content/ex1/ex1-hi.ipynb

{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# अभ्यास 1 - Toffoli gate\n",
    "\n",
    "## ऐतिहासिक पृष्ठभूमि\n",
    "\n",
    "चालीस साल पहले, 50 विचारकों के एक रैगटैग समूह ने MIT के Endicott House के लॉन पर एक तस्वीर के लिए खुद को व्यवस्थित किया। MIT और IBM द्वारा संयुक्त रूप से आयोजित Physics of Computation सम्मेलन में कुछ लोगों ने सोचा कि वे 1981 में इतिहास रच रहे हैं। यह यकीनन Physics of Computation का जन्मस्थान था, और विशेष रूप से Quantum Computing के अब-बढ़ते क्षेत्र, एक गंभीर विषय के रूप में योग्य है। एक पाठ्यपुस्तक या विश्वविद्यालय पाठ्यक्रम\n",
    "यह इस सम्मेलन में था जहां फेनमैन ने अपने अब तक के प्रसिद्ध उद्धरण का उच्चारण किया: 'प्रकृति शास्त्रीय नहीं है, लानत है, और यदि आप प्रकृति का अनुकरण करना चाहते हैं, तो आप इसे Quantum Mechanical बनाना बेहतर समझते हैं, और से यह एक अद्भुत समस्या है , क्योंकि यह इतना आसान नहीं लगता है।' [१] इस महीने की शुरुआत में, हमने इस महत्वपूर्ण सम्मेलन की ४०वीं वर्षगांठ मनाई। आप और पढ़ सकते हैं [here](https://youtu.be/GR6ANm6Z0yk).\n",
    "\n",
    "![](resources/conference-photo.jpeg)\n",
    "\n",
    "सम्मेलन में चर्चा किये गये विषयों में से एक reversible computing था, जिसके बारे में MIT में Tommaso Toffoli और Edward Fredkin पिछले कुछ वर्षों से सोच रहे थे। [२-३] Toffoli ने AND/NAND gate के reversible version के साथ आया (जो कि अब Toffoli gate, या controlled-controlled-NOT gate कहा जाता है)। क्योंकि classical computing में NAND gate यूनिवर्सल है, Toffoli गेट एक universal reversible logic gate है।  Quantum computing, reversible computing का एक विशेष रूप है; किसी भी reversible gate को quantum computer पर लागू किया जा सकता है, और इसलिए Toffoli gate भी एक क्quantum logic gateट है। हालांकि, अकेले Toffoli gate क्Quantum computing के लिए एक सार्वभौमिक gate नहीं है। \n",
    "\n",
    "इस अभ्यास में, हम क्quantum computers के लिए Toffoli gate और universal gate sets का पता लगाएंगे।\n",
    "\n",
    "### संदर्भ \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",
    "\n",
    "classical computation में अक्सर इस्तेमाल किया जाने वाला model Boolean logic या classical logical gates है। ऐसे gates Boolean functions का प्रतिनिधित्व करते हैं, केवल binary (0,1) इनपुट और आउटपुट के साथ कार्य करते हैं। Boolean logic का एक दिलचस्प पहलू यह है कि सभी संभावित Boolean functions को केवल विभिन्न logic gates की एक छोटी संख्या के संयोजन का उपयोग करके बनाया जा सकता है। ऐसे sets को क्रियात्मक रूप से functionally complete sets कहते हैं। ऐसा ही एक प्रसिद्ध सेट AND और NOT है। ये दो gates सभी संभावित functions को व्यक्त करने के लिए पर्याप्त हैं। OR और NOT के लिए भी यही सच है। छोटे सेट मौजूद हैं, जैसे कि NAND और NOR जो अकेले universal हैं, फिर भी funtions AND, NOT और OR को अक्सर classical computation के basic blocks के रूप में देखा जाता है।\n",
    "\n",
    "<div class=\"alert alert-block alert-success\">\n",
    "\n",
    "**Goal**\n",
    "\n",
    'Basis gate set का उपयोग करके IBM Quantum systems का Toffoli gate का निर्माण करें (CX, RZ, SX and X gates)\n',
    "\n",
    "</div>\n",
    "\n",
    "<div class=\"alert alert-block alert-danger\">\n",
    "    \n",
    "इस अभ्यास का उद्देश्य आपको Quantum gates की बुनियादी अवधारणाएँ और Quantum Circuits बनाने का तरीका सिखाना है\n",
    "1. Circuit Composer widget  का दृश्य रूप से उपयोग करना \n",
    "1. Qiskit का programmatic रूप से उपयोग करना \n",
    "\n",
    "यदि आप पहले से ही quantum gates और Qiskit से परिचित हैं। आप सीधे आगे बढ़ सकते हैं <a href=#problem>the problem</a>.\n",
    "</div>"
   ]
  },
{
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "# Getting rid of unnecessary warnings\n",
    "import warnings\n",
    "from matplotlib.cbook import MatplotlibDeprecationWarning\n",
    "warnings.filterwarnings('ignore', category=MatplotlibDeprecationWarning)\n",
    "\n",
    "# Importing standard Qiskit libraries\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",
    "# Useful to have pi\n",
    "import math\n",
    "pi=math.pi"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Quantum circuits क्या हैं??\n",
    "\n",
    "Quantum circuits, quantum computation के लिए model हैं जिसमें गणना quantum gates का अनुक्रम है। Quantum gates अक्सर Bloch sphere पर rotations का प्रतिनिधित्व करते हैं। आइए कुछ लोकप्रिय क्वांटम गेट्स पर एक नज़र डालें।\n",
    "\n",
    "### X gate \n",
    "\n",
    "The X-gate is represented by the Pauli-X matrix: \n",
    "\n",
    "$X = \\begin{pmatrix}\n",
    "0 & 1 \\\\\n",
    "1 & 0 \\\\\n",
    "\\end{pmatrix}$\n",
    "\n",
    "एक X gate  Bloch sphere के X-axis के चारों ओर $\\pi$ रेडियंस के rotation के बराबर है. यह map $|0\\rangle$ to $|1\\rangle$ and $|1\\rangle$ to $|0\\rangle$. यह classical computers के लिए NOT gate का quantum equivalent है और इसे कभी-कभी bit-flip भी कहा जाता है।"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "x_gate=QuantumCircuit(1) # 1 qubit के साथ एक क्QuantumCircuit बनाएं\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": [
    "### SX gate\n",
    "\n",
   "SX gate Bloch sphere के X-axis के चारों ओर एक  $\\pi/2$ rotation के बराबर है। इसे SX gate कहा जाता है यह संकेत करने के लिए कि यह X gate का square-root है। इस गेट को दो बार लागू करने से उत्पन्न होता है मानक Pauli-X gate। SX के विपरीत SX dagger है, जो विपरीत दिशा में $\\pi/2$ से घूमता है।\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": [
    "### RZ gate\n",
    "\n",
    "Rz gate Z-axis दिशा के चारों ओर $\\phi$ का rotation करता है (जहां $\\phi$ एक real number है)। इसके नीचे मैट्रिक्स है:\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": [
    "चूंकि रोटेशन Z-axis के आसपास है, इसलिए जब हम इसे default state $|0\\rangle$ पर लागू करते हैं तो कोई अंतर नहीं दिखाई देगा, इसलिए हम उस default state का उपयोग करते हैं जो इसके बजाय SX gate लगाने से उत्पन्न हुआ था और RZ को इसपे लागू करते हैं"
   ]
  },
  {
   "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": [
    "### Hadamard gate\n",
    "एक Hadamard gate X-axis और Z-axis के बीच में axis के बारे में $\\pi$ के rotation का प्रतिनिधित्व करता है।\n",
    "यह basis state $|0\\rangle$ से $\\frac{|0\\rangle + |1\\rangle}{\\sqrt{2}}$ को मैप करता है, जिसका अर्थ है कि माप की समान संभावनाएं होंगी `1` या `0` होने के नाते, states का एक 'superposition' बनाना। इस state को $|+\\rangle$ के रूप में भी लिखा जाता है। Hadamard $|0\\rangle$ $|1\\rangle$ और $|+\\rangle$ $|-\\rangle$ आधार के बीच परिवर्तन करता है।\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": [
    "# Let's do an H-gate on a |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": [
    "# Let's see the result\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": [
    "### CX gate (CNOT gate)\n",
    "\n",
    'Controlled NOT (or CNOT or CX) gate, दो qubits पर कार्य करता है। यह दूसरे qubit पर NOT operation (X gate लगाने के बराबर) करता है, जब पहला qubit $|1\\rangle$ होता है और अन्यथा इसे अपरिवर्तित छोड़ देती है। \n',
    "\n",
    "Note: Qiskit bits को दाएं से बाएं एक string में संख्या देता है।\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": [
    "### CCX gate (Toffoli gate)\n",
    "\n",
    "CCX gate (controlled controlled X Gate) को Toffoli gate भी कहा जाता है। CCX gate एक 3-bit गेट है, जिसमें दो नियंत्रण और एक लक्ष्य उनके इनपुट और आउटपुट के रूप में है। यदि पहले दो बिट $|1\\rangle$ state में हैं, तो यह तीसरे bit पर Pauli-X Gate(या NOT) लागू करता है। अन्यथा, यह कुछ नहीं करता है।\n",
    "\n",
    "Note: Qiskit bits को दाएं से बाएं एक string में संख्या देता है।\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": [
    "## Quantum gates की मदद से Logical gates बनाएं\n",
    "\n",
    "### NOT gate\n",
    "\n",
    "NOT गेट एक बिट को फ़्लिप करता है और जैसा कि पहले उल्लेख किया गया था, एक X gate को NOT gate माना जा सकता है। NOT gate के लिए truth table इस प्रकार है:\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) # Create a quantum circuit with 1 qubit and 1 classical bit\n",
    "not_gate.x(0)\n",
    "not_gate.measure(0,0)\n",
    "not_gate.draw(output='mpl')"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### AND gate\n",
    "\n",
    "AND का output तभी True केवल तभी जब दोनों input True हों। AND गेट के लिए truth table इस तरह दिखती है:\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",
    'Tofolli gate के साथ, हम दो control bits को input bits और target bit को output bit के रूप में व्याख्या करके AND gate का परिणाम प्राप्त कर सकते हैं।'
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "and_gate=QuantumCircuit(3,1) # 3 qubit और 1 classical bit के साथ एक quantum circuit बनाएं\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",
    "\n",
    "यदि कम से कम एक inout gate true है तो एक OR गेट true होता है.\n",
    "OR गेट के लिए truth table इस तरह दिखती है:\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) # C3 qubit और 1 classical bit के साथ एक quantum circuit बनाएं\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": [
    "## Circuit Composer widget का उपयोग करना।\n",
    "\n",
    "आप IBM quantum circuit composer से परिचित हो सकते हैं, जहां आप graphical interface का उपयोग करते हुए circuit उत्पन्न कर सकते हैं। नए Circuit Composer widget के साथ, ज्यूपिटर नोटबुक में समान कार्यक्षमता का उपयोग किया जा सकता है। आप widget का उपयोग करने के तरीके के बारे में अधिक जान सकते हैं [यहां](https://quantum-computing.ibm.com/lab/docs/iql/composer-widget)\n",
    "\n",
    "<div class=\"alert alert-block alert-success\">\n",
    "\n",
    "**Exercise 1a:** Build a NOR (a negated OR) gate using circuit composer\n",
    "    \n",
    "इस अभ्यास का उद्देश्य आपको Composer के साथ थोड़ा खेलने के लिए प्रोत्साहित करना है। यह वर्गीकृत नहीं है। आप चाहें तो आगे निकल सकते हैं। नीचे दिए गए cell को निष्पादित करने से आपको एक Composer दिखाई देगा, जहां आप gates को सही जगह पर खींचकर जोड़ सकते हैं।    \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": [
    "### मौजूदा Circuit के साथ काम करने के लिए Circuit Composer widget का उपयोग करें\n",
    "\n",
    "हम पहले से बनाए गए Circuit को खोलने के लिए Circuit Composer का भी उपयोग कर सकते हैं। नीचे दिए गए code से हम ऊपर बनाए गए  Circuit  को खोलेंगे, जो OR gate को दर्शाता है। आप इसका उपयोग यह जांचने के लिए कर सकते हैं कि आपने अपना OR सही ढंग से बनाया है या नहीं।\n",
    "\n",
    "<div class=\"alert alert-block alert-success\">\n",
    "\n",
    "आप खुले हुए Circuit को ड्रैग एंड ड्रॉप द्वारा edit कर सकते हैं। अंत में measurement को हटाने का प्रयास करें।\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": [
    "नीचे दिए गए उदाहरण में हम Composer से अंतिम Circuit को variable qc2 में स्टोर करते हैं और फिर output पर X लागू करते हैं और फिर से मापते हैं।\n",
    "\n",
    "पिछले editor से Circuit का उपयोग करें और output पर X लागू करें और फिर से मापें."
   ]
  },
  {
   "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": [
    "अब हमने OR का NOT यानि NOR बनाया है, यह उस circuit के समान होना चाहिए जिसे आपने पहले अभ्यास में स्वयं बनाया था। \n",
    "\n",
    "## Composite quantum gates और उनका cost \n",
    "\n",
    "एक वास्तविक quantum computer में सामान्य रूप से सभी gates. का physical implementations नहीं होता है। इसके बजाय वे base gates के एक छोटे से सेट का उपयोग करते हैं, जो एक universal gate set बनाते हैं, यह classical मामले के समान है, instructions का एक set जिसका उपयोग सभी संभावित संचालन को लागू करने के लिए किया जा सकता है।\n",
    "\n",
    "इस कारण से, quantum circuits को execution से पहले basis gates में transpile किया जाना चाहिए। यह आमतौर पर Qiskit transpiler द्वारा स्वचालित रूप से किया जाता है जब एक  quantum circuit, IBM Quantum system को भेजा जाता है। लेकिन सीखने के उद्देश्यों के लिए, आपसे basis gates का उपयोग करके हाथ से circuit का निर्माण करने की अपेक्षा की जाती है। IBM Quantum system के लिए basis gates आमतौर पर CX, ID, RZ, SX और X gates हैं। आप देख सकते हैं[`ibmq_mumbai` system](https://quantum-computing.ibm.com/services?skip=0&systems=all&system=ibmq_mumbai) for an example.\n",
    "\n",
    "आइए अब नीचे दिए गए circuit पर एक नजर डालते हैं:"
   ]
  },
  {
   "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": [
    "अब दिखाते हैं कि केवल  का उपयोग करके क्quantum computer के लिए उपरोक्त circuit का अपघटन कैसे दिख सकता है."
   ]
  },
  {
   "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": [
    "जैसा कि आप देख सकते हैं, अब हम केवल base gates का उपयोग करते हैं, लेकिन इस कारण से अधिक total gates का उपयोग किया जाता है। एक circuit में जितने अधिक gates होते हैं, उसे चलाना उतना ही जटिल होता है। इसलिए, जब हम एक circuit की cost की गणना करना चाहते हैं, तो हम उपयोग किए जाने वाले gates की संख्या पर विचार करते हैं। हालांकि, सभी gates को समान रूप से महंगा नहीं माना जाता है, इसलिए जब हम एक circuit की cost की गणना करते हैं, तो हम निम्नलिखित सूत्र का उपयोग करते हैं:\n",
    "\n",
    "$$\n",
    "Cost = 10 N_{CNOT} + N_{other}\n",
    "$$\n",
    "\n",
    "where $N_{CNOT}$ is the number of CNOT gate and $N_{other}$ is the number of other gates.\n",
    "\n",
    "### Hadamard gate\n",
    "\n",
    "जैसा कि कहा गया है कि सभी कार्यों को केवल base gates का उपयोग करके व्यक्त किया जा सकता है। एक उदाहरण के रूप में हम दिखाते हैं कि हमारे base gate सेट का उपयोग करके Hadamard gate का निर्माण कैसे किया जाता है। हमारे पास base gate नहीं है जो X-axis और Z-axis के मध्य axis के चारों ओर सीधा rotation करता है, इसलिए हम समान परिणाम प्राप्त करने के लिए X-axis और Z-axis के चारों ओर  rotation का उपयोग करते हैं . \n",
    "\n",
    "क्या आप अनुमान लगा सकते हैं कि हमें कौन-से rotations लगाने होंगे?"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "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')"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "जैसा कि आपको याद होगा, यह वह circuit है जो ऊपर था, जब हमने RZ gate के rotation की कल्पना की थी। ऊपर हमने देखा कि पहला RZ कुछ नहीं करता है, जब हम $|0\\rangle$ या $|1\\rangle$ states में होते हैं। तो यह थोड़ा बेकार लग सकता है। हालांकि, अगर हम $|+\\rangle$ और $|-\\rangle$ states में हैं तो पहले rotation का प्रभाव पड़ता है। हमारे पास विपरीत परिदृश्य है, क्योंकि SX gate लगाने के बाद हम फिर से $|0\\rangle$ या $|1\\rangle$ state में हैं और फिर दूसरे RZ का कोई प्रभाव नहीं पड़ता है।\n",
    "\n",
    "### Controlled rotation\n",
    "\n",
    "हमने Controlled NOT के ऊपर देखा है, आइए अब एक उदाहरण दिखाते हैं कि  Y-axis के चारों ओर एक Controlled rotation कैसे बनाया जा सकता है। rotation $\\theta$ कोई भी rotation हो सकता है, यह $\\pi$ होना जरूरी नहीं है, यह सिर्फ एक उदाहरण है। "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "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')"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "जब कोई इस circuit को देखता है, तो कोई भी यह समझ सकता है कि यदि पहला qubit 0 है तो दो rotation एक दूसरे को रद्द कर देते हैं और कुछ भी नहीं होता है। \n",
    "\n",
    "दूसरी ओर, यदि पहली qubit 1 है, तो हमें एक ऐसी state मिलेगी जो $\\theta / 2$ को दो बार rotation के बराबर है जो हमारे प्रारंभिक rotation $\\theta$ को बनाती है। यह काम करता है क्योंकि X-axis और वाई-अक्ष orthogonal हैं। \n",
    "\n",
    "<div class=\"alert alert-block alert-danger\">\n",
    "अन्य axes के चारों ओर rotation के लिए आपको अन्य तरकीबों का उपयोग करने की आवश्यकता हो सकती है।\n",
    "</div>\n",
    "\n",
    "### Controlled controlled rotation\n",
    "\n",
    "ऊपर हमने एक उदाहरण देखा कि कैसे कोई $Y$-axis के आसपास एक controlled rotation कर सकता है।\n",
    "अब हम मानते हैं कि हमारे पास एक controlled rotation है (हम चाहते हैं axis के चारों ओर ) और उस से एक double controlled rotation बनाना चाहते हैं, जो केवल तभी लागू होता है जब दोनों control qubits CCX gate के समान होते हैं."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "qc = QuantumCircuit(3)\n",
    "theta = pi # Theta कुछ भी हो सकता है (pi मनमाने ढंग से चुना गया)\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()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "इस परिपथ में यदि पहली और दूसरी दोनों qubit 0 हैं, तो कुछ भी नहीं होता है। यदि केवल दूसरी qubit 1 है, तो पहले हम $\\pi/2$ का एक rotation लागू करते हैं और बाद में $-\\pi/2$ का एक rotation लागू करते हैं जो एक दूसरे को रद्द कर देता है। यदि केवल पहली qubit 1 है, तो हम दूसरी qubit भी पहले CX के बाद एक होगी, इसलिए $-\\ pi/2$ द्वारा घूर्णन लागू किया जाएगा और बाद में $\\ pi/2$ द्वारा घूर्णन किया जाएगा लागू होता है और ये दो घुमाव एक दूसरे को फिर से रद्द कर देते हैं।\n",
    "\n",
    "यदि पहली और दूसरी दोनों qubit 1 हैं तो पहले $\\pi/2$ से एक rotation होगा फिर दूसरा qubit 0 हो जाएगा, इसलिए अगला rotation लागू नहीं होता है और फिर इसे 1 पर वापस flip किया जाता है। बाद में दूसरा $\\pi/2$ द्वारा  rotation  लागू किया जाता है क्योंकि पहली qubit 1 है। इसलिए हमारे पास $\\pi/2$ का दो गुना rotation है जो एक साथ $\\pi$ का rotation बनाता है।"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## The problem\n",
    "\n",
    "<div id='problem'></div>\n",
    "<div class=\"alert alert-block alert-success\">\n",
    "\n",
    "We have seen above how to construct a Hadamard gate with our base set, and now we want to build a Toffoli gate as well. Why the Toffoli gate? As mentioned above, the Toffoli gate is also a universal gate for classical computation the same way the NAND gate is, but it is reversible. Further  it builds a simple universal gate set for quantum computation if combined with the Hadamard gate. \n",
    "\n",
    "हमने basis gates का उपयोग करके अधिक जटिल gates को व्यक्त करने के कुछ उदाहरण देखे हैं, अब हम प्राप्त ज्ञान का उपयोग केवल अपने basis gates का उपयोग करके Toffoli gate के निर्माण के लिए करना चाहते हैं। इस अभ्यास को हल करने के लिए controlled rotations का निर्माण और उपयोग कैसे करें, इस पर उपरोक्त उदाहरण काम आएंगे। सबसे बड़ी चुनौती आवश्यक controlled rotations का निर्माण करना है।\n",
    "    \n",
    "आप नीचे दिए गए कोड का उपयोग कर सकते हैं, Composer widget का उपयोग करके यहां अपना circuit बनाएं\n",
    "    \n",
    "</div>\n",
    "\n",
    "\n",
    "<div class=\"alert alert-block alert-danger\">\n",
    "\n",
    "एक अनुस्मारक के रूप में, IBM Quantum systems  के  basis gates CX, RZ, SX और X gates हैं, इसलिए किसी अन्य gates की अनुमति नहीं है।\n",
    "\n",
    "बेशक हम cost को कम करने का भी प्रयास करना चाहते हैं। \n",
    "    \n",
    "$$\n",
    "Cost = 10 N_{CNOT} + N_{other}\n",
    "$$\n",
    "    \n",
    "</div>\n"
   ]
  },
  {
   "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",
    "##### Composer widget का उपयोग करके यहां अपना quantum circuit बनाएं।"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "# आप Qiskit code का उपयोग करके programmatically अपना circuit भी बना सकते हैं\n",
    "\n",
    "circuit = QuantumCircuit(3)\n",
    "\n",
    "# इन पंक्तियों के बीच अपना code लिखें- START\n",
    "\n",
    "\n",
    "\n",
    "\n",
    "\n",
    "# इन पंक्तियों के बीच अपना code लिखें - END"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "# Execute the circuit by qasm_simulator\n",
    "qc = editorEx.circuit \n",
    "#qc = circuit # यदि आप Qiskit कोड का उपयोग करके निर्मित circuit को submit करना चाहते हैं तो इस लाइन को Uncomment करें \n",
    "\n",
    "qc.draw(output='mpl')"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "# निम्नलिखित code का उपयोग करके अपने उत्तर की जाँच करें\n",
    "from qc_grader import grade_ex1\n",
    "grade_ex1(qc)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "# अपना उत्तर सबमिट करें। आप किसी भी समय पुनः सबमिट कर सकते हैं।\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",
    "**Version:** 1.0.0"
   ]
  }
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