content/ex1/ex1-fr.ipynb

{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Exercice 1 - Toffoli gate\n",
    "\n",
    "## Contexte historique\n",
    "\n",
    "Il y a quarante ans, un groupe hétéroclite de 50 penseurs s’est arrangé pour une photo sur la pelouse de la maison Endicott du MIT. Peu de gens à la conférence sur la physique de l'informatique, organisée conjointement par le MIT et IBM, pensaient faire l'histoire en 1981. C'était sans doute le berceau de la physique de l'informatique, et en particulier du domaine en plein essor de l'informatique quantique, en tant que sujet sérieux digne d'un manuel ou d'un cours universitaire.\n",
    "\n",
    "C'est lors de cette conférence que Feynman a prononcé sa citation désormais célèbre: «La nature n'est pas classique, bon sang, et si vous voulez faire une simulation de la nature, vous feriez mieux de la faire de la mécanique quantique, et par golly, c'est un merveilleux problème, car cela n'a pas l'air si facile. »[1] Plus tôt ce mois-ci, nous avons célébré le 40e anniversaire de cette importante conférence. Vous pouvez en savoir plus [ici](https://youtu.be/GR6ANm6Z0yk).\n",
    "\n",
    "![](resources/conference-photo.jpeg)\n",
    "\n",
    "L'un des thèmes abordés lors de la conférence était l'informatique réversible, à laquelle Tommaso Toffoli et Edward Fredkin du MIT ont réfléchi ces dernières années. [2-3] Toffoli a proposé une version réversible de la porte AND / NAND (qui sont maintenant appelées Toffoli gate, ou porte contrôlée-contrôlée-NOT). Parce que la porte NAND est universelle dans le calcul classique, Toffoli gate est une porte logique réversible universelle. L'informatique quantique est une forme particulière de calcul réversible; toute porte réversible peut être mise en œuvre sur un ordinateur quantique, et par conséquent Toffoli gate est également une porte à logique quantique. Cependant, Toffoli gate n'est pas à elle seule une porte universelle pour l'informatique quantique.\n",
    "\n",
    "Dans cet exercice, nous explorerons Toffoli gate et universal gate  pour les ordinateurs quantiques.\n",
    "\n",
    "### References\n",
    "1. Feynman, Richard P. \"Simulating physics with computers.\" Int. J. Theor. Phys 21.6/7 (1982).\n",
    "1. Toffoli, Tommaso. \"Reversible computing.\" International colloquium on automata, languages, and programming. Springer, Berlin, Heidelberg, 1980.\n",
    "1. Fredkin, Edward, and Tommaso Toffoli. \"Conservative logic.\" International Journal of theoretical physics 21.3 (1982): 219-253."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Classical logic gates\n",
    "Dans le calcul classique, un modèle souvent utilisé est la logique booléenne ou les portes logiques classiques. Ces portes représentent des fonctions booléennes, des fonctions avec uniquement une entrée et une sortie binaires (0,1). Un aspect intéressant de la logique booléenne est que toutes les fonctions binaires possibles peuvent être formées en utilisant uniquement une combinaison d'un petit nombre de portes logiques différentes. Ces ensembles sont appelés ensembles fonctionnellement complets. Un tel ensemble célèbre est ET et PAS. Ces deux portes suffisent à exprimer toutes les fonctions possibles. La même chose est vraie pour OU et NON. Il existe des ensembles plus petits, tels que NAND et NOR qui seuls sont universels, néanmoins, les fonctions AND, NOT et OR sont souvent considérées comme les blocs de base du calcul classique.\n",
    "\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "\n",
    "<div class=\"alert alert-block alert-success\">\n",
    "\n",
    "**L'ambition**\n",
    "\n",
    "Construisez une porte Toffoli à l'aide du jeu de portes de base (portes CX, RZ, SX et X) des systèmes IBM Quantum.\n",
    "</div>\n",
    "\n",
    "<div class=\"alert alert-block alert-danger\">\n",
    "\n",
    "Cet exercice vise à vous apprendre les concepts de base des portes quantiques et comment construire des circuits quantiques\n",
    "\n",
    "1. visuellement à l'aide du widget Circuit Composer\n",
    "2. par programme en utilisant Qiskit.\n",
    "\n",
    "Si vous connaissez déjà les portes quantiques et Qiskit.\n",
    "Vous pouvez accéder directement au <a href= #problem> problèm </a>\n",
    "</div>\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {},
   "outputs": [],
   "source": [
    "# Se débarrasser des avertissements inutiles\n",
    "import warnings\n",
    "from matplotlib.cbook import MatplotlibDeprecationWarning\n",
    "warnings.filterwarnings('ignore', category=MatplotlibDeprecationWarning)\n",
    "\n",
    "\n",
    "# Importation de bibliothèques Qiskit standard\n",
    "from qiskit import QuantumCircuit, execute, Aer, IBMQ, QuantumRegister, ClassicalRegister\n",
    "from qiskit.compiler import transpile, assemble\n",
    "from qiskit.tools.jupyter import *\n",
    "from qiskit.visualization import *\n",
    "from ibm_quantum_widgets import *\n",
    "\n",
    "\n",
    "# Utile pour avoir pi\n",
    "import math\n",
    "pi=math.pi"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Que sont les circuits quantiques?\n",
    "\n",
    "Les circuits quantiques sont des modèles de calcul quantique dans lesquels un calcul est une séquence de portes quantiques. Les portes quantiques représentent souvent des rotations sur la sphère de Bloch. Jetons un coup d'œil à certaines des portes quantiques les plus populaires.\n",
    "\n",
    "### Porte X\n",
    "\n",
    "La porte X est représentée par la matrice de Pauli-X:\n",
    "\n",
    "$X = \\begin{pmatrix}\n",
    "0 & 1 \\\\\n",
    "1 & 0 \\\\\n",
    "\\end{pmatrix}$\n",
    "\n",
    "Une porte X équivaut à une rotation autour de l'axe X de la sphère de Bloch de $ \\ pi $ radians. Il mappe $|0\\rangle$ to $|1\\rangle$ and $|1\\rangle$ to $|0\\rangle$. C'est l'équivalent quantique de la porte NOT pour les ordinateurs classiques et est parfois appelé un bit-flip.\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "x_gate=QuantumCircuit(1) # Créer un circuit quantique avec 1 qubit\n",
    "x_gate.x(0)\n",
    "x_gate.draw(output='mpl')"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "backend = Aer.get_backend('statevector_simulator')\n",
    "result = execute(x_gate, backend).result().get_statevector()\n",
    "plot_bloch_multivector(result)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Porte SX\n",
    "\n",
    "La porte SX équivaut à une rotation autour de l'axe X de la sphère de Bloch de $\\pi/2$. Elle est appelée porte SX pour indiquer qu'il s'agit de la racine carrée de la porte X. L'application de cette porte deux fois produit la porte Pauli-X standard. Le contraire du SX est le poignard SX, qui est une rotation de $\\pi/2$ dans la direction opposée.\n",
    "\n",
    "$SX = \\frac{1}{\\sqrt{2}}\\begin{pmatrix}\n",
    "1+i & 1-i \\\\\n",
    "1-i & 1+i \\\\\n",
    "\\end{pmatrix}$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "sx_gate = QuantumCircuit(1)\n",
    "sx_gate.sx(0)  \n",
    "sx_gate.draw(output='mpl')"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "backend = Aer.get_backend('statevector_simulator')\n",
    "result = execute(sx_gate, backend).result().get_statevector()\n",
    "plot_bloch_multivector(result)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Portail RZ\n",
    "\n",
    "La porte Rz effectue une rotation de $\\phi$ autour de la direction de l'axe Z (où $\\phi$ est un nombre réel). Il a la matrice ci-dessous:\n",
    "\n",
    "$RZ = \\begin{pmatrix}\n",
    "1 & 0 \\\\\n",
    "0 & e ^{i \\phi } \\\\\n",
    "\\end{pmatrix}$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "rz_gate = QuantumCircuit(1)\n",
    "rz_gate.rz(pi/2, 0)\n",
    "rz_gate.draw(output='mpl')"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "backend = Aer.get_backend('statevector_simulator')\n",
    "result = execute(rz_gate, backend).result().get_statevector()\n",
    "plot_bloch_multivector(result)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Puisque la rotation est autour de l'axe Z, on ne verrait pas de différence lorsque nous l'appliquons à l'état par défaut $|0\\rangle$, nous utilisons donc l'état qui a été généré en appliquant la porte SX à la place et lui appliquons le RZ ."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "rz_gate.sx(0)\n",
    "rz_gate.rz(pi/2, 0)\n",
    "rz_gate.draw(output='mpl')"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "backend = Aer.get_backend('statevector_simulator')\n",
    "result = execute(rz_gate, backend).result().get_statevector()\n",
    "plot_bloch_multivector(result)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Porte Hadamard\n",
    "Une porte Hadamard représente une rotation de $\\pi$ autour de l'axe qui se trouve au milieu de l'axe X et de l'axe Z.\n",
    "Il mappe l'état de base $|0\\rangle$à$\\frac{|0\\rangle+|1\\rangle}{\\sqrt{2}}$, ce qui signifie qu'une mesure aura des probabilités égales d'être `1` ou «0», créant une «superposition» d'états. Cet état s'écrit également $|+\\rangle$. Ce que fait Hadamard est de transformer entre le $|0\\rangle$ $|1\\rangle$ et le$|+\\rangle$ $|-\\rangle$ base.\n",
    "\n",
    "$H = \\frac{1}{\\sqrt{2}}\\begin{pmatrix}\n",
    "1 & 1 \\\\\n",
    "1 & -1 \\\\\n",
    "\\end{pmatrix}$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "# Faisons une porte H sur un | 0> qubit\n",
    "h_gate = QuantumCircuit(1)\n",
    "h_gate.h(0)\n",
    "h_gate.draw(output='mpl')"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "\n",
    "# Voyons le résultat\n",
    "backend = Aer.get_backend('statevector_simulator')\n",
    "result = execute(h_gate, backend).result().get_statevector()\n",
    "plot_bloch_multivector(result)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Porte CX (porte CNOT)\n",
    "\n",
    "La porte contrôlée NOT (ou CNOT ou CX) agit sur deux qubits. Il effectue l'opération NOT (équivalente à l'application d'une porte X) sur le deuxième qubit uniquement lorsque le premier qubit est $|1\\rangle$ et le laisse autrement inchangé.\n",
    "\n",
    "Remarque: Qiskit numérote les bits d'une chaîne de droite à gauche.\n",
    "\n",
    "$CX = \\begin{pmatrix}\n",
    "1 & 0 & 0 & 0  \\\\\n",
    "0 & 1 & 0 & 0 \\\\\n",
    "0 & 0 & 0 & 1 \\\\\n",
    "0 & 0 & 1 & 0 \\\\\n",
    "\\end{pmatrix}$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "cx_gate = QuantumCircuit(2)\n",
    "cx_gate.cx(0,1)\n",
    "cx_gate.draw(output='mpl')"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Porte CCX (porte Toffoli)\n",
    "\n",
    "La porte CCX (X Gate contrôlée contrôlée) est également appelée porte Toffoli. La porte CCX est une porte à trois bits, avec deux commandes et une cible comme entrée et sortie. Si les deux premiers bits sont dans l'état $|1\\rangle$, il applique un Pauli-X (ou NOT) sur le troisième bit. Sinon, ça ne fait rien.\n",
    "\n",
    "Remarque: Qiskit numérote les bits d'une chaîne de droite à gauche.\n",
    "\n",
    "$CCX = \\begin{pmatrix}\n",
    "1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\\\\n",
    "0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 \\\\\n",
    "0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 \\\\\n",
    "0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 \\\\\n",
    "0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 \\\\\n",
    "0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 \\\\\n",
    "0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 \\\\\n",
    "0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 \\\\\n",
    "\\end{pmatrix}$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "ccx_gate = QuantumCircuit(3)\n",
    "ccx_gate.ccx(0,1,2)\n",
    "ccx_gate.draw(output='mpl')"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Créez des portes logiques à l'aide de portes quantiques\n",
    "\n",
    "### NOT de porte\n",
    "\n",
    "La porte NOT inverse la valeur d'un bit et, comme cela a été mentionné précédemment, une porte X peut être considérée comme une porte NOT. La table de vérité pour une porte NOT ressemble à ceci:\n",
    "\n",
    "| Input | Output |\n",
    "| --- | --- | \n",
    "| 1 | 0 |\n",
    "| 0 | 1 |"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "not_gate=QuantumCircuit(1,1) \n",
    "# Créer un circuit quantique avec 1 qubit et 1 bit classique\n",
    "not_gate.x(0)\n",
    "not_gate.measure(0,0)\n",
    "not_gate.draw(output='mpl')"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### AND de porte\n",
    "\n",
    "La sortie d'un ET est vraie si et seulement si les deux entrées sont vraies. La table de vérité pour une porte AND ressemble à ceci:\n",
    "\n",
    "| A (Input) | B (Input) | Output |\n",
    "| --- | --- | --- |\n",
    "| 0 | 0 | 0 | \n",
    "| 0 | 1 | 0 |\n",
    "| 1 | 0 | 0 |\n",
    "| 1 | 1 | 1 |\n",
    "\n",
    "Avec une porte Toffoli, nous pouvons obtenir le résultat d'une porte ET en interprétant les deux bits de contrôle comme les bits d'entrée et le bit cible comme le bit de sortie.\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "and_gate=QuantumCircuit(3,1) # Créer un circuit quantique avec 3 qubits et 1 bit classique\n",
    "and_gate.ccx(0,1,2)\n",
    "and_gate.measure(2,0)\n",
    "and_gate.draw(output='mpl')"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### OR gate\n",
    "Une porte OR renvoie vrai si au moins une des portes d'entrée est vraie.\n",
    "La table de vérité pour une porte OU ressemble à ceci:\n",
    "\n",
    "| A (Input) | B (Input) | Output |\n",
    "| --- | --- | --- |\n",
    "| 0 | 0 | 0 | \n",
    "| 0 | 1 | 1 |\n",
    "| 1 | 0 | 1 |\n",
    "| 1 | 1 | 1 |"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "scrolled": true
   },
   "outputs": [],
   "source": [
    "or_gate=QuantumCircuit(3,1) # Créer un circuit quantique avec 3 qubits et 1 bit classique\n",
    "or_gate.cx(1,2)\n",
    "or_gate.cx(0,2)\n",
    "or_gate.ccx(0,1,2)\n",
    "or_gate.measure(2,0)\n",
    "or_gate.draw(output='mpl')"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Utilisation du widget Circuit Composer\n",
    "\n",
    "Vous connaissez peut-être le composeur de circuits quantiques IBM, dans lequel vous pouvez générer des circuits tout en utilisant une interface graphique. Avec le nouveau widget de composition de circuits, la même fonctionnalité peut être utilisée dans un notebook jupyter. Vous pouvez en savoir plus sur l'utilisation du widget [ici] (https://quantum-computing.ibm.com/lab/docs/iql/composer-widget)\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "\n",
    "<div class = \"alert alert-block alert-success\">\n",
    "\n",
    "** Exercice 1a: ** Construire un NON-OU (a nié OR) porte à l'aide du circuit compositeur\n",
    "    \n",
    "Cet exercice a pour but de vous encourager à jouer un peu avec le compositeur. Il n'est pas noté. Vous pouvez avancer si vous le souhaitez. L'exécution de la cellule ci-dessous vous montrera un compositeur, où vous pouvez ajouter des portes en les faisant glisser vers le bon endroit.\n",
    "</div>"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "from ibm_quantum_widgets import CircuitComposer\n",
    "editor = CircuitComposer()\n",
    "editor"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Utilisez le widget Circuit Composer pour travailler avec un circuit existant\n",
    "\n",
    "Nous pouvons également utiliser le compositeur de circuits pour ouvrir un circuit précédemment créé. Avec le code ci-dessous, nous ouvrirons le circuit créé ci-dessus, qui représente la porte OR. Vous pouvez l'utiliser pour vérifier si vous avez correctement construit votre porte OR.\n",
    "\n",
    "<div class = \"alert alert-block alert-success\">\n",
    "\n",
    "Vous pouvez éditer le circuit ouvert par glisser-déposer. Essayez de supprimer la mesure à la fin.\n",
    "    \n",
    "</div>"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "scrolled": true
   },
   "outputs": [],
   "source": [
    "from ibm_quantum_widgets import CircuitComposer\n",
    "editor2 = CircuitComposer(circuit=or_gate)\n",
    "editor2"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Dans l'exemple ci-dessous, nous stockons le dernier circuit du composeur dans la variable qc2, puis appliquons un X à la sortie et mesurons à nouveau.\n",
    "\n",
    "Utilisez le circuit du dernier éditeur et appliquez un X à la sortie et mesurez à nouveau."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "outputs": [],
   "source": [
    "qc2 = editor2.circuit\n",
    "\n",
    "qc2.x(2)\n",
    "qc2.measure((2), (0))\n",
    "\n",
    "qc2.draw(output='mpl')"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Nous avons maintenant créé un NOR la ​​négation d'un OU, il doit être identique au circuit que vous avez vous-même construit dans le premier exercice.\n",
    "\n",
    "## Portes quantiques composites et leur coût\n",
    "\n",
    "Un véritable ordinateur quantique n'a normalement pas d'implémentations physiques de toutes les portes. Au lieu de cela, ils utilisent un petit ensemble de portes de base, qui forment un ensemble de portes universelles, c'est, similaire au cas classique, un ensemble d'instructions qui peuvent être utilisées pour implémenter toutes les opérations possibles.\n",
    "\n",
    "Pour cette raison, les circuits quantiques doivent être transpilés en portes de base avant exécution. Ceci est généralement effectué automatiquement par le transpileur Qiskit lorsqu'un circuit quantique est envoyé à un système IBM Quantum. Mais à des fins d'apprentissage, vous devez construire le circuit à la main en utilisant les portes de base. Les portes de base du système IBM Quantum sont généralement les portes CX, ID, RZ, SX et X. Vous pouvez voir [système `ibmq_mumbai`] (https://quantum-computing.ibm.com/services?skip=0&systems=all&system=ibmq_mumbai) pour un exemple.\n",
    "\n",
    "Jetons maintenant un œil au circuit ci-dessous:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "qc = QuantumCircuit(2)\n",
    "qc.sxdg(0)\n",
    "qc.t(1)\n",
    "qc.draw(output='mpl')"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Voyons maintenant à quoi pourrait ressembler une décomposition du circuit ci-dessus pour un ordinateur quantique, utilisant uniquement les portes de base."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "qc = QuantumCircuit(2)\n",
    "qc.sx(0)\n",
    "qc.sx(0)\n",
    "qc.sx(0)\n",
    "qc.rz(pi/4,1)\n",
    "qc.draw(output='mpl')"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Comme vous pouvez le voir, nous n'utilisons plus que les portes de base, mais pour cette raison, plus de portes totales sont utilisées. Plus un circuit a de portes, plus son fonctionnement est complexe. Ainsi, lorsque nous voulons calculer le coût d'un circuit, nous considérons le nombre de portes utilisées. Cependant, toutes les portes ne sont pas considérées comme égales coûteuses, donc lorsque nous calculons le coût d'un circuit, nous utilisons la formule suivante:\n",
    "\n",
    "$$\n",
    "Coût = 10 N_{CNOT} + N_{autre}\n",
    "$$\n",
    "\n",
    "où $ N_{CNOT} $ est le nombre de portes CNOT et $ N_{other} $ est le nombre d'autres portes.\n",
    "\n",
    "### Porte Hadamard\n",
    "\n",
    "Comme dit, toutes les opérations peuvent être exprimées en utilisant simplement les portes de base. À titre d'exemple, nous montrons comment construire une porte Hadamard en utilisant notre ensemble de portes de base. Nous n'avons pas de porte de base qui effectue une rotation directe autour de l'axe qui se trouve au milieu de l'axe X et de l'axe Z, nous utilisons donc plutôt des rotations autour de l'axe X et de l'axe Z pour obtenir le même résultat .\n",
    "\n",
    "Pouvez-vous deviner quelles rotations nous devons faire?"
   ]
  },
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   "execution_count": null,
   "metadata": {},
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   "source": [
    "q=QuantumRegister(1)\n",
    "c=ClassicalRegister(1)\n",
    "qc=QuantumCircuit(q,c)\n",
    "qc.rz(pi/2, 0)\n",
    "qc.sx(0)\n",
    "qc.rz(pi/2, 0)\n",
    "qc.draw(output='mpl')"
   ]
  },
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   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "\n",
    "Comme vous vous en souvenez peut-être, c'est le circuit que nous avions ci-dessus, lorsque nous avons visualisé la rotation de la porte RZ. Ci-dessus, nous avons vu que le premier RZ ne fait rien, lorsque nous sommes dans les états $|0\\rangle$ ou $|1\\rangle$. Cela peut donc sembler un peu inutile. Cependant, si nous sommes dans les états $|+\\rangle$ et $|-\\rangle$, la première rotation a un effet. Nous avons le scénario inverse, car après avoir appliqué la porte SX, nous sommes à nouveau dans l'état $|0\\rangle$ ou $|1\\rangle$ et le deuxième RZ n'a aucun effet.\n",
    "\n",
    "### Rotation contrôlée\n",
    "\n",
    "Nous avons vu ci-dessus le NOT contrôlé, montrons maintenant un exemple sur la façon dont on peut construire une rotation contrôlée autour de l'axe Y. La rotation $\\theta$ peut être n'importe quelle rotation, il n'est pas nécessaire qu'elle soit $\\pi$, ce n'est qu'un exemple. "
   ]
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   "source": [
    "qc = QuantumCircuit(2)\n",
    "theta = pi # Theta can be anything (pi chosen arbitrarily)\n",
    "qc.ry(theta/2,1)\n",
    "qc.cx(0,1)\n",
    "qc.ry(-theta/2,1)\n",
    "qc.cx(0,1)\n",
    "qc.draw(output='mpl')"
   ]
  },
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   "source": [
    "Quand on parcourt ce circuit, on peut voir que si le premier qubit est 0 alors les deux rotations s'annulent et rien ne se passe.\n",
    "\n",
    "Par contre, si le premier qubit est 1, nous obtiendrons un état égal à l'application de la rotation $\\theta/2$ deux fois qui forme notre rotation initiale $\\theta$. Cela fonctionne puisque les axes X et Y sont orthogonaux.\n",
    "\n",
    "<div class = \"alert alert-block alert-danger\">\n",
    "Pour la rotation autour d'autres axes, vous devrez peut-être utiliser d'autres astuces.\n",
    "</div>\n",
    "\n",
    "### Rotation contrôlée contrôlée\n",
    "\n",
    "Ci-dessus, nous avons vu un exemple sur la façon dont on peut faire une rotation contrôlée autour de l'axe $ Y $.\n",
    "Maintenant, nous supposons que nous avons une rotation contrôlée (autour de l'axe que nous voulons) et que nous voulons construire à partir de cela une double rotation contrôlée, qui n'est appliquée que si les deux qubits de contrôle sont 1 similaires à la porte CCX. "
   ]
  },
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    "qc = QuantumCircuit(3)\n",
    "theta = pi # Theta peut être n'importe quoi (pi choisi arbitrairement)\n",
    "qc.cp(theta/2,1,2)\n",
    "qc.cx(0,1)\n",
    "qc.cp(-theta/2,1,2)\n",
    "qc.cx(0,1)\n",
    "qc.cp(theta/2,0,2)\n",
    "qc.draw()"
   ]
  },
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   "cell_type": "markdown",
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   "source": [
    "Dans ce circuit, si le premier et le deuxième qubit sont à 0, alors rien ne se passe du tout. Si seul le deuxième qubit est un, on applique d'abord une rotation de $\\pi/2$ et ensuite une rotation de $-\\pi/2$ qui s'annulent. Si seul le premier qubit vaut 1, alors nous le deuxième qubit sera également un après le premier CX donc une rotation de $ -\\pi/2$ sera appliquée et ensuite une rotation de $\\pi/2$ sera appliquée et ces deux rotations s'annulent à nouveau.\n",
    "\n",
    "Si le premier et le deuxième qubit sont tous deux 1, il y aura d'abord une rotation de $\\pi/2$, puis le deuxième qubit deviendra 0 donc la rotation suivante ne s'applique pas et ensuite il est retourné à 1. Ensuite, une autre rotation by $\\pi/2$ est appliqué puisque le premier qubit est 1. On a donc deux fois une rotation de $\\pi/2$ qui forment ensemble une rotation de $\\pi$."
   ]
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   "cell_type": "markdown",
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   "source": [
    "\n",
    "## Le problème\n",
    "\n",
    "<div id = 'problem'> </div>\n",
    "<div class = \"alert alert-block alert-success\">\n",
    "\n",
    "Nous avons vu ci-dessus comment construire une porte Hadamard avec notre ensemble de base, et maintenant nous voulons également construire une porte Toffoli. Pourquoi la porte de Toffoli? Comme mentionné ci-dessus, la porte de Toffoli est également une porte universelle pour le calcul classique de la même manière que la porte NAND, mais elle est réversible. En outre, il construit un ensemble de portes universelles simples pour le calcul quantique s'il est combiné avec la porte Hadamard.\n",
    "\n",
    "Nous avons vu quelques exemples sur la façon d'exprimer des portes plus complexes en utilisant des portes de base, nous voulons maintenant utiliser les connaissances acquises pour construire une porte de Toffoli en utilisant uniquement nos portes de base. Afin de résoudre cet exercice, les exemples ci-dessus sur la façon de construire et d'utiliser des rotations contrôlées vous seront utiles. Le plus grand défi est de construire les rotations contrôlées nécessaires.\n",
    "    \n",
    "Vous pouvez utiliser le code ci-dessous en utilisant le widget compositeur pour construire votre circuit.\n",
    "    \n",
    "</div>\n",
    "\n",
    "\n",
    "<div class = \"alert alert-block alert-danger\">\n",
    "\n",
    "Pour rappel, les portes de base des systèmes IBM Quantum sont les portes CX, RZ, SX et X, donc aucune autre porte n'est autorisée.\n",
    "\n",
    "Bien sûr, nous voulons également essayer de minimiser les coûts.\n",
    "    \n",
    "$$\n",
    "Cost = 10 N_{CNOT} + N_{other}\n",
    "$$\n",
    "    \n",
    "</div>\n"
   ]
  },
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   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "from qiskit import QuantumRegister, ClassicalRegister, QuantumCircuit\n",
    "from qiskit import IBMQ, Aer, execute\n",
    "from ibm_quantum_widgets import CircuitComposer\n",
    "editorEx = CircuitComposer() \n",
    "editorEx\n",
    "##### Construisez votre circuit quantique ici en utilisant le widget composeur."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "# Vous pouvez également créer votre circuit par programmation en utilisant le code Qiskit\n",
    "circuit = QuantumCircuit(3)\n",
    "\n",
    "# ÉCRIVEZ VOTRE CODE ENTRE CES LIGNES - COMMENCEZ\n",
    "\n",
    "\n",
    "\n",
    "\n",
    "\n",
    "# ECRIVEZ VOTRE CODE ENTRE CES LIGNES - FIN"
   ]
  },
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   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "\n",
    "# Exécuter le circuit par qasm_simulator\n",
    "qc = editorEx.circuit\n",
    "#qc = circuit # Décommentez cette ligne si vous souhaitez soumettre le circuit construit en utilisant le code Qiskit\n",
    "qc.draw(output='mpl')"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "# Vérifiez votre réponse en utilisant le code suivant\n",
    "from qc_grader import grade_ex1\n",
    "grade_ex1(qc)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "# Soumettez votre réponse. Vous pouvez soumettre à nouveau à tout moment.\n",
    "from qc_grader import submit_ex1\n",
    "submit_ex1(qc)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Additional information\n",
    "\n",
    "**Created by:** Marcel Pfaffhauser, Brahmani Thota, Junye Huang\n",
    "\n",
    "**Translation:** Anush Krishna V\n",
    "\n",
    "**Version:** 1.0.1"
   ]
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