{ "cells": [ { "attachments": {}, "cell_type": "markdown", "id": "a94f5c51-348c-4fb0-9488-70ad4a19ace0", "metadata": {}, "source": [ "---\n", "title: \"Simulación cuántica\"\n", "description: \"Este curso trata sobre la simulación cuántica, incluida la trotterización del hamiltoniano de Ising de campo transversal\"\n", "---\n", "\n", "{/* cspell:ignore sharex */}\n", "\n", "\n", "\n", "# Simulación cuántica\n", "\n", "\n", " Yukio Kawashima (30 de mayo de 2024)\n", "\n", " [Descargue el pdf](https://ibm.ent.box.com/s/kzzsmxhw38vph1ioohczaet53euwi310) de la conferencia original. Tenga en cuenta que algunos fragmentos de código pueden quedar obsoletos, ya que se trata de imágenes estáticas.\n", "\n", " *El tiempo aproximado de QPU para ejecutar este experimento es de 7 segundos.*\n", "\n", " (Este cuaderno está tomado en su mayor parte de un [cuaderno tutorial](https://github.com/qiskit-community/qiskit-algorithms/blob/main/docs/tutorials/13_trotterQRTE.ipynb) de Algoritmos Qiskit, ahora obsoleto)\n", "\n", "\n" ] }, { "cell_type": "markdown", "id": "aef7b02a-2b09-4409-85a0-8cf8346df430", "metadata": {}, "source": [ "\n", "\n", "## 1. Introducción\n", "\n", "Como técnica de evolución en tiempo real, la trotterización consiste en la aplicación sucesiva de una o varias compuertas cuánticas, elegidas para aproximar la evolución temporal de un sistema para una franja de tiempo. Siguiendo la ecuación de Schrödinger, la evolución temporal de un sistema inicialmente en el estado $\\vert\\psi(0)\\rangle$ toma la forma:\n", "\n", "$$\n", "\\vert \\psi(t) \\rangle = e^{-i H t} \\vert \\psi(0) \\rangle \\text{,}\n", "$$\n", "\n", "donde $H$ es el Hamiltoniano independiente del tiempo que gobierna el sistema. Consideramos un Hamiltoniano que puede escribirse como una suma ponderada de términos de Pauli $H=\\sum_j a_j P_j$, con $P_j$ representando un producto tensorial de términos de Pauli actuando sobre $n$ qubits. En particular, estos términos de Pauli pueden conmutar entre sí o no. Dado un estado en el tiempo $t=0$, ¿cómo obtenemos el estado del sistema en un tiempo posterior $|\\psi(t)\\rangle$ utilizando un ordenador cuántico? La exponencial de un operador se puede entender más fácilmente a través de su serie de Taylor:\n", "\n", "$$\n", "e^{-i H t} = 1-iHt-\\frac{1}{2}H^2t^2+...\n", "$$\n", "\n", "Algunos exponenciales muy básicos, como $e^{iZ}$, pueden implementarse fácilmente en ordenadores cuánticos utilizando un conjunto compacto de puertas cuánticas. La mayoría de los Hamiltonianos de interés no tendrán un solo término, sino que tendrán muchos términos. Observe lo que ocurre si $H = H_1+H_2$ :\n", "\n", "$$\n", "e^{-i H t} = 1-i(H_1+H_2)t-\\frac{1}{2}(H_1+H_2)^2t^2+...\n", "$$\n", "\n", "Cuando $H_1$ y $H_2$ conmutan, tenemos el caso familiar (que también es cierto para los números, y las variables $a$ y $b$ a continuación):\n", "\n", "$$\n", "e^{-i (a+b) t} = e^{-i a t}e^{-i b t}\n", "$$\n", "\n", "Pero cuando los operadores no conmutan, los términos no se pueden reordenar en la serie de Taylor para simplificar de esta manera. Por tanto, expresar hamiltonianos complicados en puertas cuánticas es todo un reto.\n", "\n", "Una solución es considerar un tiempo muy pequeño $t$, de forma que domine el término de primer orden de la expansión de Taylor. Bajo ese supuesto:\n", "\n", "$$\n", "e^{-i (H_1+H_2) t} \\approx 1-i(H_1+H_2)t \\approx (1-i H_1 t)(1-i H_2 t) \\approx e^{-i H_1 t}e^{-i H_2 t}\n", "$$\n", "\n", "Por supuesto, puede que necesitemos evolucionar nuestro estado durante más tiempo. Eso se consigue dando muchos pequeños pasos en el tiempo. Este proceso se denomina trotterización:\n", "\n", "$$\n", "\\vert \\psi(t) \\rangle \\approx \\left(\\prod_j e^{-i a_j P_j t/r} \\right)^r \\vert\\psi(0) \\rangle \\text{,}\n", "$$\n", "\n", "Aquí $t/r$ es la franja de tiempo (paso de evolución) que estamos eligiendo. Como resultado, se crea una puerta que se aplicará $r$ veces. Un paso de tiempo más pequeño conduce a una aproximación más precisa. Sin embargo, esto también conduce a circuitos más profundos que, en la práctica, provocan una mayor acumulación de errores (una preocupación no desdeñable en los dispositivos cuánticos a corto plazo).\n", "\n", "Hoy estudiaremos la evolución temporal del [modelo de Ising](https://en.wikipedia.org/wiki/Ising_model) en celosías lineales de $N=2$ y $N=6$ sitios. Estas celosías consisten en una matriz de espines $\\sigma_i$ que interactúan sólo con sus vecinos más cercanos. Estos espines pueden tener dos orientaciones: $\\uparrow$ y $\\downarrow$, que corresponden a una magnetización de $+1$ y $-1$ respectivamente.\n", "\n", "$$\n", "H = - J \\sum_{i=0}^{N-2} Z_i Z_{i+1} - h \\sum_{i=0}^{N-1} X_i \\text{,}\n", "$$\n", "\n", "donde $J$ describe la energía de interacción, y $h$ la magnitud de un campo externo (en la dirección x anterior, pero modificaremos esto). Escribamos esta expresión utilizando matrices de Pauli, y considerando que el campo externo tiene un ángulo $\\alpha$ con respecto a la dirección transversal,\n", "\n", "$$\n", "H = -J \\sum_{i=0}^{N-2} Z_i Z_{i+1} -h \\sum_{i=0}^{N-1} (\\sin\\alpha Z_i + \\cos\\alpha X_i) \\text{.}\n", "$$\n", "\n", "Este Hamiltoniano es útil porque nos permite estudiar fácilmente los efectos de un campo externo. En la base computacional, el sistema se codificará como sigue:\n", "\n", "| Estado cuántico | Representación giratoria |\n", "| :----------------------: | :----------------------------------------: |\n", "| $\\lvert 0 0 0 0 \\rangle$ | $\\uparrow\\uparrow\\uparrow\\uparrow$ |\n", "| $\\lvert 1 0 0 0 \\rangle$ | $\\downarrow\\uparrow\\uparrow\\uparrow$ |\n", "| $\\ldots$ | $\\ldots$ |\n", "| $\\lvert 1 1 1 1 \\rangle$ | $\\downarrow\\downarrow\\downarrow\\downarrow$ |\n", "\n", "Empezaremos investigando la evolución temporal de un sistema cuántico de este tipo. Más concretamente, visualizaremos la evolución temporal de ciertas propiedades del sistema, como la magnetización.\n", "\n" ] }, { "cell_type": "markdown", "id": "8d26b146-12b4-4882-a0c8-3f7b5b15f04d", "metadata": {}, "source": [ "\n", "\n", "### 1.1 Requisitos\n", "\n", "Antes de empezar este tutorial, asegúrate de tener instalado lo siguiente:\n", "\n", "* Qiskit SDK v1.2 o posterior ( `pip install qiskit` )\n", "* Qiskit Runtime v0.30 o posterior ( `pip install qiskit-ibm-runtime` )\n", "* Numpy v1.24.1 o posterior \\< 2 ( `pip install numpy` )\n", "\n" ] }, { "cell_type": "markdown", "id": "e73c8a4a-a540-4637-b179-670da6b1a6e6", "metadata": {}, "source": [ "\n", "\n", "### 1.2 Importar las bibliotecas\n", "\n", "Observe que se incluyen algunas bibliotecas que podrían ser útiles ( MatrixExponential, QDrift) aunque no se utilicen en este cuaderno actual. Puede probarlos si tiene tiempo\n", "\n" ] }, { "cell_type": "code", "execution_count": 1, "id": "2edce0da-4b37-411f-ac13-191590d99e1b", "metadata": {}, "outputs": [ { "data": { "text/plain": [ "'2.0.2'" ] }, "execution_count": 1, "metadata": {}, "output_type": "execute_result" } ], "source": [ "# Check the version of Qiskit\n", "import qiskit\n", "\n", "qiskit.__version__" ] }, { "cell_type": "code", "execution_count": 2, "id": "803f9fab-abab-482e-b8f5-90d817f2452f", "metadata": { "scrolled": true, "tags": [] }, "outputs": [], "source": [ "# Import the qiskit library\n", "import numpy as np\n", "import matplotlib.pylab as plt\n", "import warnings\n", "\n", "from qiskit import QuantumCircuit\n", "from qiskit.circuit.library import PauliEvolutionGate\n", "from qiskit.primitives import StatevectorEstimator\n", "from qiskit.quantum_info import Statevector, SparsePauliOp\n", "from qiskit.synthesis import (\n", " SuzukiTrotter,\n", " LieTrotter,\n", ")\n", "from qiskit.transpiler.preset_passmanagers import generate_preset_pass_manager\n", "\n", "from qiskit_ibm_runtime import QiskitRuntimeService, SamplerV2\n", "\n", "warnings.filterwarnings(\"ignore\")" ] }, { "cell_type": "markdown", "id": "64dbeb76-bb61-469d-bdce-65f4b5cb65fb", "metadata": {}, "source": [ "\n", "\n", "## 2. Identificar el problema\n", "\n", "\n", "\n", "### 2.1 Definición del hamiltoniano de Ising de campo transversal\n", "\n", "Consideramos aquí el modelo de Ising de campo transversal 1-D.\n", "\n", "En primer lugar, crearemos una función que reciba los parámetros del sistema $N$, $J$, $h$ y $\\alpha$, y devuelva nuestro Hamiltoniano como `SparsePauliOp`. A [SparsePauliOp](/docs/api/qiskit/qiskit.quantum_info.SparsePauliOp) es una representación dispersa de un operador en términos de términos de [Pauli](/docs/api/qiskit/qiskit.quantum_info.Pauli) ponderados.\n", "\n" ] }, { "cell_type": "code", "execution_count": 3, "id": "440e0b22-d221-485c-845f-9fefa1d10ae6", "metadata": {}, "outputs": [], "source": [ "def get_hamiltonian(nqubits, J, h, alpha):\n", " # List of Hamiltonian terms as 3-tuples containing\n", " # (1) the Pauli string,\n", " # (2) the qubit indices corresponding to the Pauli string,\n", " # (3) the coefficient.\n", " ZZ_tuples = [(\"ZZ\", [i, i + 1], -J) for i in range(0, nqubits - 1)]\n", " Z_tuples = [(\"Z\", [i], -h * np.sin(alpha)) for i in range(0, nqubits)]\n", " X_tuples = [(\"X\", [i], -h * np.cos(alpha)) for i in range(0, nqubits)]\n", "\n", " # We create the Hamiltonian as a SparsePauliOp, via the method\n", " # `from_sparse_list`, and multiply by the interaction term.\n", " hamiltonian = SparsePauliOp.from_sparse_list(\n", " [*ZZ_tuples, *Z_tuples, *X_tuples], num_qubits=nqubits\n", " )\n", " return hamiltonian.simplify()" ] }, { "cell_type": "markdown", "id": "b56952ca-f354-4719-8bd9-674f71f681a4", "metadata": {}, "source": [ "\n", "\n", "#### Definir el hamiltoniano\n", "\n", "El sistema que ahora consideramos tiene un tamaño de $N=6$, $J=0.2$, $h=1.2$ y $\\alpha=\\frac{\\pi}{8.0}$ como ejemplo.\n", "\n" ] }, { "cell_type": "code", "execution_count": 4, "id": "0eca59f6-8aec-4c58-a22b-9a237b5787b8", "metadata": {}, "outputs": [ { "data": { "text/plain": [ "SparsePauliOp(['IIIIZZ', 'IIIZZI', 'IIZZII', 'IZZIII', 'ZZIIII', 'IIIIIZ', 'IIIIZI', 'IIIZII', 'IIZIII', 'IZIIII', 'ZIIIII', 'IIIIIX', 'IIIIXI', 'IIIXII', 'IIXIII', 'IXIIII', 'XIIIII'],\n", " coeffs=[-0.2 +0.j, -0.2 +0.j, -0.2 +0.j, -0.2 +0.j,\n", " -0.2 +0.j, -0.45922012+0.j, -0.45922012+0.j, -0.45922012+0.j,\n", " -0.45922012+0.j, -0.45922012+0.j, -0.45922012+0.j, -1.10865544+0.j,\n", " -1.10865544+0.j, -1.10865544+0.j, -1.10865544+0.j, -1.10865544+0.j,\n", " -1.10865544+0.j])" ] }, "execution_count": 4, "metadata": {}, "output_type": "execute_result" } ], "source": [ "n_qubits = 6\n", "\n", "hamiltonian = get_hamiltonian(nqubits=n_qubits, J=0.2, h=1.2, alpha=np.pi / 8.0)\n", "hamiltonian" ] }, { "cell_type": "markdown", "id": "5a9fa555-e4ad-4e4c-acc1-bd7302b9361e", "metadata": {}, "source": [ "\n", "\n", "### 2.2 Establezca los parámetros de la simulación de evolución temporal\n", "\n", "Aquí consideraremos tres técnicas diferentes de trotterización:\n", "\n", "* Lie-Trotter (primer orden)\n", "* suzuki-Trotter de segundo orden\n", "* suzuki-Trotter de cuarto orden\n", "\n", "Los dos últimos se utilizarán en el ejercicio y en el apéndice.\n", "\n" ] }, { "cell_type": "code", "execution_count": 5, "id": "9749e548-3238-4d0a-bcbf-c7b6699942d3", "metadata": {}, "outputs": [], "source": [ "num_timesteps = 60\n", "evolution_time = 30.0\n", "dt = evolution_time / num_timesteps\n", "product_formula_lt = LieTrotter()\n", "product_formula_st2 = SuzukiTrotter(order=2)\n", "product_formula_st4 = SuzukiTrotter(order=4)" ] }, { "cell_type": "markdown", "id": "d910b0e0-5e95-4899-b433-d13710eb4ada", "metadata": {}, "source": [ "\n", "\n", "### 2.3 Prepare el circuito cuántico 1 (estado inicial)\n", "\n", "Crear un estado inicial. Aquí empezaremos con una configuración de espín de $\\uparrow\\uparrow\\downarrow\\downarrow\\uparrow\\uparrow$.\n", "\n" ] }, { "cell_type": "code", "execution_count": 6, "id": "c9d78393-1e0c-46bb-b909-ada0cadfd59d", "metadata": { "scrolled": true, "tags": [] }, "outputs": [ { "data": { "text/plain": [ "\"Output" ] }, "execution_count": 6, "metadata": {}, "output_type": "execute_result" } ], "source": [ "initial_circuit = QuantumCircuit(n_qubits)\n", "initial_circuit.prepare_state(\"001100\")\n", "# Change reps and see the difference when you decompose the circuit\n", "initial_circuit.decompose(reps=1).draw(\"mpl\")" ] }, { "cell_type": "markdown", "id": "2e31b0bd-ebbb-4e74-b95d-a3ae2e3fff62", "metadata": {}, "source": [ "\n", "\n", "### 2.4 Prepare el circuito cuántico 2 (circuito único para la evolución temporal)\n", "\n", "Aquí construimos un circuito para un solo paso de tiempo utilizando Lie-Trotter.\n", "\n", "La fórmula del producto de Lie (de primer orden) se implementa en la clase [LieTrotter](/docs/api/qiskit/qiskit.synthesis.LieTrotter) clase Una fórmula de primer orden consiste en la aproximación indicada en la introducción, donde la exponencial matricial de una suma se aproxima por un producto de exponenciales matriciales:\n", "\n", "$$\n", "e^{H_1+H_2} \\approx e^{H_1} e^{H_2}\n", "$$\n", "\n", "Como ya se ha dicho, los circuitos muy profundos conducen a la acumulación de errores y causan problemas a los ordenadores cuánticos modernos. Dado que las puertas de dos qubits tienen tasas de error más altas que las puertas de un qubit, una cantidad de particular interés es la profundidad del circuito de dos qubits. Lo que realmente importa es la profundidad del circuito de dos qubits después de la transpilación (ya que es el circuito que el ordenador cuántico ejecuta realmente). Pero acostumbrémonos a contar las operaciones de este circuito, incluso ahora utilizando el simulador.\n", "\n" ] }, { "cell_type": "code", "execution_count": 8, "id": "e1ce17c2-02f2-4270-b271-55661ca14112", "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "\n", "Trotter step with Lie-Trotter\n", "-----------------------------\n", "Depth: 17\n", "Gate count: 27\n", "Nonlocal gate count: 10\n", "Gate breakdown: U3: 12, CX: 10, U1: 5\n", "\n" ] }, { "data": { "text/plain": [ "\"Output" ] }, "execution_count": 8, "metadata": {}, "output_type": "execute_result" } ], "source": [ "single_step_evolution_gates_lt = PauliEvolutionGate(\n", " hamiltonian, dt, synthesis=product_formula_lt\n", ")\n", "single_step_evolution_lt = QuantumCircuit(n_qubits)\n", "single_step_evolution_lt.append(\n", " single_step_evolution_gates_lt, single_step_evolution_lt.qubits\n", ")\n", "\n", "print(\n", " f\"\"\"\n", "Trotter step with Lie-Trotter\n", "-----------------------------\n", "Depth: {single_step_evolution_lt.decompose(reps=3).depth()}\n", "Gate count: {len(single_step_evolution_lt.decompose(reps=3))}\n", "Nonlocal gate count: {single_step_evolution_lt.decompose(reps=3).num_nonlocal_gates()}\n", "Gate breakdown: {\", \".join([f\"{k.upper()}: {v}\" for k, v in single_step_evolution_lt.decompose(reps=3).count_ops().items()])}\n", "\"\"\"\n", ")\n", "single_step_evolution_lt.decompose(reps=3).draw(\"mpl\", fold=-1)" ] }, { "cell_type": "markdown", "id": "6710865f-8e47-485f-bab8-e281d37bfbcd", "metadata": {}, "source": [ "\n", "\n", "### 2.5 Configure los operadores para medir\n", "\n", "Definamos un *operador de magnetización* $\\sum_i \\langle Z_i \\rangle / N$, y un *operador de correlación de espín medio* $\\sum_i \\langle Z_i Z_{i+1} \\rangle/ (N - 1)$.\n", "\n" ] }, { "cell_type": "code", "execution_count": 9, "id": "008188d1-047a-40b9-aaea-cf6baa42c434", "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "magnetization : SparsePauliOp(['IIIIIZ', 'IIIIZI', 'IIIZII', 'IIZIII', 'IZIIII', 'ZIIIII'],\n", " coeffs=[0.16666667+0.j, 0.16666667+0.j, 0.16666667+0.j, 0.16666667+0.j,\n", " 0.16666667+0.j, 0.16666667+0.j])\n", "correlation : SparsePauliOp(['IIIIZZ', 'IIIZZI', 'IIZZII', 'IZZIII', 'ZZIIII'],\n", " coeffs=[0.2+0.j, 0.2+0.j, 0.2+0.j, 0.2+0.j, 0.2+0.j])\n" ] } ], "source": [ "magnetization = (\n", " SparsePauliOp.from_sparse_list(\n", " [(\"Z\", [i], 1.0) for i in range(0, n_qubits)], num_qubits=n_qubits\n", " )\n", " / n_qubits\n", ")\n", "correlation = SparsePauliOp.from_sparse_list(\n", " [(\"ZZ\", [i, i + 1], 1.0) for i in range(0, n_qubits - 1)], num_qubits=n_qubits\n", ") / (n_qubits - 1)\n", "print(\"magnetization : \", magnetization)\n", "print(\"correlation : \", correlation)" ] }, { "cell_type": "markdown", "id": "9a54c9da-d0e6-423d-aa4f-be0808159fa3", "metadata": {}, "source": [ "\n", "\n", "### 2.6 Realizar simulación de evolución temporal\n", "\n", "Controlaremos la energía (valor de expectativa del Hamiltoniano), la magnetización (valor de expectativa del operador de magnetización) y la correlación media de espín (valor de expectativa del operador de correlación media de espín). La primitiva de Qiskit `StatevectorEstimator` ( EstimatorV2 ) estima los valores de expectativa de los observables, $\\langle\\psi\\vert\\hat{O}\\vert\\psi\\rangle$.\n", "\n" ] }, { "cell_type": "code", "execution_count": 10, "id": "0045114c-9f92-4e10-92d5-9cd478628781", "metadata": {}, "outputs": [], "source": [ "# Initiate the circuit\n", "evolved_state = QuantumCircuit(initial_circuit.num_qubits)\n", "# Start from the initial spin configuration\n", "evolved_state.append(initial_circuit, evolved_state.qubits)\n", "# Initiate Estimator (V2)\n", "estimator = StatevectorEstimator()\n", "# Set number of shots\n", "shots = 10000\n", "# Translate the precision required from the number of shots\n", "precision = np.sqrt(1 / shots)\n", "energy_list = []\n", "mag_list = []\n", "corr_list = []\n", "# Estimate expectation values for t=0.0\n", "job = estimator.run(\n", " [(evolved_state, [hamiltonian, magnetization, correlation])], precision=precision\n", ")\n", "# Get estimated expectation values\n", "evs = job.result()[0].data.evs\n", "energy_list.append(evs[0])\n", "mag_list.append(evs[1])\n", "corr_list.append(evs[2])\n", "# Start time evolution\n", "for n in range(num_timesteps):\n", " # Expand the circuit to describe delta-t\n", " evolved_state.append(single_step_evolution_gates_lt, evolved_state.qubits)\n", " # Estimate expectation values at delta-t\n", " job = estimator.run(\n", " [(evolved_state, [hamiltonian, magnetization, correlation])],\n", " precision=precision,\n", " )\n", " # Retrieve results (expectation values)\n", " evs = job.result()[0].data.evs\n", " energy_list.append(evs[0])\n", " mag_list.append(evs[1])\n", " corr_list.append(evs[2])\n", "# Transform the list of expectation values (at each time step) to arrays\n", "energy_array = np.array(energy_list)\n", "mag_array = np.array(mag_list)\n", "corr_array = np.array(corr_list)" ] }, { "cell_type": "markdown", "id": "91003bdc-0f5d-47cb-b664-2a483062a7ac", "metadata": {}, "source": [ "\n", "\n", "### 2.7 Trazar la evolución temporal de los observables\n", "\n", "Trazamos los valores de las expectativas que hemos medido en función del tiempo.\n", "\n" ] }, { "cell_type": "code", "execution_count": 11, "id": "3e3cd58b-0705-4cdf-a63b-01d1e3f2315b", "metadata": {}, "outputs": [ { "data": { "text/plain": [ "Text(0.5, 0.98, 'Observable evolution')" ] }, "execution_count": 11, "metadata": {}, "output_type": "execute_result" }, { "data": { "text/plain": [ "\"Output" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "fig, axes = plt.subplots(3, sharex=True)\n", "times = np.linspace(0, evolution_time, num_timesteps + 1) # includes initial state\n", "axes[0].plot(\n", " times,\n", " energy_array,\n", " label=\"First order\",\n", " marker=\"x\",\n", " c=\"darkmagenta\",\n", " ls=\"-\",\n", " lw=0.8,\n", ")\n", "axes[1].plot(\n", " times, mag_array, label=\"First order\", marker=\"x\", c=\"darkmagenta\", ls=\"-\", lw=0.8\n", ")\n", "axes[2].plot(\n", " times, corr_array, label=\"First order\", marker=\"x\", c=\"darkmagenta\", ls=\"-\", lw=0.8\n", ")\n", "axes[0].set_ylabel(\"Energy\")\n", "axes[1].set_ylabel(\"Magnetization\")\n", "axes[2].set_ylabel(\"Mean spin correlation\")\n", "axes[2].set_xlabel(\"Time\")\n", "fig.suptitle(\"Observable evolution\")" ] }, { "cell_type": "markdown", "id": "481f1e48-0650-4f53-b3ce-7d0b8c1e171e", "metadata": {}, "source": [ "\n", "\n", "## 3. Ejercicio 1. Realizar simulación utilizando Suzuki-Trotter de segundo orden\n", "\n", "Ahora vamos a intentar realizar la simulación con Suzuki-Trotter de segundo orden siguiendo el ejemplo de Lie-Trotter mostrado anteriormente.\n", "\n", "El Suzuki-Trotter de segundo orden puede utilizarse en Qiskit mediante la [clase SuzukiTrotter](/docs/api/qiskit/qiskit.synthesis.SuzukiTrotter). Utilizando esta fórmula, se obtiene una descomposición de segundo orden:\n", "\n", "$$\n", "e^{H_1+H_2} \\approx e^{H_1/2}e^{H_2}e^{H_1/2}\n", "$$\n", "\n" ] }, { "cell_type": "markdown", "id": "1e47dd47-ce42-4c9b-88f3-7a44145edf31", "metadata": {}, "source": [ "\n", "\n", "### 3.1 Construya un circuito para un solo paso de tiempo\n", "\n", "Utilice product\\_formula\\_st2 ( SuzukiTrotter(order=2 )) y construya un circuito para un solo paso de tiempo utilizando Suzuki-Trotter de segundo orden. Además, cuenta el número de puertas y la profundidad del circuito y compáralo con Lie-Trotter.\n", "\n" ] }, { "cell_type": "code", "execution_count": 13, "id": "8b6389e3-9c44-4618-9d19-7854f122715a", "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "\n", "Trotter step with second-order Suzuki-Trotter\n", "-----------------------------\n", "Depth: 34\n", "Gate count: 53\n", "Nonlocal gate count: 20\n", "Gate breakdown: U3: 23, CX: 20, U1: 10\n", "\n" ] }, { "data": { "text/plain": [ "\"Output" ] }, "execution_count": 13, "metadata": {}, "output_type": "execute_result" } ], "source": [ "# Modify the line below (Use PauliEvolutionGate)\n", "single_step_evolution_gates_st2 = PauliEvolutionGate(\n", " hamiltonian, dt, synthesis=product_formula_st2\n", ")\n", "single_step_evolution_st2 = QuantumCircuit(n_qubits)\n", "single_step_evolution_st2.append(\n", " single_step_evolution_gates_st2, single_step_evolution_st2.qubits\n", ")\n", "# Let us print some stats\n", "print(\n", " f\"\"\"\n", "Trotter step with second-order Suzuki-Trotter\n", "-----------------------------\n", "Depth: {single_step_evolution_st2.decompose(reps=3).depth()}\n", "Gate count: {len(single_step_evolution_st2.decompose(reps=3))}\n", "Nonlocal gate count: {single_step_evolution_st2.decompose(reps=3).num_nonlocal_gates()}\n", "Gate breakdown: {\", \".join([f\"{k.upper()}: {v}\" for k, v in single_step_evolution_st2.decompose(reps=3).count_ops().items()])}\n", "\"\"\"\n", ")\n", "single_step_evolution_st2.decompose(reps=2).draw(\"mpl\", fold=-1)" ] }, { "cell_type": "markdown", "id": "1748cc5d-360b-415f-8281-ab908d71c057", "metadata": {}, "source": [ "\n", "\n", "### 3.2 Realizar simulación de evolución temporal\n", "\n", "Realiza la evolución temporal mediante Suzuki-Trotter de segundo orden.\n", "\n" ] }, { "cell_type": "code", "execution_count": 14, "id": "59ceae10-5675-4af2-b88f-6b928df35606", "metadata": {}, "outputs": [], "source": [ "# Initiate the circuit\n", "evolved_state = QuantumCircuit(initial_circuit.num_qubits)\n", "# Start from the initial spin configuration\n", "evolved_state.append(initial_circuit, evolved_state.qubits)\n", "# Initiate Estimator (V2)\n", "estimator = StatevectorEstimator()\n", "# Set number of shots\n", "shots = 10000\n", "# Translate the precision required from the number of shots\n", "precision = np.sqrt(1 / shots)\n", "energy_list_st2 = []\n", "mag_list_st2 = []\n", "corr_list_st2 = []\n", "# Estimate expectation values for t=0.0\n", "job = estimator.run(\n", " [(evolved_state, [hamiltonian, magnetization, correlation])], precision=precision\n", ")\n", "# Get estimated expectation values\n", "evs = job.result()[0].data.evs\n", "energy_list_st2.append(evs[0])\n", "mag_list_st2.append(evs[1])\n", "corr_list_st2.append(evs[2])\n", "# Start time evolution\n", "for n in range(num_timesteps):\n", " # Expand the circuit to describe delta-t\n", " evolved_state.append(single_step_evolution_gates_st2, evolved_state.qubits)\n", " # Estimate expectation values at delta-t\n", " job = estimator.run(\n", " [(evolved_state, [hamiltonian, magnetization, correlation])],\n", " precision=precision,\n", " )\n", " # Retrieve results (expectation values)\n", " evs = job.result()[0].data.evs\n", " energy_list_st2.append(evs[0])\n", " mag_list_st2.append(evs[1])\n", " corr_list_st2.append(evs[2])\n", "# Transform the list of expectation values (at each time step) to arrays\n", "energy_array_st2 = np.array(energy_list_st2)\n", "mag_array_st2 = np.array(mag_list_st2)\n", "corr_array_st2 = np.array(corr_list_st2)" ] }, { "cell_type": "markdown", "id": "cd6468c8-bd78-4043-ab54-b6036178399f", "metadata": {}, "source": [ "\n", "\n", "### 3.3 Trazar los resultados de segundo orden de Suzuki-Trotter\n", "\n" ] }, { "cell_type": "code", "execution_count": 15, "id": "b333b560-a6ae-4ab0-a900-c098e545c9c8", "metadata": {}, "outputs": [ { "data": { "text/plain": [ "\"Output" ] }, "execution_count": 15, "metadata": {}, "output_type": "execute_result" } ], "source": [ "axes[0].plot(\n", " times,\n", " energy_array_st2,\n", " label=\"Second Order\",\n", " marker=\"x\",\n", " c=\"limegreen\",\n", " ls=\"-\",\n", " lw=0.8,\n", ")\n", "axes[1].plot(\n", " times,\n", " mag_array_st2,\n", " label=\"Second Order\",\n", " marker=\"x\",\n", " c=\"limegreen\",\n", " ls=\"-\",\n", " lw=0.8,\n", ")\n", "axes[2].plot(\n", " times,\n", " corr_array_st2,\n", " label=\"Second Order\",\n", " marker=\"x\",\n", " c=\"limegreen\",\n", " ls=\"-\",\n", " lw=0.8,\n", ")\n", "\n", "# Replace the legend\n", "# legend.remove()\n", "legend = fig.legend(\n", " *axes[0].get_legend_handles_labels(),\n", " bbox_to_anchor=(1.0, 0.5),\n", " loc=\"center left\",\n", " framealpha=0.5,\n", ")\n", "fig" ] }, { "cell_type": "markdown", "id": "e12ad5f0-eff1-4238-97fb-593e7caf67ba", "metadata": {}, "source": [ "\n", "\n", "### 3.4 Comparar con resultados exactos\n", "\n", "Los datos siguientes son los resultados exactos precalculados del ordenador clásico.\n", "\n" ] }, { "cell_type": "code", "execution_count": 16, "id": "912e69a9-fa10-4c34-ab8c-6d970a461f8f", "metadata": {}, "outputs": [], "source": [ "exact_times = np.array(\n", " [\n", " 0.0,\n", " 0.3,\n", " 0.6,\n", " 0.8999999999999999,\n", " 1.2,\n", " 1.5,\n", " 1.7999999999999998,\n", " 2.1,\n", " 2.4,\n", " 2.6999999999999997,\n", " 3.0,\n", " 3.3,\n", " 3.5999999999999996,\n", " 3.9,\n", " 4.2,\n", " 4.5,\n", " 4.8,\n", " 5.1,\n", " 5.3999999999999995,\n", " 5.7,\n", " 6.0,\n", " 6.3,\n", " 6.6,\n", " 6.8999999999999995,\n", " 7.199999999999999,\n", " 7.5,\n", " 7.8,\n", " 8.1,\n", " 8.4,\n", " 8.7,\n", " 9.0,\n", " 9.299999999999999,\n", " 9.6,\n", " 9.9,\n", " 10.2,\n", " 10.5,\n", " 10.799999999999999,\n", " 11.1,\n", " 11.4,\n", " 11.7,\n", " 12.0,\n", " 12.299999999999999,\n", " 12.6,\n", " 12.9,\n", " 13.2,\n", " 13.5,\n", " 13.799999999999999,\n", " 14.1,\n", " 14.399999999999999,\n", " 14.7,\n", " 15.0,\n", " 15.299999999999999,\n", " 15.6,\n", " 15.899999999999999,\n", " 16.2,\n", " 16.5,\n", " 16.8,\n", " 17.099999999999998,\n", " 17.4,\n", " 17.7,\n", " 18.0,\n", " 18.3,\n", " 18.599999999999998,\n", " 18.9,\n", " 19.2,\n", " 19.5,\n", " 19.8,\n", " 20.099999999999998,\n", " 20.4,\n", " 20.7,\n", " 21.0,\n", " 21.3,\n", " 21.599999999999998,\n", " 21.9,\n", " 22.2,\n", " 22.5,\n", " 22.8,\n", " 23.099999999999998,\n", " 23.4,\n", " 23.7,\n", " 24.0,\n", " 24.3,\n", " 24.599999999999998,\n", " 24.9,\n", " 25.2,\n", " 25.5,\n", " 25.8,\n", " 26.099999999999998,\n", " 26.4,\n", " 26.7,\n", " 27.0,\n", " 27.3,\n", " 27.599999999999998,\n", " 27.9,\n", " 28.2,\n", " 28.5,\n", " 28.799999999999997,\n", " 29.099999999999998,\n", " 29.4,\n", " 29.7,\n", " 30.0,\n", " ]\n", ")\n", "exact_energy = np.array(\n", " [\n", " -1.1184402376762155,\n", " -1.1184402376762157,\n", " -1.1184402376762157,\n", " -1.1184402376762148,\n", " -1.1184402376762153,\n", " -1.1184402376762155,\n", " -1.1184402376762148,\n", " -1.118440237676216,\n", " -1.118440237676216,\n", " -1.1184402376762166,\n", " -1.1184402376762148,\n", " -1.118440237676216,\n", " -1.1184402376762153,\n", " -1.1184402376762148,\n", " -1.118440237676217,\n", " -1.118440237676215,\n", " -1.1184402376762161,\n", " -1.1184402376762157,\n", " -1.118440237676217,\n", " -1.1184402376762161,\n", " -1.1184402376762137,\n", " -1.1184402376762161,\n", " -1.1184402376762161,\n", " -1.118440237676218,\n", " -1.1184402376762155,\n", " -1.1184402376762166,\n", " -1.1184402376762155,\n", " -1.1184402376762137,\n", " -1.1184402376762186,\n", " -1.1184402376762215,\n", " -1.1184402376762148,\n", " -1.118440237676216,\n", " -1.1184402376762166,\n", " -1.1184402376762148,\n", " -1.1184402376762121,\n", " -1.1184402376762166,\n", " -1.1184402376762181,\n", " -1.1184402376762137,\n", " -1.1184402376762148,\n", " -1.1184402376762193,\n", " -1.1184402376762108,\n", " -1.1184402376762144,\n", " -1.118440237676217,\n", " -1.1184402376762197,\n", " -1.1184402376762153,\n", " -1.1184402376762161,\n", " -1.1184402376762184,\n", " -1.1184402376762126,\n", " -1.118440237676214,\n", " -1.118440237676214,\n", " -1.1184402376762161,\n", " -1.118440237676212,\n", " -1.1184402376762164,\n", " -1.118440237676217,\n", " -1.1184402376762121,\n", " -1.1184402376762157,\n", " -1.1184402376762212,\n", " -1.1184402376762217,\n", " -1.1184402376762206,\n", " -1.118440237676222,\n", " -1.1184402376762166,\n", " -1.118440237676212,\n", " -1.1184402376762137,\n", " -1.11844023767622,\n", " -1.1184402376762206,\n", " -1.118440237676219,\n", " -1.1184402376762153,\n", " -1.1184402376762164,\n", " -1.118440237676209,\n", " -1.1184402376762144,\n", " -1.1184402376762161,\n", " -1.118440237676216,\n", " -1.1184402376762173,\n", " -1.118440237676214,\n", " -1.1184402376762093,\n", " -1.1184402376762184,\n", " -1.1184402376762126,\n", " -1.118440237676213,\n", " -1.1184402376762195,\n", " -1.1184402376762095,\n", " -1.1184402376762075,\n", " -1.1184402376762197,\n", " -1.1184402376762141,\n", " -1.1184402376762146,\n", " -1.1184402376762184,\n", " -1.118440237676218,\n", " -1.1184402376762224,\n", " -1.118440237676219,\n", " -1.118440237676218,\n", " -1.1184402376762206,\n", " -1.1184402376762168,\n", " -1.118440237676221,\n", " -1.118440237676218,\n", " -1.1184402376762148,\n", " -1.1184402376762106,\n", " -1.1184402376762173,\n", " -1.118440237676216,\n", " -1.118440237676216,\n", " -1.1184402376762113,\n", " -1.1184402376762275,\n", " -1.1184402376762195,\n", " ]\n", ")\n", "exact_magnetization = np.array(\n", " [\n", " 0.3333333333333333,\n", " 0.26316769633415005,\n", " 0.0912947227110664,\n", " -0.09317712543141576,\n", " -0.20391854332115245,\n", " -0.19318196655046493,\n", " -0.06411527074401464,\n", " 0.12558269854206197,\n", " 0.28252754464640606,\n", " 0.3264196194042506,\n", " 0.2361586169847769,\n", " 0.060894367906122224,\n", " -0.10842387093076275,\n", " -0.18636359582538073,\n", " -0.1338364343947887,\n", " 0.020284606520827753,\n", " 0.19151142743926025,\n", " 0.2905341647678381,\n", " 0.2723014646745304,\n", " 0.15147481733047252,\n", " -0.008179102877790292,\n", " -0.1242999208732406,\n", " -0.1372529247781061,\n", " -0.04083616185958952,\n", " 0.11066094926716476,\n", " 0.23140661570567636,\n", " 0.2587109403786205,\n", " 0.1868237670027325,\n", " 0.061201779383143744,\n", " -0.051391248969654205,\n", " -0.09843899603365061,\n", " -0.061297056158849166,\n", " 0.04199010081939773,\n", " 0.15861461430963147,\n", " 0.22336830674799552,\n", " 0.20179555623336537,\n", " 0.11407111438609417,\n", " 0.01609419104778282,\n", " -0.04239611796730001,\n", " -0.04249123521065924,\n", " 0.008850291714888112,\n", " 0.08780898151558082,\n", " 0.1561486776507056,\n", " 0.17627348772811832,\n", " 0.13870676179652253,\n", " 0.07205869195282538,\n", " 0.018300003064909465,\n", " 0.0001095640839572417,\n", " 0.015157929316037586,\n", " 0.05077755280969454,\n", " 0.09245534457650838,\n", " 0.12206907551110702,\n", " 0.12284950557969157,\n", " 0.09570215398601932,\n", " 0.06294378255078983,\n", " 0.045503313813986014,\n", " 0.043389819499542556,\n", " 0.046725117769796744,\n", " 0.054956411358382404,\n", " 0.0713814528253614,\n", " 0.08743689703248492,\n", " 0.08951216359166674,\n", " 0.07878386475305985,\n", " 0.06955669116405788,\n", " 0.06639892435963689,\n", " 0.05890378761746903,\n", " 0.04541796525844558,\n", " 0.0414221088331947,\n", " 0.05499634106912299,\n", " 0.07409418836014572,\n", " 0.08371859070160165,\n", " 0.08211623987959302,\n", " 0.07615055161378328,\n", " 0.06702584458783024,\n", " 0.051891407742740085,\n", " 0.038049378383635625,\n", " 0.03825614149768043,\n", " 0.054183218463525695,\n", " 0.0753534475741016,\n", " 0.08853147112587295,\n", " 0.08767917178542013,\n", " 0.07709383184439536,\n", " 0.06308595032042386,\n", " 0.0498812359204284,\n", " 0.04299040064096167,\n", " 0.04769159891460652,\n", " 0.06483569572288776,\n", " 0.08698035745435016,\n", " 0.10047391641776235,\n", " 0.09747255683203637,\n", " 0.08098863187287358,\n", " 0.05959496723987331,\n", " 0.04383882265040485,\n", " 0.04232138798062125,\n", " 0.05720514169944535,\n", " 0.08201306299870219,\n", " 0.10274898262000469,\n", " 0.10707552455080133,\n", " 0.09210856128265357,\n", " 0.06379922105742579,\n", " 0.03624325103307953,\n", " ]\n", ")\n", "exact_correlation = np.array(\n", " [\n", " 0.2,\n", " 0.1247704225763532,\n", " 0.01943938494098705,\n", " 0.03854917181332821,\n", " 0.11196616231067426,\n", " 0.0906546700356683,\n", " 0.01629373561896267,\n", " 0.011352652889791095,\n", " 0.0636185676540077,\n", " 0.09543834437789013,\n", " 0.10058518161011307,\n", " 0.11829217731417431,\n", " 0.1397812224038133,\n", " 0.12316460402216707,\n", " 0.08541383059335775,\n", " 0.06144846844403662,\n", " 0.020246372880505827,\n", " -0.02693683090021662,\n", " 0.003919250903281282,\n", " 0.1117419430168554,\n", " 0.19676155181256794,\n", " 0.18594408880783336,\n", " 0.1002673802566004,\n", " 0.03821525827438024,\n", " 0.04485205090247377,\n", " 0.05348102743040269,\n", " 0.03160026140008638,\n", " 0.033437649060464834,\n", " 0.10486939975320728,\n", " 0.20249469538955758,\n", " 0.19735507621013149,\n", " 0.0553097261765083,\n", " -0.04889114490131667,\n", " 0.011685690974970964,\n", " 0.11705971535823065,\n", " 0.11681165998194759,\n", " 0.06637091239560744,\n", " 0.10936684225958895,\n", " 0.20225454101061405,\n", " 0.16284420833341812,\n", " -0.0025823294931362067,\n", " -0.0763416631752919,\n", " 0.02985268630418397,\n", " 0.15234468006771007,\n", " 0.14606385406970995,\n", " 0.0935341856492092,\n", " 0.12325421854361143,\n", " 0.17130422930386324,\n", " 0.10383730044042278,\n", " -0.031333159406547614,\n", " -0.05241572078596815,\n", " 0.07722509925347705,\n", " 0.17642188574256007,\n", " 0.12765340239966838,\n", " 0.06309968945093776,\n", " 0.11574687130499339,\n", " 0.16978282647206913,\n", " 0.0736143632571229,\n", " -0.05356602733119409,\n", " -0.0009649396796768892,\n", " 0.15921620111869142,\n", " 0.17760366431811037,\n", " 0.04736297330213485,\n", " 0.012122870263181897,\n", " 0.13268065586830521,\n", " 0.1728473023503636,\n", " 0.03999259331072221,\n", " -0.036997053070222885,\n", " 0.06951528580242439,\n", " 0.1769169993516561,\n", " 0.12290448295710298,\n", " 0.012897784654866427,\n", " 0.02859435620982225,\n", " 0.12895847695150875,\n", " 0.13629536955485938,\n", " 0.05394621059822597,\n", " 0.02298040588184324,\n", " 0.07036499900317271,\n", " 0.11706448623132719,\n", " 0.10435285842074606,\n", " 0.055721236329964965,\n", " 0.04676334743672697,\n", " 0.08417924910022263,\n", " 0.10611161955304965,\n", " 0.089304171047322,\n", " 0.06098589533081194,\n", " 0.06314519797488709,\n", " 0.09431492621892917,\n", " 0.09667836915967139,\n", " 0.0651298357290882,\n", " 0.05176966009147416,\n", " 0.06727229484222669,\n", " 0.08871788283607947,\n", " 0.09907054249093444,\n", " 0.09785167773502176,\n", " 0.09277216140054353,\n", " 0.07520999642062785,\n", " 0.05894392248382922,\n", " 0.07236135251622376,\n", " 0.08608284185200156,\n", " 0.07282922961856123,\n", " ]\n", ")" ] }, { "cell_type": "code", "execution_count": 17, "id": "d626c298-49e5-4cba-9cf4-ba0935b13f95", "metadata": { "scrolled": true }, "outputs": [ { "data": { "text/plain": [ "\"Output" ] }, "execution_count": 17, "metadata": {}, "output_type": "execute_result" } ], "source": [ "axes[0].plot(exact_times, exact_energy, c=\"k\", ls=\":\", label=\"Exact\")\n", "axes[1].plot(exact_times, exact_magnetization, c=\"k\", ls=\":\", label=\"Exact\")\n", "axes[2].plot(exact_times, exact_correlation, c=\"k\", ls=\":\", label=\"Exact\")\n", "# Replace the legend\n", "legend.remove()\n", "# Select the labels of only the first axis\n", "legend = fig.legend(\n", " *axes[0].get_legend_handles_labels(),\n", " bbox_to_anchor=(1.0, 0.5),\n", " loc=\"center left\",\n", " framealpha=0.5,\n", ")\n", "fig.tight_layout()\n", "fig" ] }, { "cell_type": "markdown", "id": "6c72f95e-2259-4b76-9e06-ad296e0cfa7d", "metadata": {}, "source": [ "\n", "\n", "## 4. Ejecución en el hardware cuántico\n", "\n", "A continuación, ejecutamos la simulación de evolución temporal en el hardware cuántico. Trabajaremos en un problema más pequeño, de tamaño de celosía N=2. Variamos el parámetro $\\alpha$ y vemos la diferencia en la dinámica de la función de onda.\n", "\n" ] }, { "cell_type": "markdown", "id": "9a5bb8bf-9a96-4464-8bcc-eeaee193c23a", "metadata": {}, "source": [ "\n", "\n", "### 4.1 Paso 1. Asignar entradas clásicas a un problema cuántico\n", "\n", "Elige la configuración inicial de la simulación:\n", "\n" ] }, { "cell_type": "code", "execution_count": 19, "id": "40ac8e04-7eab-4e92-8964-6987d707cc04", "metadata": {}, "outputs": [], "source": [ "n_qubits_2 = 2\n", "dt_2 = 1.6\n", "product_formula = LieTrotter(reps=1)" ] }, { "cell_type": "markdown", "id": "1ad01369-146b-41c6-a557-5abae8eb0122", "metadata": {}, "source": [ "A continuación, establece el circuito inicial:\n", "\n", "La configuración inicial de giro será \"abajo-arriba\"\n", "\n" ] }, { "cell_type": "code", "execution_count": 20, "id": "b7ed1249-1a69-4d62-97a3-102cf905a295", "metadata": {}, "outputs": [ { "data": { "text/plain": [ "\"Output" ] }, "execution_count": 20, "metadata": {}, "output_type": "execute_result" } ], "source": [ "# We prepare an initial state ↓↑ (10).\n", "# Note that Statevector and SparsePauliOp interpret the qubits from right to left\n", "initial_circuit_2 = QuantumCircuit(n_qubits_2)\n", "initial_circuit_2.prepare_state(\"10\")\n", "# Change reps and see the difference when you decompose the circuit\n", "initial_circuit_2.decompose(reps=1).draw(\"mpl\")" ] }, { "cell_type": "markdown", "id": "b383a0e6-bebe-4d47-865d-50b0258a8fd9", "metadata": {}, "source": [ "Ahora calcula el valor de referencia utilizando un simulador de vector de estado ideal.\n", "\n" ] }, { "cell_type": "code", "execution_count": 21, "id": "d15f5cef-59a4-4946-9a3f-6bee81dd2587", "metadata": {}, "outputs": [ { "data": { "text/plain": [ "" ] }, "execution_count": 21, "metadata": {}, "output_type": "execute_result" }, { "data": { "text/plain": [ "\"Output" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "bar_width = 0.1\n", "# initial_state = Statevector.from_label(\"10\")\n", "final_time = 1.6\n", "eps = 1e-5\n", "\n", "# We create the list of angles in radians, with a small epsilon\n", "# the exactly longitudinal field, which would present no dynamics at all\n", "alphas = np.linspace(-np.pi / 2 + eps, np.pi / 2 - eps, 5)\n", "\n", "for i, alpha in enumerate(alphas):\n", " evolved_state_2 = QuantumCircuit(initial_circuit_2.num_qubits)\n", " evolved_state_2.append(initial_circuit_2, evolved_state_2.qubits)\n", " hamiltonian_2 = get_hamiltonian(nqubits=2, J=0.2, h=1.0, alpha=alpha)\n", " single_step_evolution_gates_2 = PauliEvolutionGate(\n", " hamiltonian_2, dt_2, synthesis=product_formula\n", " )\n", " evolved_state_2.append(single_step_evolution_gates_2, evolved_state_2.qubits)\n", " evolved_state_2 = Statevector(evolved_state_2)\n", " # Dictionary of probabilities\n", " amplitudes_dict = evolved_state_2.probabilities_dict()\n", " labels = list(amplitudes_dict.keys())\n", " values = list(amplitudes_dict.values())\n", " # Convert angle to degrees\n", " alpha_str = f\"$\\\\alpha={int(np.round(alpha * 180 / np.pi))}^\\\\circ$\"\n", " plt.bar(np.arange(4) + i * bar_width, values, bar_width, label=alpha_str, alpha=0.7)\n", "\n", "plt.xticks(np.arange(4) + 2 * bar_width, labels)\n", "plt.xlabel(\"Measurement\")\n", "plt.ylabel(\"Probability\")\n", "plt.suptitle(\n", " f\"Measurement probabilities at $t={final_time}$, for various field angles $\\\\alpha$\\n\"\n", " f\"Initial state: 10, Linear lattice of size $L=2$\"\n", ")\n", "plt.legend()" ] }, { "cell_type": "markdown", "id": "0588757d-5dcf-4ee3-8d62-1b14070156eb", "metadata": {}, "source": [ "Hemos preparado un sistema inicialmente con una secuencia de espines $\\downarrow\\uparrow$, que corresponde a $\\vert\\psi(0)\\rangle = \\vert10\\rangle$. Después de dejarlo evolucionar durante $t=1.6$ bajo un campo transversal ( $\\alpha=0^\\circ$ ), tenemos casi garantizado medir $\\uparrow\\downarrow$, es decir, tener un intercambio de espines. (Obsérvese que las etiquetas se interpretan de derecha a izquierda). Si el campo es longitudinal ( $\\alpha=\\pm90^\\circ$ ), no tendremos evolución, por lo que mediremos el sistema tal y como estaba preparado inicialmente, $\\downarrow\\uparrow$. Con ángulos intermedios, en $\\alpha=\\pm45^\\circ$, podremos medir todas las combinaciones con diferentes probabilidades, siendo un intercambio de espín la más probable con una probabilidad del 67%.\n", "\n" ] }, { "cell_type": "markdown", "id": "36ef1a7e-1ab7-48ed-89ad-efbef819c871", "metadata": {}, "source": [ "\n", "\n", "#### Construir circuito para experimento de HW\n", "\n" ] }, { "cell_type": "code", "execution_count": 24, "id": "ef73cba1-3469-472d-a100-26fb1e103840", "metadata": {}, "outputs": [], "source": [ "circuit_list = []\n", "for i, alpha in enumerate(alphas):\n", " evolved_state_2 = QuantumCircuit(initial_circuit_2.num_qubits)\n", " evolved_state_2.append(initial_circuit_2, evolved_state_2.qubits)\n", " hamiltonian_2 = get_hamiltonian(nqubits=2, J=0.2, h=1.0, alpha=alpha)\n", " single_step_evolution_gates_2 = PauliEvolutionGate(\n", " hamiltonian_2, dt_2, synthesis=product_formula\n", " )\n", " evolved_state_2.append(single_step_evolution_gates_2, evolved_state_2.qubits)\n", " evolved_state_2.measure_all()\n", " circuit_list.append(evolved_state_2)" ] }, { "cell_type": "markdown", "id": "0cf2010d-2621-46fa-9828-f06979faad60", "metadata": {}, "source": [ "\n", "\n", "### 4.2 Paso 2. Optimizar para el hardware objetivo\n", "\n", "Especificamos un backend.\n", "\n" ] }, { "cell_type": "code", "execution_count": null, "id": "0a7734a6-6ec3-48a8-91cb-8995a15ada88", "metadata": {}, "outputs": [ { "data": { "text/plain": [ "'ibm_strasbourg'" ] }, "execution_count": 25, "metadata": {}, "output_type": "execute_result" } ], "source": [ "service = QiskitRuntimeService()\n", "backend = service.least_busy(operational=True, simulator=False)\n", "backend.name" ] }, { "cell_type": "markdown", "id": "1d4be180-a241-4e84-a730-66e1a4fc9e9a", "metadata": {}, "source": [ "A continuación, transpilamos el circuito para el backend seleccionado.\n", "\n" ] }, { "cell_type": "code", "execution_count": 26, "id": "b154db84-439c-40d0-bf14-d2a6b584b593", "metadata": {}, "outputs": [], "source": [ "pm = generate_preset_pass_manager(backend=backend, optimization_level=3)\n", "circuit_isa = pm.run(circuit_list)" ] }, { "cell_type": "markdown", "id": "ce78ebe6-1ed5-470d-a000-490cfb6cc9e3", "metadata": {}, "source": [ "Comprueba el circuito.\n", "\n" ] }, { "cell_type": "code", "execution_count": 28, "id": "04de26ef-4854-4004-846b-3e2d7a2a3d0e", "metadata": {}, "outputs": [ { "data": { "text/plain": [ "\"Output" ] }, "execution_count": 28, "metadata": {}, "output_type": "execute_result" } ], "source": [ "circuit_isa[1].draw(\"mpl\", idle_wires=False)" ] }, { "cell_type": "markdown", "id": "d08cdec5-9127-434b-b435-7525d7379158", "metadata": {}, "source": [ "\n", "\n", "### 4.3 Paso 3. Ejecutar con primitivas de Qiskit Runtime\n", "\n", "La primitiva `Sampler` ( V2 ) de Qiskit proporciona los recuentos de las cadenas de bits medidas.\n", "\n" ] }, { "cell_type": "code", "execution_count": 29, "id": "836b8fb2-6799-4c93-8c00-278c4dbb2744", "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "job id: d13pswfmya70008ek070\n" ] } ], "source": [ "sampler = SamplerV2(mode=backend)\n", "job = sampler.run(circuit_isa)\n", "job_id = job.job_id()\n", "print(\"job id:\", job_id)" ] }, { "cell_type": "markdown", "id": "382d29d0-c9fc-452e-9584-d4276df6d93a", "metadata": {}, "source": [ "Guardar los resultados\n", "\n" ] }, { "cell_type": "code", "execution_count": 30, "id": "8a2fc35c-8801-49bd-b715-b2906e3e422e", "metadata": {}, "outputs": [], "source": [ "results = job.result()" ] }, { "cell_type": "markdown", "id": "b67e9e11-794d-4159-be82-d1935988f5af", "metadata": {}, "source": [ "\n", "\n", "### 4.4 Paso 4. Resultados del procesamiento posterior\n", "\n", "Construye el histograma de las cadenas de bits, que corresponde al análisis de la función de onda, y compáralas con los valores ideales mostrados anteriormente.\n", "\n" ] }, { "cell_type": "code", "execution_count": null, "id": "336e6b34-deee-467f-bd8f-6d1a8ed1ccba", "metadata": {}, "outputs": [ { "data": { "text/plain": [ "" ] }, "execution_count": 32, "metadata": {}, "output_type": "execute_result" }, { "data": { "text/plain": [ "\"Output" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "list_temp = [\"00\", \"01\", \"10\", \"11\"]\n", "\n", "for i, alpha in enumerate(alphas):\n", " # Dictionary of probabilities\n", " amplitudes_dict = results[i].data.meas.get_counts()\n", " values = []\n", " for str_temp in list_temp:\n", " values.append(\n", " amplitudes_dict[str_temp] / 4096.0\n", " ) # divided by default number of shots\n", " # Convert angle to degrees\n", " alpha_str = f\"$\\\\alpha={int(np.round(alpha * 180 / np.pi))}^\\\\circ$\"\n", " plt.bar(np.arange(4) + i * bar_width, values, bar_width, label=alpha_str, alpha=0.7)\n", "\n", "plt.xticks(np.arange(4) + 2 * bar_width, labels)\n", "plt.xlabel(\"Measurement\")\n", "plt.ylabel(\"Probabilities\")\n", "plt.suptitle(\n", " f\"Measurement probabilities at $t={final_time}$, for various field angles $\\\\alpha$\\n\"\n", " f\"Initial state: 10, Linear lattice of size $L=2$\"\n", ")\n", "plt.legend()" ] }, { "cell_type": "markdown", "id": "8960b35a-ab79-48ad-a9a9-6024ad767699", "metadata": {}, "source": [ "Aquí mostramos un ejemplo para construir un circuito utilizando Suzuki-Trotter de orden superior (cuarto orden).\n", "Ahora vamos a intentar construir una simulación de circuito con Suzuki-Trotter de cuarto orden siguiendo los ejemplos mostrados anteriormente.\n", "\n", "El Suzuki-Trotter de cuarto orden puede utilizarse en Qiskit mediante la [clase SuzukiTrotter](/docs/api/qiskit/qiskit.synthesis.SuzukiTrotter). El cuarto orden puede evaluarse mediante la siguiente relación de recursión. Tenga en cuenta que el orden de Suzuki-Trotter se denota como \" 2k \" en las siguientes ecuaciones.\n", "\n", "$$\n", "\\hat{U}_{ST(2k)}\\left(t\\right) = \\left[ \\hat{U}_{ST(2k-2)}\\left(p_k t\\right) \\right]^2 \\hat{U}_{ST(2k-2)}\\left( (1- 4 p_k) t\\right)\\left[ \\hat{U}_{ST(2k-2)}\\left(p_k t\\right) \\right]^2\n", "$$\n", "\n", "$$\n", "p_k = 1 / \\left(4-4^{\\frac{1}{2k-1}}\\right)\n", "$$\n", "\n" ] }, { "cell_type": "markdown", "id": "67041754-db9b-457a-b326-059e0e565d20", "metadata": {}, "source": [ "\n", "\n", "#### Construya un circuito para un solo paso de tiempo\n", "\n", "Utilice product\\_formula\\_st4 ( SuzukiTrotter(order=4 )) y construya un circuito para un solo paso de tiempo utilizando Suzuki-Trotter de cuarto orden. Además, cuenta el número de puertas y la profundidad del circuito y compáralo con Lie-Trotter y Suzuki-Trotter de segundo orden.\n", "\n" ] }, { "cell_type": "code", "execution_count": 33, "id": "be1a7a71-1f42-47c2-bee5-0f510f54d064", "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "\n", "Trotter step with second-order Suzuki-Trotter\n", "-----------------------------\n", "Depth: 170\n", "Gate count: 265\n", "Nonlocal gate count: 100\n", "Gate breakdown: U3: 115, CX: 100, U1: 50\n", "\n" ] }, { "data": { "text/plain": [ "\"Output" ] }, "execution_count": 33, "metadata": {}, "output_type": "execute_result" } ], "source": [ "# Modify the line below (Use PauliEvolutionGate)\n", "single_step_evolution_gates_st4 = PauliEvolutionGate(\n", " hamiltonian, dt, synthesis=product_formula_st4\n", ")\n", "single_step_evolution_st4 = QuantumCircuit(n_qubits)\n", "single_step_evolution_st4.append(\n", " single_step_evolution_gates_st4, single_step_evolution_st4.qubits\n", ")\n", "# Let us print some stats\n", "print(\n", " f\"\"\"\n", "Trotter step with second-order Suzuki-Trotter\n", "-----------------------------\n", "Depth: {single_step_evolution_st4.decompose(reps=3).depth()}\n", "Gate count: {len(single_step_evolution_st4.decompose(reps=3))}\n", "Nonlocal gate count: {single_step_evolution_st4.decompose(reps=3).num_nonlocal_gates()}\n", "Gate breakdown: {\", \".join([f\"{k.upper()}: {v}\" for k, v in single_step_evolution_st4.decompose(reps=3).count_ops().items()])}\n", "\"\"\"\n", ")\n", "single_step_evolution_st4.decompose(reps=2).draw(\"mpl\", fold=-1)" ] }, { "cell_type": "code", "execution_count": 34, "id": "87364170-60b0-4d7b-9dd1-a12676b1f80f", "metadata": {}, "outputs": [ { "data": { "text/plain": [ "'2.0.2'" ] }, "execution_count": 34, "metadata": {}, "output_type": "execute_result" } ], "source": [ "# Check Qiskit version\n", "import qiskit\n", "\n", "qiskit.__version__" ] }, { "cell_type": "markdown", "metadata": {}, "id": "a1b8767d", "source": "© IBM Corp., 2017-2026" } ], "metadata": { "kernelspec": { "display_name": "Python 3", "language": "python", "name": "python3" }, "language_info": { "codemirror_mode": { "name": "ipython", "version": 3 }, "file_extension": ".py", "mimetype": "text/x-python", "name": "python", "nbconvert_exporter": "python", "pygments_lexer": "ipython3", "version": "3" }, "widgets": { "application/vnd.jupyter.widget-state+json": { "state": {}, "version_major": 2, "version_minor": 0 } } }, "nbformat": 4, "nbformat_minor": 4 }