channels <i>\"2\"</i> and <i>\"3\"</i> define qubit <i>\"1\"</i>. We can encode photonic states into qubits with this map, <p>"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "id": "0b8efeba",
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Photonic states:\n",
      "\n",
      "\u001b[36m | 0, 1, 0, 1 >\u001b[0m:  1.00000000 + 0.00000000 j\n",
      "\n",
      "\u001b[36m | 0, 1, 1, 0 >\u001b[0m:  1.00000000 + 0.00000000 j\n",
      "\n",
      "\u001b[36m | 1, 0, 0, 1 >\u001b[0m:  1.00000000 + 0.00000000 j\n",
      "\n",
      "\u001b[36m | 1, 0, 1, 0 >\u001b[0m:  1.00000000 + 0.00000000 j\n",
      "\n",
      "\n",
      "Equivalent logic states:\n",
      "\n",
      "\u001b[36m | 0, 0 >\u001b[0m:  1.00000000 + 0.00000000 j\n",
      "\n",
      "\u001b[36m | 0, 1 >\u001b[0m:  1.00000000 + 0.00000000 j\n",
      "\n",
      "\u001b[36m | 1, 0 >\u001b[0m:  1.00000000 + 0.00000000 j\n",
      "\n",
      "\u001b[36m | 1, 1 >\u001b[0m:  1.00000000 + 0.00000000 j\n",
      "\n"
     ]
    }
   ],
   "source": [
    "# Define states\n",
    "input0101 = qoptkit.state(4,st=[[1.0,[0,1,0,1]]]) #0101\n",
    "input0110 = qoptkit.state(4,st=[[1.0,[0,1,1,0]]]) #0110\n",
    "input1001 = qoptkit.state(4,st=[[1.0,[1,0,0,1]]]) #1001\n",
    "input1010 = qoptkit.state(4,st=[[1.0,[1,0,1,0]]]) #1010\n",
    "\n",
    "# Encode\n",
    "encoded00=input0101.encode(qmap,hom) #00\n",
    "encoded01=input0110.encode(qmap,hom) #01\n",
    "encoded10=input1001.encode(qmap,hom) #10\n",
    "encoded11=input1010.encode(qmap,hom) #11\n",
    "\n",
    "# Print comparison\n",
    "print(\"Photonic states:\\n\")\n",
    "input0101.prnt_state()\n",
    "input0110.prnt_state()\n",
    "input1001.prnt_state()\n",
    "input1010.prnt_state()\n",
    "print(\"\")\n",
    "print(\"Equivalent logic states:\\n\")\n",
    "encoded00.prnt_state()\n",
    "encoded01.prnt_state()\n",
    "encoded10.prnt_state()\n",
    "encoded11.prnt_state()"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "3c6e50c0",
   "metadata": {},
   "source": [
    "It is also possible to decode them back to photonic states,"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "id": "683bae23",
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Photonic states:\n",
      "\n",
      "\u001b[36m | 0, 1, 0, 1 >\u001b[0m:  1.00000000 + 0.00000000 j\n",
      "\n",
      "\u001b[36m | 0, 1, 1, 0 >\u001b[0m:  1.00000000 + 0.00000000 j\n",
      "\n",
      "\u001b[36m | 1, 0, 0, 1 >\u001b[0m:  1.00000000 + 0.00000000 j\n",
      "\n",
      "\u001b[36m | 1, 0, 1, 0 >\u001b[0m:  1.00000000 + 0.00000000 j\n",
      "\n"
     ]
    }
   ],
   "source": [
    "# Decode\n",
    "decoded00=encoded00.decode(qmap,[],hom) #00\n",
    "decoded01=encoded01.decode(qmap,[],hom) #01\n",
    "decoded10=encoded10.decode(qmap,[],hom) #10\n",
    "decoded11=encoded11.decode(qmap,[],hom) #11\n",
    "print(\"Photonic states:\\n\")\n",
    "decoded00.prnt_state()\n",
    "decoded01.prnt_state()\n",
    "decoded10.prnt_state()\n",
    "decoded11.prnt_state()"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "b8bcf6d5",
   "metadata": {},
   "source": [
    "### 2.2 Rotations"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "17895cdc",
   "metadata": {},
   "source": [
    "<p style='text-align: justify'>  Now, we can define circuits and states, and we know how to interpret those states as qubits therefore we are in position to define some basic gates. </p>"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "9e96d551",
   "metadata": {},
   "source": [
    "### 2.2.1 Rx"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "198fedd3",
   "metadata": {},
   "source": [
    "For example, we can define a $Rx(\\pi/2)$ gate using optical elements,"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "id": "e73a0a41",
   "metadata": {},
   "outputs": [
    {
     "data": {
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",
      "text/plain": [
       "<Figure size 1900x320 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "qmap=[[0, 1]]\n",
    "rx = qoptkit.qocircuit(2);\n",
    "rx.bs(0,1,45.0,180.0/2.0)\n",
    "rx.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "c6bd44a9",
   "metadata": {},
   "source": [
    "We can test this gate for an input qubit initialized to zero,"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "id": "d856a454",
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "\u001b[36m | 1 >\u001b[0m: -0.00000000 - 0.70710678 j\n",
      "\u001b[36m | 0 >\u001b[0m:  0.70710678 + 0.00000000 j\n",
      "\n"
     ]
    }
   ],
   "source": [
    "qubit = qoptkit.state(1,st=[[1.0,[0]]])\n",
    "input=qubit.decode(qmap,[],rx) \n",
    "output=sim.run(input,rx)\n",
    "result=output.encode(qmap,rx)\n",
    "result.prnt_state()"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "b54fd6f5",
   "metadata": {},
   "source": [
    "And finally, we can also check the correctness of this result with the equivalent simulation of this gate in QISKIT,"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "id": "a7b74f72",
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "QISKIT State vector:\n",
      " | 1 >:    0.0 - 0.70710678 j\n",
      " | 0 >:    0.70710678 + 0.0 j\n"
     ]
    }
   ],
   "source": [
    "qiskitsim = Aer.get_backend('aer_simulator') \n",
    "qc = QuantumCircuit(1)\n",
    "qc.rx(np.pi/2,0)\n",
    "qc.save_statevector() \n",
    "result = qiskitsim.run(qc).result()\n",
    "state = result.get_statevector(decimals=8)\n",
    "print(\"QISKIT State vector:\")\n",
    "print(\" | 1 >: \", ' '  if np.real(state[1]) >= 0.0 else '', np.real(state[1]), '+'  if np.imag(state[1]) >= 0.0 else '-', abs(np.imag(state[1])),'j' )\n",
    "print(\" | 0 >: \", ' '  if np.real(state[0]) >= 0.0 else '', np.real(state[0]), '+'  if np.imag(state[0]) >= 0.0 else '-', abs(np.imag(state[0])),'j' )"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "36bd35a8",
   "metadata": {},
   "source": [
    "### 2.2.2 Ry"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "0448ca55",
   "metadata": {},
   "source": [
    "We can do the same for the $Ry(\\pi/2)$ gate,"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "id": "b63b7bbd",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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",
      "text/plain": [
       "<Figure size 1900x320 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "qmap=[[0, 1]]\n",
    "ry = qoptkit.qocircuit(2);\n",
    "ry.bs(0,1,-45.0,0)\n",
    "ry.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "a15f90a5",
   "metadata": {},
   "source": [
    "Again test the gate with an input qubit initialized to zero,"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "id": "1bfae457",
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "\u001b[36m | 1 >\u001b[0m:  0.70710678 + 0.00000000 j\n",
      "\u001b[36m | 0 >\u001b[0m:  0.70710678 + 0.00000000 j\n",
      "\n"
     ]
    }
   ],
   "source": [
    "qubit = qoptkit.state(1,st=[[1.0,[0]]])\n",
    "input=qubit.decode(qmap,[],ry) \n",
    "output=sim.run(input,ry)\n",
    "result=output.encode(qmap,ry)\n",
    "result.prnt_state()"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "3e15b725",
   "metadata": {},
   "source": [
    "And check the result with the equivalent QISKIT simulation,"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "id": "937af3e0",
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "QISKIT State vector:\n",
      " | 1 >:    0.70710678 + 0.0 j\n",
      " | 0 >:    0.70710678 + 0.0 j\n"
     ]
    }
   ],
   "source": [
    "qiskitsim = Aer.get_backend('aer_simulator') \n",
    "qc = QuantumCircuit(1)\n",
    "qc.ry(np.pi/2,0)\n",
    "qc.save_statevector() \n",
    "result = qiskitsim.run(qc).result()\n",
    "state = result.get_statevector(decimals=8)\n",
    "print(\"QISKIT State vector:\")\n",
    "print(\" | 1 >: \", ' '  if np.real(state[1]) >= 0.0 else '', np.real(state[1]), '+'  if np.imag(state[1]) >= 0.0 else '-', abs(np.imag(state[1])),'j' )\n",
    "print(\" | 0 >: \", ' '  if np.real(state[0]) >= 0.0 else '', np.real(state[0]), '+'  if np.imag(state[0]) >= 0.0 else '-', abs(np.imag(state[0])),'j' )"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "bace000b",
   "metadata": {},
   "source": [
    "### 2.2.3 Rz"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "c4a2f059",
   "metadata": {},
   "source": [
    "Finally, we can also define a $Rz(\\pi/2)$ gate,"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 15,
   "id": "f0c06be7",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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",
      "text/plain": [
       "<Figure size 1900x320 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "qmap=[[0, 1]]\n",
    "rz = qoptkit.qocircuit(2);\n",
    "rz.ph(0,  45)\n",
    "rz.ph(1, -45)\n",
    "rz.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "943137b0",
   "metadata": {},
   "source": [
    "<p style='text-align: justify'>  In this case, instead of using a beamsplitter gate we have used two phase shifters. A phase shifter adds a phase $e^{i\\theta}$ to a state for each photon present in the channel in which it is defined. Once more, we test the gate for an input qubit initialized to zero. The reader may test those gates with other inputs or create rotation gates with different angles. </p>"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 16,
   "id": "7e4942b7",
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "\u001b[36m | 0 >\u001b[0m:  0.70710678 - 0.70710678 j\n",
      "\n"
     ]
    }
   ],
   "source": [
    "qubit = qoptkit.state(1,st=[[1.0,[0]]])\n",
    "input=qubit.decode(qmap,[],rz) \n",
    "output=sim.run(input,rz)\n",
    "result=output.encode(qmap,rz)\n",
    "result.prnt_state()"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "4110f3b3",
   "metadata": {},
   "source": [
    "Finally, we can check again the result with QISKIT."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 17,
   "id": "21024fa4",
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "QISKIT State vector:\n",
      " | 0 >:    0.70710678 - 0.70710678 j\n",
      " | 1 >:    0.0 + 0.0 j\n"
     ]
    }
   ],
   "source": [
    "qiskitsim = Aer.get_backend('aer_simulator') \n",
    "qc = QuantumCircuit(1)\n",
    "qc.rz(np.pi/2,0)\n",
    "qc.save_statevector() \n",
    "result = qiskitsim.run(qc).result()\n",
    "state = result.get_statevector(decimals=8)\n",
    "print(\"QISKIT State vector:\")\n",
    "print(\" | 0 >: \", ' '  if np.real(state[0]) >= 0.0 else '', np.real(state[0]), '+'  if np.imag(state[0]) >= 0.0 else '-', abs(np.imag(state[0])),'j' )\n",
    "print(\" | 1 >: \", ' '  if np.real(state[1]) >= 0.0 else '', np.real(state[1]), '+'  if np.imag(state[1]) >= 0.0 else '-', abs(np.imag(state[1])),'j' )\n"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "11c6e039",
   "metadata": {},
   "source": [
    "<p style='text-align: justify'> It is not very difficult to see how these operations can be generalized to any rotation. Therefore, we can implement with optical circuits any single qubit operation. But what about two qubits operations? </p>"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "43cdf05d",
   "metadata": {},
   "source": [
    "## 3 Post-selection"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "0020e06c",
   "metadata": {},
   "source": [
    "<p style='text-align: justify'> Unfortunately, a linear optical circuit by itself can not implement every possible quantum gate. Some non-linearity is needed to achieve this purpose. It has been demonstrated [2] that the required non-linearity can be introduced by a partial measurement of the circuit output called post-selection. The price to pay is that gates that make use of post-selection only work with a certain probability of success. Let's see an example. </p>"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "7ee530b9",
   "metadata": {},
   "source": [
    "### The $NS_{-1}$ gate"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "759f29d0",
   "metadata": {},
   "source": [
    "<p style='text-align: justify'>  For example, we can find in the literature [3] a gate that changes the sign of an input state if the number of photons is two. This is, we want a photonic gate that implements the operation,\n",
    "$$ |0\\rangle + |1\\rangle + |2\\rangle \\rightarrow |0\\rangle + |1\\rangle - |2\\rangle$$ </p>"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 18,
   "id": "269d907f",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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",
      "text/plain": [
       "<Figure size 1200x344 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "# Create circuit\n",
    "NSx = qoptkit.qocircuit(3);\n",
    "# Build circuit\n",
    "NSx.ph(0,180.0)\n",
    "NSx.bs(1,2,22.5,0.0)\n",
    "NSx.bs(0,1,65.5302,0.0)\n",
    "NSx.bs(1,2,-22.5,0.0)\n",
    "# Show circuit\n",
    "NSx.show(depth=8,sizexy=80,font=16)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "6933f569",
   "metadata": {},
   "source": [
    "<p style='text-align: justify'> This gate has three channels instead of one. It adds two auxiliary channels to perform the operation. We will assume, for the moment, that these two extra channels are initialized to 1 and 0 photons respectively and we will require from those channels to have the same values at the output. We will explain the reason below. Therefore,\n",
    "$$ |0,1,0 \\rangle + |1,1,0\\rangle + |2,1,0\\rangle \\rightarrow |0,1,0\\rangle + |1,1,0\\rangle - |2,1,0\\rangle$$ </p>\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 19,
   "id": "31d24211",
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "\u001b[36m | 0, 1, 0 >\u001b[0m:  1.00000000 + 0.00000000 j\n",
      "\u001b[36m | 1, 1, 0 >\u001b[0m:  1.00000000 + 0.00000000 j\n",
      "\u001b[36m | 2, 1, 0 >\u001b[0m:  1.00000000 + 0.00000000 j\n",
      "\n"
     ]
    }
   ],
   "source": [
    "# Create input state\n",
    "state=[ [1.0,[0,1,0]],\n",
    "        [1.0,[1,1,0]],\n",
    "        [1.0,[2,1,0]],\n",
    "      ]\n",
    "inputst = qoptkit.state(NSx.num_ch(),st=state,nph=3)\n",
    "inputst.prnt_state()"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "876677bd",
   "metadata": {},
   "source": [
    "<p style='text-align: justify'> We have left the state intentionally unnormalized as it will simplify an explanation below. Next, we can calculate the output of the circuit with QoptKIT. </p>"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 20,
   "id": "5e7e91a9",
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "\u001b[36m | 1, 0, 0 >\u001b[0m: -0.84089642 + 0.00000000 j\n",
      "\u001b[36m | 0, 1, 0 >\u001b[0m:  0.49999999 + 0.00000000 j\n",
      "\u001b[36m | 0, 0, 1 >\u001b[0m:  0.20710678 + 0.00000000 j\n",
      "\u001b[36m | 2, 0, 0 >\u001b[0m:  0.49258571 - 0.00000000 j\n",
      "\u001b[36m | 1, 1, 0 >\u001b[0m:  0.50000001 - 0.00000000 j\n",
      "\u001b[36m | 1, 0, 1 >\u001b[0m: -0.37867966 + 0.00000000 j\n",
      "\u001b[36m | 0, 2, 0 >\u001b[0m: -0.59460355 + 0.00000000 j\n",
      "\u001b[36m | 0, 0, 2 >\u001b[0m:  0.10201784 - 0.00000000 j\n",
      "\u001b[36m | 3, 0, 0 >\u001b[0m: -0.24989165 + 0.00000000 j\n",
      "\u001b[36m | 2, 1, 0 >\u001b[0m: -0.50000000 + 0.00000000 j\n",
      "\u001b[36m | 2, 0, 1 >\u001b[0m:  0.27817459 - 0.00000000 j\n",
      "\u001b[36m | 1, 2, 0 >\u001b[0m: -0.24629288 + 0.00000000 j\n",
      "\u001b[36m | 1, 1, 1 >\u001b[0m:  0.34831071 - 0.00000000 j\n",
      "\u001b[36m | 1, 0, 2 >\u001b[0m: -0.16177851 + 0.00000000 j\n",
      "\u001b[36m | 0, 3, 0 >\u001b[0m:  0.61237243 - 0.00000000 j\n",
      "\u001b[36m | 0, 2, 1 >\u001b[0m: -0.14644660 + 0.00000000 j\n",
      "\u001b[36m | 0, 1, 2 >\u001b[0m: -0.06066017 + 0.00000000 j\n",
      "\u001b[36m | 0, 0, 3 >\u001b[0m:  0.04351997 - 0.00000000 j\n",
      "\n"
     ]
    }
   ],
   "source": [
    "sim=qoptkit.simulator()\n",
    "outputst=sim.run(inputst,NSx)\n",
    "outputst.prnt_state()"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "0b0510c8",
   "metadata": {},
   "source": [
    "<p style='text-align: justify'> On a first glance the result looks quite different from what we were expecting. However, the intended result is obtained if after measuring channels <i>\"1\"</i> and  <i>\"2\"</i>  we require the detection of one photon only in channel <i>\"1\"</i>. If this condition is met, we conclude the gate has worked, otherwise we reject the result in channel <i>\"0\"</i> . Post-selected channels <i>\"1\"</i> and  <i>\"2\"</i> are erased from the result. </p>"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 21,
   "id": "84391ad0",
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "\u001b[36m | 0 >\u001b[0m:  0.49999999 + 0.00000000 j\n",
      "\u001b[36m | 1 >\u001b[0m:  0.50000001 - 0.00000000 j\n",
      "\u001b[36m | 2 >\u001b[0m: -0.50000000 + 0.00000000 j\n",
      "\n"
     ]
    }
   ],
   "source": [
    "finalst=outputst.post_selection([-1,1,0])\n",
    "finalst.prnt_state()"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "7ca6e958",
   "metadata": {},
   "source": [
    "<p style='text-align: justify'> Physically post-selection means that we only accept the quantum state defined by channel zero if after a partial measurement of the rest of the channels the result meets the pre-established conditions. The price to pay is that this condition may not be always met. This gate has a success probability of 1/4. This means that only 1 in 4 times on average the \"10\" condition will be measured in channels 2 and 3. Therefore only around 1 of 4 times this gate will work as intended. This can also be seen in the normalization of the end state that is 1/4. (This is the reason we didn't normalize the state before. It is easier to see the success probability this way). </p>"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "e3cf7bc3",
   "metadata": {},
   "source": [
    "### CZ gate"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "e23c1ff2",
   "metadata": {},
   "source": [
    "<p style='text-align: justify'>  Post-selection is one of the more difficult concepts in quantum computation with photonic circuits but once mastered the sky is the limit. We can use the $NS_{-1}$  gate to build a two qubit CZ gate as shown in [2]. This can be achieved with a eight channel circuit. Four channel for the qubit definitions and four extra ancilla channels for post-selection because this circuit uses two $NS_{-1}$ gates.</p>"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 22,
   "id": "759570d3",
   "metadata": {},
   "outputs": [],
   "source": [
    "qmap=[[0, 1], [2, 3]]\n",
    "csign = qoptkit.qocircuit(8)\n",
    "csign.bs(0,2,45.0,0.0)\n",
    "csign.gate([ 0, 4, 5 ],NSx,\"NSX\")\n",
    "csign.gate([ 2, 6, 7 ],NSx,\"NSX\")\n",
    "csign.bs(0,2,-45.0,0.0)  "
   ]
  },
  {
   "cell_type": "markdown",
   "id": "aaa8915e",
   "metadata": {},
   "source": [
    "<div>\n",
    "<img src=\"./CZ.png\" width=\"1200\"/>\n",
    "</div>\n",
    "<p style='text-align: center;'> <b>CZ asdescribed in ref. [2] </b></p>"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "dd85d504",
   "metadata": {},
   "source": [
    "In this case, we choose an input state with all the possibilities in superposition with equal probability."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 23,
   "id": "418d522f",
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "\u001b[36m | 0, 0 >\u001b[0m:  0.50000000 + 0.00000000 j\n",
      "\u001b[36m | 0, 1 >\u001b[0m:  0.50000000 + 0.00000000 j\n",
      "\u001b[36m | 1, 0 >\u001b[0m:  0.50000000 + 0.00000000 j\n",
      "\u001b[36m | 1, 1 >\u001b[0m:  0.50000000 + 0.00000000 j\n",
      "\n"
     ]
    }
   ],
   "source": [
    "qubit= qoptkit.state(2,st=[ [0.5,[0,0]], [0.5,[0,1]], [0.5,[1,0]], [0.5,[1,1]] ],nph=2)\n",
    "qubit.prnt_state()"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "b644d2c2",
   "metadata": {},
   "source": [
    "<p style='text-align: justify'> First, we decode the qubit state into its equivalent photonic state. Next, we obtain the output of the circuit and\n",
    "apply the post-selection conditions required by the $NS_{-1}$ gates. Finally, we encode the output into a qubit representation. </p>"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 24,
   "id": "fe3503bb",
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "\u001b[36m | 0, 0 >\u001b[0m:  0.49999999 + 0.00000000 j\n",
      "\u001b[36m | 0, 1 >\u001b[0m:  0.50000001 - 0.00000000 j\n",
      "\u001b[36m | 1, 0 >\u001b[0m:  0.50000001 - 0.00000000 j\n",
      "\u001b[36m | 1, 1 >\u001b[0m: -0.49999999 + 0.00000000 j\n",
      "\n"
     ]
    }
   ],
   "source": [
    "# Decode input into photonic state\n",
    "inputst=qubit.decode(qmap,[1,0,1,0],csign)\n",
    "# Create a simulator\n",
    "sim=qoptkit.simulator()\n",
    "# Calculate the output\n",
    "outputst=sim.run(inputst,csign)\n",
    "# Apply post-selection\n",
    "pselected=outputst.post_selection([-1,-1,-1,-1, 1,0,1,0])\n",
    "# Encode output photonic state into qubit state \n",
    "encoded=pselected.encode(qmap,csign)\n",
    "# Normalize\n",
    "encoded.normalize()\n",
    "# Print the resulting state\n",
    "encoded.prnt_state()"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "c7fc515f",
   "metadata": {},
   "source": [
    "<p style='text-align: justify'> We can see how a phase flip happens for the outcome $|1,1\\rangle$. We have normalized the output, otherwise we would see that the success probability of this gate is 1/16. Note that, in this particular case, the post-selected states only contain outcomes with 0 or 1 photons by channel. </p>"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 25,
   "id": "3c9d856a",
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "\u001b[36m | 0, 1, 0, 1 >\u001b[0m:  0.12500000 + 0.00000000 j\n",
      "\u001b[36m | 0, 1, 1, 0 >\u001b[0m:  0.12500000 - 0.00000000 j\n",
      "\u001b[36m | 1, 0, 0, 1 >\u001b[0m:  0.12500000 - 0.00000000 j\n",
      "\u001b[36m | 1, 0, 1, 0 >\u001b[0m: -0.12500000 + 0.00000000 j\n",
      "\n"
     ]
    }
   ],
   "source": [
    "pselected.prnt_state()"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "b6e0be96",
   "metadata": {},
   "source": [
    "<p style='text-align: justify'> Therefore, the encoding is straightforward. The encoding operation implemented in QOKIT will ignore outcomes with more than one photon by qubit. This may be used on purpose like a sort of an additional post-selection condition although it has some disadvantages. See our next example. </p>"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "8f07439a",
   "metadata": {},
   "source": [
    "### CNOT gate"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "9f36c112",
   "metadata": {},
   "source": [
    "Here we show a CNOT implementation as suggested in ref. [4]. "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 26,
   "id": "b887f392",
   "metadata": {},
   "outputs": [
    {
     "data": {
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      "text/plain": [
       "<Figure size 1740x456 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "qmap=[[1, 2], [3, 4]]\n",
    "cnot=qoptkit.qocircuit(6);\n",
    "cnot.bs(3,4, -45.0,0.0)\n",
    "cnot.barrier()\n",
    "cnot.bs(0,1,180*acos(1.0/sqrt(3.0))/pi,0.0)\n",
    "cnot.bs(2,3,180*acos(1.0/sqrt(3.0))/pi,0.0)\n",
    "cnot.bs(4,5,180*acos(1.0/sqrt(3.0))/pi,0.0)\n",
    "cnot.barrier()\n",
    "cnot.bs(3,4, -45.0,0.0)\n",
    "cnot.barrier()\n",
    "cnot.ph(1, 180)\n",
    "cnot.ph(3, 180)\n",
    "cnot.show(depth=15,sizexy=60,font=18)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "30337c04",
   "metadata": {},
   "source": [
    "Below it is shown how to perform the simulation for the case of a $|10\\rangle$ input state. We invite the reader to check the rest of possible inputs."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 27,
   "id": "fe675fff",
   "metadata": {},
   "outputs": [],
   "source": [
    "# Input state (qubit definition)\n",
    "qubit= qoptkit.state(2,st=[[1.0,[1,0]]])\n",
    "# Decode input into photonic state\n",
    "input=qubit.decode(qmap,[0,0],cnot)\n",
    "# Create a simulator\n",
    "sim=qoptkit.simulator()\n",
    "# Calculate the output\n",
    "output=sim.run(input,cnot)\n",
    "# Apply post-selection\n",
    "pselected=output.post_selection([0,-1,-1,-1,-1, 0])\n",
    "# Encode output photonic state into qubit state \n",
    "encoded=pselected.encode(qmap,cnot)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "e11393a2",
   "metadata": {},
   "source": [
    "Now, we can show the output. However, this time we will plot the probabilities instead of printing the states,"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 28,
   "id": "4a52d7c9",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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",
      "text/plain": [
       "<Figure size 400x400 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "encoded.show(sizex=4,sizey=4)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "1296ec88",
   "metadata": {},
   "source": [
    "<p style='text-align: justify'> Note that the normalization 1/9 correspond with the success probability of this gate. Here we have printed the outcome probabilities after the encoding of the photonic state into qubits. However, if we plot the outcome probabilities of the photonic state directly we can see that there are two possible different outcomes. A \"1010\" outcome with success probability 1/9 that corresponds with a 11 qubit configuration and a \"1100\" outcome that has no valid encoding. </p>"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 29,
   "id": "4d63c9c2",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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",
      "text/plain": [
       "<Figure size 400x400 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "pselected.show(sizex=4,sizey=4)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "78e2d758",
   "metadata": {},
   "source": [
    "<p style='text-align: justify'> We reject outcomes that can not be encoded. This can be understood as a sort of an additional post-selection condition. Of course, that means that this CNOT gate\n",
    "can work as a CNOT in an isolated way but can not be used to design large circuits like the CZ because these extra outcomes will play a role once multiple gates are interconnected.  </p>"
   ]
  },
  {
   "cell_type": "markdown",
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   "source": [
    "## Bibliography"
   ]
  },
  {
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   "source": [
    "**[1]** C. K. Hong, Z. Y. Ou and L. Mandel, **Phys. Rev. Lett. 59 (18): 2044–2046.** (1987) \n",
    "\n",
    "**[2]** E. Knill, R. Laflamme, G. J. Milburn, **Nature 409 46-52** (2001) \n",
    "\n",
    "**[3]** C. Myers et Al. **Quantum Computers, Algorithms and Chaos, vol. 162, IOS Press, pp. 45–93** (2006) \n",
    "\n",
    "**[4]** J L O'Brien, G J Pryde, A G White, T C Ralph, D Branning, **Nature 426:264** (2003)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "8e668366",
   "metadata": {},
   "source": [
    "**THIS CODE IS PART OF QoptKIT** <br>\n",
    "\n",
    "**Copyright:** <br>\n",
    "Copyright © 2023 National University of Ireland Maynooth, Maynooth University.\n",
    "\n",
    "Licensed under the Apache License, Version 2.0 (the \"License\"); you may not use this file except in compliance with the <a href=\"./LICENSE.TXT\">License</a>"
   ]
  }
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