content/ex1/ex1-es.ipynb

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    "# Ejercicio 1 - Compuerta de Toffoli\n",
    "\n",
    "\n",
    "## Antecedente histórico\n",
    "\n",
    "Hace ya 40 años, un grupo revoltoso de 50 pensadores se organizaron para una foto sobre el césped de la casa del Endicott en el MIT. Algunos en la conferencia de la Física de la Computación organizada conjuntamente por el MIT y IBM pensaron que estaban haciendo historia. Era sin duda alguna el nacimiento de la física de la computación que ahora es conocida como un área de investigación en auge, conocida como computación cuántica que se ha convertido en tema digno de merecer libros y cursos universitarios.\n",
    "\n",
    "Fue en esta conferencia donde Feynman pronunció su famosa fraseÑ \"La naturaleza no es cuántica demonios, y si quieres hacer una simulación de la naturaleza más te vale hacerla mecanico cuántica, y por Dios que es un extraordinario problema, porque no parece fácil de resolver\".[1] Al comienzo de este mes, hemos celebrado el 40º aniversario de esta importante conferencia. Puedes leer más [aquí](https://youtu.be/GR6ANm6Z0yk).\n",
    "\n",
    "![](resources/conference-photo.jpeg)\n",
    "\n",
    "Uno de los temas discutidos en la conferencia fue sobre computación reversible, en el cual TOmmaso Toffoli y Edward Frekin en el MIT han estado pensando en los últimos años previos.[2-3] Toffoli se le ocurrio una versión reversible de las compuertas AND/NAND (ahora conocidas por \"compuerta Toffoli\" o \"compuerta controlled-controlled-NOT\"). Debido a que la compuerta NAND es universal en computación clásica, la compuerta Toffoli es también una compuerta lógica universal reversible). La Computación Cuántica es una forma especial de computación reversible, cualquier compuerta reversible puede se implementada en una computadora cuántica, y por ello la compuerta de Toffoli es también una compuerta lógica cuántica; sin embargo, la compuerta Toffoli solamente no es una compuerta universal para computación cuántica.\n",
    "\n",
    "En este ejercicio, exploraremos la compuerta de Toffoli y un conjuntos universales de compuertas para computadoras cuánticas.\n",
    "\n",
    "### Referencias (en inglés)\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."
   ]
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    "## Compuertas lógicas clásicas\n",
    "\n",
    "En la teoría de la computación clásica usualmente se usa el modelo de lógica Booleana or compuertas clásicas lógicas. Tales compuertas representan una funciones Booleanas, que son funciones con valores binarios (i.e., 1,0) de entrada/salida. Un aspecto interesante de la lógica Booleana es que todas las posibles funciones binarias pueden ser realizadas utilizando una combinación de un número reducido de compuertas lógicas distintas. \n",
    "\n",
    "Tales conjuntos son llamados conjuntos funcionalmente completos. Un conjunto famoso es el compuesto por las compuertas AND y NOT. Estas dos compuertas son suficientes para expresar todas las posibles funciones. Lo mismo es cierto para el conjunto OR y NOT. También existen conjuntos más pequeños tales como NAND y NOR, los cuales en sí mismos son universales, más sin embargo las compuertas AND, NOT y OR son usualmente vistos como los elementos fundamentales de la computación clásica.\n",
    "\n",
    "<div class=\"alert alert-block alert-success\">\n",
    "\n",
    "**Objetivo**\n",
    "\n",
    "Construir una compuerta de Toffoli utilizando el conjunto básico de compuertas {CX, RZ, SX y X} de IBM Quantum systems.\n",
    "\n",
    "</div>\n",
    "\n",
    "<div class=\"alert alert-block alert-danger\">\n",
    "    \n",
    "Este ejercicio tiene como propósito enseñarte los conceptos básicos de las compuertas cuánticas y como construir circuitos cuánticos:\n",
    "\n",
    "1. Visualmente, usando el widget del Compositor de Circuitos o por su nombre original en inglés  \"Circuit Composer widget\".\n",
    "1. Programáticamente, usando Qiskit. \n",
    "\n",
    "Si ya estas familiarizado con compuertas cuánticas y Qiskit, puedes saltar directamente a <a href=#problem>el problema</a>.\n",
    "</div>"
   ]
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  {
   "cell_type": "code",
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   "source": [
    "# Obviar advertencias innecesarias\n",
    "import warnings\n",
    "from matplotlib.cbook import MatplotlibDeprecationWarning\n",
    "warnings.filterwarnings('ignore', category=MatplotlibDeprecationWarning)\n",
    "\n",
    "# Importar librerias estandar de Qiskit\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",
    "# Asignar valor pi\n",
    "import math\n",
    "pi=math.pi"
   ]
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    "## ¿Qué son los circuitos cuánticos?\n",
    "\n",
    "Los Circuitos cuánticos es el modelo utilizado para la computación cuántica en el desarrollo de cálculos como una secuencia de compuertas cuánicas. Las compuertas cuánticas usualmente representan rotaciones del vector estado en la esfera de Bloch. Hagamos una revisión de algunas de las compuertas cuánticas más populares.\n",
    "\n",
    "### Compuerta X\n",
    "\n",
    "La compuerta X es representada por la matriz de Pauli-X: \n",
    "\n",
    "$X = \\begin{pmatrix}\n",
    "0 & 1 \\\\\n",
    "1 & 0 \\\\\n",
    "\\end{pmatrix}$\n",
    "\n",
    "Una compuerta X equivale a una rotación alrededor del eje-X de la esfera de Bloch por un ángulo de $\\pi$-radianes. Esta realiza un mapa de $|0\\rangle$ a $|1\\rangle$ y de $|1\\rangle$ a $|0\\rangle$. Esta compuerta es equivalente a la compuerta NOT de las computadoras clásicas y algunas veces es llamada \"bit-flip\"."
   ]
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   "source": [
    "x_gate=QuantumCircuit(1) # Crea un circuito cuantico con 1 \"qubit\"\n",
    "x_gate.x(0)\n",
    "x_gate.draw(output='mpl')"
   ]
  },
  {
   "cell_type": "code",
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   "source": [
    "backend = Aer.get_backend('statevector_simulator')\n",
    "result = execute(x_gate, backend).result().get_statevector()\n",
    "plot_bloch_multivector(result)"
   ]
  },
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   "source": [
    "### Compuerta SX\n",
    "\n",
    "La compuerta SX equivale a una rotación alrededor del eje-X en la esfera de Bloch por un ángulo de $\\pi/2$. Esta compuerta es llamada SX para indicar que es una raíz-cuadrada de la compuerta-X. Aplicando esta compuerta dos veces de manera consecutiva produce una compuerta estándar Pauli-X. La operación inversa a SX es SX daga (i.e., operación transpuesta conjugada), el cual es una rotación de $\\pi/2$ en dirección opuesta.\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,
   "id": "41012687",
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    "hidden": true
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   "source": [
    "sx_gate = QuantumCircuit(1)\n",
    "sx_gate.sx(0)  \n",
    "sx_gate.draw(output='mpl')"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "566763f5",
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   "source": [
    "backend = Aer.get_backend('statevector_simulator')\n",
    "result = execute(sx_gate, backend).result().get_statevector()\n",
    "plot_bloch_multivector(result)"
   ]
  },
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   "source": [
    "### RZ gate\n",
    "\n",
    "La compuerta Rz realiza una rotación de un ángulo de $\\phi$ alrededor del eje-Z, donde $\\phi$ es un número real. Se asocia a la siguiente matriz:\n",
    "\n",
    "$RZ = \\begin{pmatrix}\n",
    "1 & 0 \\\\\n",
    "0 & e ^{i \\phi } \\\\\n",
    "\\end{pmatrix}$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "92d8695e",
   "metadata": {
    "hidden": true
   },
   "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,
   "id": "c8b2ac74",
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   "source": [
    "backend = Aer.get_backend('statevector_simulator')\n",
    "result = execute(rz_gate, backend).result().get_statevector()\n",
    "plot_bloch_multivector(result)"
   ]
  },
  {
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   "source": [
    "Debido a que la rotación es alrededor del eje-Z, uno podría no observar alguna diferencia cuando se aplica a un estado inicial en $|0\\rangle$, de este modo usamos el estado el cual fue generado de la aplicación de una compuerta SX seguido de una compuerta RZ en su lugar."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "1464c920",
   "metadata": {
    "hidden": true
   },
   "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,
   "id": "a6571646",
   "metadata": {
    "hidden": true
   },
   "outputs": [],
   "source": [
    "backend = Aer.get_backend('statevector_simulator')\n",
    "result = execute(rz_gate, backend).result().get_statevector()\n",
    "plot_bloch_multivector(result)"
   ]
  },
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   "source": [
    "### Compuerta de Hadamard\n",
    "\n",
    "La compuerta de Hadamard representa una rotación de $\\pi$ alrededor del eje en el medio entre el eje-X y el eje-Z. Esta realiza un mapa entre los autoestado $|0\\rangle$ al estado $\\frac{|0\\rangle + |1\\rangle}{\\sqrt{2}}$, que significa que la medida tendrá igual probabilidad de estar en `1` o `0`, creando una 'superposición' de estados. Este estado puede ser también escrito como $|+\\rangle$. En otras palabras, lo que la compuerta de Hadamard hace es transformar entre la base de estados $\\lbrace|0\\rangle,\\, |1\\rangle \\rbrace$ y la base de estados $\\lbrace|+\\rangle,\\, |-\\rangle\\rbrace$. \n",
    "\n",
    "$H = \\frac{1}{\\sqrt{2}}\\begin{pmatrix}\n",
    "1 & 1 \\\\\n",
    "1 & -1 \\\\\n",
    "\\end{pmatrix}$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
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   "source": [
    "# Hagamos una compuerta-H sobre el |0> qubit\n",
    "h_gate = QuantumCircuit(1)\n",
    "h_gate.h(0)\n",
    "h_gate.draw(output='mpl')"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "03a84f6d",
   "metadata": {
    "hidden": true
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   "outputs": [],
   "source": [
    "# Veamos el resultado\n",
    "backend = Aer.get_backend('statevector_simulator')\n",
    "result = execute(h_gate, backend).result().get_statevector()\n",
    "plot_bloch_multivector(result)"
   ]
  },
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   "cell_type": "markdown",
   "id": "0f1174ca",
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   "source": [
    "### Compuerta CX (compuerta CNOT)\n",
    "\n",
    "La compuerta control-NOT (o CNOT o CX) actúa sobre 2 qubits. Realiza la operación NOT (equivalente a aplicar una compuerta X) en el segundo qubit, solamente cuando el primer qubit esta en el estado $|1\\rangle$ y sino lo deja sin alterar en caso contrario. \n",
    "\n",
    "Nota: Qiskit enumera los bits en una cadena de caracteres de derecha a izquierda.\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,
   "id": "3d515e6a",
   "metadata": {
    "hidden": true
   },
   "outputs": [],
   "source": [
    "cx_gate = QuantumCircuit(2)\n",
    "cx_gate.cx(0,1)\n",
    "cx_gate.draw(output='mpl')"
   ]
  },
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   "source": [
    "### Compuerta CCX (compuerta Toffoli)\n",
    "\n",
    "La compuerta CCX (compuerta control-control X) es también conocida como compuerta Toffoli. La compuerta CCX es una compuerta de tres qubits, con dos qubit-control y un qubit target como su entrada/salida. Si los primero dos qubits estan en el estado $|1\\rangle$, entonces se aplicará una compuerta Pauli-X (o NOT) en el tercer qubit, de lo contrario, nada ocurre. \n",
    "\n",
    "Nota: Qiskit enumera los bits en una cadena de caracteres de derecha a izquierda.\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,
   "id": "7f1d30a0",
   "metadata": {
    "hidden": true
   },
   "outputs": [],
   "source": [
    "ccx_gate = QuantumCircuit(3)\n",
    "ccx_gate.ccx(0,1,2)\n",
    "ccx_gate.draw(output='mpl')"
   ]
  },
  {
   "cell_type": "markdown",
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   "source": [
    "## Crear compuertas lógicas con ayuda de compuertas cuánticas\n",
    "\n",
    "### Compuerta NOT\n",
    "\n",
    "La compuerta NOT cambia el valor de un bit como hemos mencionado anteriormente, una compuerta X puede ser considerada una compuerta NOT. La table de la verdad para compuertas NOT se parece a esta:\n",
    "\n",
    "| Entrada | Salida |\n",
    "| --- | --- | \n",
    "| 1 | 0 |\n",
    "| 0 | 1 |"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "af86e78a",
   "metadata": {
    "hidden": true
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   "outputs": [],
   "source": [
    "not_gate=QuantumCircuit(1,1) # Crea un circuito cuantico con 1 qubit y 1 bit clasico\n",
    "not_gate.x(0)\n",
    "not_gate.measure(0,0)\n",
    "not_gate.draw(output='mpl')"
   ]
  },
  {
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   "source": [
    "### Compuerta AND\n",
    "\n",
    "La salida de una compuerta AND es \"Verdadera\" (\"True\" en inglés) si y solo si ambas entradas son verdaderas. La tabla de la verdad para una compuerta AND se parece a la siguiente:\n",
    "\n",
    "| A (Entrada) | B (Entrada) | Salida |\n",
    "| --- | --- | --- |\n",
    "| 0 | 0 | 0 | \n",
    "| 0 | 1 | 0 |\n",
    "| 1 | 0 | 0 |\n",
    "| 1 | 1 | 1 |\n",
    "\n",
    "Con una compuerta Toffoli podemos obtener el resultado de una compuerta AND, interpretando los dos bits de control como bits de entrada y el bit de destino como bit de salida."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "7adf8fd7",
   "metadata": {
    "hidden": true
   },
   "outputs": [],
   "source": [
    "and_gate=QuantumCircuit(3,1) # Crea un circuito cuantico con 3 qubits y 1 bit clasico\n",
    "and_gate.ccx(0,1,2)\n",
    "and_gate.measure(2,0)\n",
    "and_gate.draw(output='mpl')"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "3f099e6a",
   "metadata": {
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   "source": [
    "### Compuerta OR\n",
    "\n",
    "Una compuerta OR devuelve un valor de verdadero, si al menos uno de los bits de entrada es verdadero. La tabla de la verdad para la compuerta OR es la siguiente:\n",
    "\n",
    "| A (Entrada) | B (Entrada) | Salida |\n",
    "| --- | --- | --- |\n",
    "| 0 | 0 | 0 | \n",
    "| 0 | 1 | 1 |\n",
    "| 1 | 0 | 1 |\n",
    "| 1 | 1 | 1 |"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "e352c485",
   "metadata": {
    "hidden": true,
    "scrolled": true
   },
   "outputs": [],
   "source": [
    "or_gate=QuantumCircuit(3,1) # Crea un circuito cuantico con 3 qubits y 1 bit clasico\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",
   "id": "f70d7694",
   "metadata": {},
   "source": [
    "## Usando el widget de Compositor de Circuito (Circuit Composer widget)\n",
    "\n",
    "Quizá ya estas familiarizado con el Compositor de circuitos de IBM (IBM quantum circuit composer), donde puedes generar circuitos usando interfaces gráficas. Con el nuevo widget de Compositor de Circuitos, se puede utilizar la misma funcionalidad en los cuadernos de Jupyter (jupyter notebooks). Puedes aprendar más al respecto en como usar el widget [aquí](https://quantum-computing.ibm.com/lab/docs/iql/composer-widget)\n",
    "\n",
    "<div class=\"alert alert-block alert-success\">\n",
    "\n",
    "**Ejercicio 1a:** Construir una compuerta NOR (una compuerta OR negada) utilizando el compositor de circuitos.\n",
    "\n",
    "Este ejercicio tiene la intención invitarte a jugar un poco con el compositor. Esta parte no es evaluada. Puedes saltarte esta sección si así lo crees. Ejecutando la celda debajo lograrás mostrar el compositor, donde puedes añadir compuertas arrastrándolas y soltándolas con el puntero en los lugares correctos.   \n",
    "</div>"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "e344074c",
   "metadata": {},
   "outputs": [],
   "source": [
    "from ibm_quantum_widgets import CircuitComposer\n",
    "editor = CircuitComposer()\n",
    "editor"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "75eba4f7",
   "metadata": {},
   "source": [
    "### Usa el widget de Compositor de Circuito para trabajar con un circuito ya existente\n",
    "\n",
    "También podemos utilizar el Compositor de Circuito para abir un circuito creado previamente. En la celda inferior, el código abre el circuito creado en la celda superior, que representa la compuerta OR. Puedes usar esto para comprobar si construiste tu compuerta OR correctamente.\n",
    "\n",
    "<div class=\"alert alert-block alert-success\">\n",
    "\n",
    "Puedes editar el circuito abierto arrastrando y soltando con el puntero. Intenta elimnar las medidas al final del circuito.\n",
    "    \n",
    "</div>"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "2208b6dc",
   "metadata": {
    "scrolled": true
   },
   "outputs": [],
   "source": [
    "from ibm_quantum_widgets import CircuitComposer\n",
    "editor2 = CircuitComposer(circuit=or_gate)\n",
    "editor2"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "5269a0f3",
   "metadata": {},
   "source": [
    "En este ejemplo debajo, guardamos el último circuito del compositor en la variable ```qc2``` y luego aplicamos una compuerta X a la salida y medimos nuevamente.\n",
    "\n",
    "Usa el circuito de la última edición y aplica X a la salida y mide nuevamente."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "e4c607f9",
   "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",
   "id": "21c843ac",
   "metadata": {},
   "source": [
    "Ahora hemos creado una compuerta NOR, que es una compuerta OR negada. Debería ser identica al circuito que construiste en el primer ejercicio.\n",
    "\n",
    "## Compuertas cuánticas compuestas y su costo\n",
    "\n",
    "Normalmente la computadoras cuánticas reales no tienen implementaciones físicas de todas las compuertas. En su lugar ellas usan un pequeño conjunto de compuertas bases, las cuales forman un conjunto universal de compuertas, esto es parecido al caso clásico donde un conjunto de instrucciones puede ser utilizado para todas las posibles operaciones.\n",
    "\n",
    "Por esta razón, los circuitos cuánticos deben ser \"transpilados\" (transpiled en inglés) a una base de compuertas antes de su ejecución. Esto es usualmente hecho automáticamente por el transpilador de Qiskit (Qiskit transpiler en inglés) cuando un circuito cuántico es enviado a los sistemas cuánticos de IBM. Pero, por razones pedagógicas, se espera que construyas un circuito a mano usando las compuertas básicas. Las compuertas básicas de IBM Quantum System son típicamente las compuertas. Puedes ver más en: [`ibmq_mumbai` system](https://quantum-computing.ibm.com/services?skip=0&systems=all&system=ibmq_mumbai) como un ejemplo.\n",
    "\n",
    "Veamos ahora el circuito a continuación:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "8b40d2d5",
   "metadata": {},
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   "source": [
    "qc = QuantumCircuit(2)\n",
    "qc.sxdg(0)\n",
    "qc.t(1)\n",
    "qc.draw(output='mpl')"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "5f5dd6ad",
   "metadata": {},
   "source": [
    "Mostremos ahora la descomposición del circuito anterior por una computadora cuántica, usando solamente las compuertas básicas."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "f4f8c736",
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   "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",
   "id": "46d66f1c",
   "metadata": {},
   "source": [
    "Como puedes ver, ahora solo hemos utilizado las compuertas básicas; sin embargo, por esta razón se han utilizado más compuertas en total. Entre más compuertas tenga un circuito, más complejo será ejecutarlo. Entonces, cuando calculemos el costo del circuito, vamos a considerar el número de compuertas utilizadas. Pero no todos las compuertas son consideradas igual de costosas, de modo que cuando calculamos el costo del circuito, vamos a usar la siguiente fórmula:\n",
    "\n",
    "$$\n",
    "\\text{Cost} = 10 N_{\\text{CNOT}} + N_{\\text{otras}}\n",
    "$$\n",
    "\n",
    "donde $N_{\\text{CNOT}}$ es el número de compuertas CNOT y $N_{\\text{otras}}$ es el número de otras compuertas.\n",
    "\n",
    "### Compuerta Hadamard\n",
    "\n",
    "Como se ha mencioando, todas las operaciones pueden ser expresadas solamente usando las compuertas básicas. Como un ejemplo, hemos mostrado como construir una compuerta Hadamard utilizando nuestro conjunto de compuertas básicas. No tenemos una compuerta básica que haga una rotación directa alrededor del eje que esta en el medio entre el eje-X y el eje-Z, en su lugar usamos rotaciones alrededor del eje-X y el eje-Z para obtener el mismo resultado.\n",
    "\n",
    "Puedes adivinar que rotaciones necesitamos hacer?"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "5837a416",
   "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",
   "id": "90670e1a",
   "metadata": {},
   "source": [
    "Como podrás recordar, este es el circuito que teníamos arriba cuando visualizamos la rotación de la compuerta RZ. Anteriormente, observamos que la primera compuerta RZ no tiene ningún efecto cuando estamos en el estado $|0\\rangle$ o $|1\\rangle$. Por lo que se siente que no tiene utilidad. Sin embargo, si estamos en los estados $|+\\rangle$ and $|-\\rangle$ la primera rotación tiene un efecto. Tenemos el escenario contrario, ya que luego de aplicar la compuerta SX, estamos nuevamente en los estados $|0\\rangle$ o $|1\\rangle$ y así la segunda compuerta RZ no tiene efecto.\n",
    "\n",
    "### Rotación Controlada\n",
    "\n",
    "Previamente hemos visto la compuerta control-NOT, ahora veamos un ejemplo sobre como podemos construir una rotación controlada alrededor del eje-Y. La rotación de un ángulo $\\theta$ puede ser cualquier rotación, no necesita ser necesariamente $\\pi$, esta elección es a modo de ejemplo."
   ]
  },
  {
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   "source": [
    "qc = QuantumCircuit(2)\n",
    "theta = pi # El angulo theta puede tiener cualquier valor (se escogio arbitrariamente pi)\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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   "cell_type": "markdown",
   "id": "b5a21c4a",
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   "source": [
    "Cuando uno revisa este circuito, uno puede observar que si el primer qubit es 0 entonces las dos rotaciones se cancelan entre ellas y nada ocurre.\n",
    "\n",
    "Por otro lado, si el primer qubit es 1 vamos a obtener un estado que es igual a aplicar la rotación $\\theta / 2$ dos veces que forma nuestra rotación inicial de $\\theta$. Esto funciones ya que el eje-X y el eje-Y son ortogonales.\n",
    "\n",
    "<div class=\"alert alert-block alert-danger\">\n",
    "Para rotar alrededor de otros ejes quizás necesites usar otros trucos.\n",
    "</div>\n",
    "\n",
    "### Rotación controlada-controlada\n",
    "\n",
    "Arriba hemos visto como ejemplos sobre como podemos hacer una rotación controlada alrededor del eje-Y.\n",
    "\n",
    "Ahora vamos a asumir que tenemos una rotación controlada (alrededor del eje que queremos) y queremos construir desde ello una rotación doblemente controlada, que es solamente aplicada si ambos qubits de control son idénticos a 1, del mismo modo que ocurre con la compuerta CCX."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
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   "source": [
    "qc = QuantumCircuit(3)\n",
    "theta = pi # El angulo theta puede tiener cualquier valor (se escogio arbitrariamente pi)\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",
   "id": "3699c2ef",
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   "source": [
    "En este circuito si ambos el primero y segundo qubit son 0, entonces nada ocurre. Si solamente el segundo qubit es uno, primero aplicamos una rotación de $\\pi/2$ y posteiormente una rotación de $-\\pi/2$ que se cancelan conjuntamente. Si solamente el primer qubit es 1, entonces tenemos que el segundo qubit va también a ser uno posteriormente de la primera compuerta CX. Así una rotación de $-\\pi/2$ será aplicada y por último una rotación de $\\pi/2$ que las cancelará nuevamente.\n",
    "\n",
    "Si ambas, el primer y segundo qubit son 1, entonces primero habrá una rotación por $\\pi/2$ luego el segundo qubit pasará a ser 0 de modo que la siguiente rotación no se aplique y sea cambiado de vuelta a 1. Finalmente, otra rotación de $\\pi/2$ es aplicada ya que el primer qubit es 1. De ese modo tenemos dos veces la rotación de $\\pi/2$ que forma conjuntamente una rotación de $\\pi$."
   ]
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   "source": [
    "## El problema\n",
    "\n",
    "<div id='problem'></div>\n",
    "<div class=\"alert alert-block alert-success\">\n",
    "\n",
    "Hemos visto anteriormente como construir una compuerta de Hadamard empleando un conjunto base de compuertas. Ahora queremos construir igualmente una compuerta de Toffoli. ¿Porqué una compuerta de Toffoli? Como mencionamos anteriormente, la compuerta de Toffoli es tambien una compuerta universal para compuertas clásicas en la misma manera que lo es la compuerta NAND, pero con la diferencia de que es reversible. Más aún, esta construye un conjunto universal de compuertas para computación cuántica si es combinada con una compuerta de Hadamard.\n",
    "    \n",
    "Adicionalmente, hemos visto algunos ejemplos sobre como crear compuertas cuánticas más complejas utilizando compuertas básicas. Ahora queremos utilizar este conocimiento para construir una compuerta de Toffoli solamente usando nuestras compuertas cuánticas básicas. Con la finalidad de resolver este ejercicio, los ejemplos anteriores sobre como construir y usar rotaciones controladas serán de utilidad. El mayor reto es el de construir la rotaciones controladas necesarias.\n",
    "    \n",
    "Puedes usar el código en la parte inferior para emplear el widget del compositor y así crear tu circuito.    \n",
    "</div>\n",
    "\n",
    "\n",
    "<div class=\"alert alert-block alert-danger\">\n",
    "\n",
    "Como recordatorio, las compuertas básicas para los IBM Quantum systems son las compuertas CX, RZ, SX y X; así pues, otras compuertas no están permitidas.\n",
    "\n",
    "Por supuesto, también queremos intentar minimizar el costo.\n",
    "    \n",
    "$$\n",
    "\\text{Costo} = 10 N_{\\text{CNOT} } + N_{\\text{ otros}}\n",
    "$$\n",
    "    \n",
    "</div>\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "5fb3baca",
   "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",
    "##### Construir tu circuito cuantico aquí usando el widget del compositor."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "18cd2b57",
   "metadata": {},
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   "source": [
    "# Tambien puedes construir tu circuito programaticamente usando codigo de Qiskit\n",
    "\n",
    "circuit = QuantumCircuit(3)\n",
    "\n",
    "# ESCRIBE TU CODIGO ENTRE ESTAS LINEAS - COMIENZO\n",
    "\n",
    "\n",
    "\n",
    "\n",
    "\n",
    "# ESCRIBE TU CODIGO ENTRE ESTAS LINEAS - FIN"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "7ee8b864",
   "metadata": {},
   "outputs": [],
   "source": [
    "# Ejecuta el circuito utilizando el qasm_simulator\n",
    "qc = editorEx.circuit \n",
    "#qc = circuit # Descomenta esta linea si quieres enviar el circuito construido usando codigo de Qiskit\n",
    "\n",
    "qc.draw(output='mpl')"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "288fabba",
   "metadata": {},
   "outputs": [],
   "source": [
    "# Revisa tu respuesta usando el siguiente codigo\n",
    "from qc_grader import grade_ex1\n",
    "grade_ex1(qc)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "2e7d551a",
   "metadata": {},
   "outputs": [],
   "source": [
    "# Envia tus respuestas. Puedes re-enviar tus respuestas en cualquier momento.\n",
    "from qc_grader import submit_ex1\n",
    "submit_ex1(qc)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "58519aa3",
   "metadata": {
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   "source": [
    "## Additional information\n",
    "\n",
    "**Created by:** Marcel Pfaffhauser, Brahmani Thota, Junye Huang\n",
    "\n",
    "**Latin-American spanish translation:** Manuel Morgado, Mauricio Gómez Viloria\n",
    "\n",
    "**Version:** 1.0.0"
   ]
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