{ "cells": [ { "cell_type": "markdown", "id": "frontmatter", "metadata": {}, "source": [ "---\n", "title: \"Utilizar diferentes normas Lp para o truncamento do termo de Pauli\"\n", "description: \"Utilizar diferentes normas Lp para o truncamento do termo de Pauli na versão mais recente da retropropagação de operadores (OBP)\"\n", "---\n", "\n" ] }, { "cell_type": "markdown", "id": "8a7c9df5-d836-4ecb-8aec-8a1f2faa017a", "metadata": {}, "source": [ "\n", "\n", "# Utilizar diferentes normas Lp para o truncamento do termo de Pauli\n", "\n" ] }, { "cell_type": "markdown", "id": "5958e635-64c1-4599-82ca-225b83312906", "metadata": {}, "source": [ "**Observação:** Antes de ler este guia, você deve ler o guia [sobre termos de Pauli truncados](/docs/addons/qiskit-addon-obp/guides/truncate-operator-terms), que descreve o truncamento de termos de Pauli de baixo peso incorporado ao método [de](/docs/api/qiskit-addon-obp/qiskit-addon-obp#backpropagate) retropropagação com base em um valor de “ [TruncationErrorBudget](/docs/api/qiskit-addon-obp/utils-truncating#truncationerrorbudget) ” especificado.\n", "\n" ] }, { "cell_type": "markdown", "id": "2780c587-cd2a-4398-8faa-de5e0c661a10", "metadata": {}, "source": [ "Neste guia, você aprenderá sobre o argumento de palavra-chave [backpropagate](/docs/api/qiskit-addon-obp/qiskit-addon-obp#backpropagate) `p_norm`, que pode ser usado para alterar a norma Lp utilizada para estimar o erro decorrente dos termos de Pauli truncados.\n", "\n" ] }, { "cell_type": "markdown", "id": "872aeae7-0312-41db-8ebb-f7b02838b54d", "metadata": {}, "source": [ "\n", "\n", "## Construa um circuito de exemplo\n", "\n", "Este guia utiliza o mesmo circuito de exemplo apresentado no guia sobre [os termos truncados de Pauli](/docs/addons/qiskit-addon-obp/guides/truncate-operator-terms) :\n", "\n" ] }, { "cell_type": "code", "execution_count": null, "id": "d6f281d4-706e-41ad-b7b5-bc7ebe106996", "metadata": {}, "outputs": [ { "data": { "text/plain": [ "\"Output" ] }, "execution_count": 1, "metadata": {}, "output_type": "execute_result" } ], "source": [ "import rustworkx.generators\n", "from qiskit.synthesis import LieTrotter\n", "from qiskit_addon_utils.problem_generators import (\n", " PauliOrderStrategy,\n", " generate_time_evolution_circuit,\n", " generate_xyz_hamiltonian,\n", ")\n", "from qiskit_addon_utils.slicing import combine_slices, slice_by_gate_types\n", "\n", "# Generate a linear chain of 10 qubits\n", "num_qubits = 10\n", "linear_chain = rustworkx.generators.path_graph(num_qubits)\n", "\n", "# Use an arbitrary XY model\n", "hamiltonian = generate_xyz_hamiltonian(\n", " linear_chain,\n", " coupling_constants=(0.05, 0.02, 0.0),\n", " ext_magnetic_field=(0.02, 0.08, 0.0),\n", " pauli_order_strategy=PauliOrderStrategy.InteractionThenColor,\n", ")\n", "# Evolve for some time\n", "circuit = generate_time_evolution_circuit(\n", " hamiltonian, synthesis=LieTrotter(reps=3), time=2.0\n", ")\n", "# slice the circuit by gate type\n", "slices = slice_by_gate_types(circuit)\n", "\n", "# For visualization purposes only, recombine the slices with barriers between them and draw the resulting circuit\n", "combine_slices(slices, include_barriers=True).draw(\"mpl\", fold=50, scale=0.6)" ] }, { "cell_type": "markdown", "id": "7a7c7299-0c82-4279-b3dc-4f39e8dddd7e", "metadata": {}, "source": [ "Defina uma observável para a magnetização total:\n", "\n" ] }, { "cell_type": "code", "execution_count": 2, "id": "ea9e7adc-b003-4762-a889-10e8d84a63d5", "metadata": {}, "outputs": [], "source": [ "from qiskit.quantum_info import SparsePauliOp\n", "\n", "obs = SparsePauliOp.from_sparse_list(\n", " [(\"Z\", [i], 1.0) for i in range(num_qubits)], num_qubits=num_qubits\n", ")" ] }, { "cell_type": "markdown", "id": "423150d8-80d2-4b80-9665-033808d30272", "metadata": {}, "source": [ "A título de referência, calcule o valor exato da esperança:\n", "\n" ] }, { "cell_type": "code", "execution_count": 3, "id": "4dc72426-2eb5-4f8d-8bba-79650da06b54", "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "9.318197859862146\n" ] } ], "source": [ "from qiskit.primitives import StatevectorEstimator\n", "\n", "estimator = StatevectorEstimator()\n", "job = estimator.run([(circuit, obs)])\n", "res = job.result()\n", "exact_exp = res[0].data.evs\n", "print(exact_exp)" ] }, { "cell_type": "markdown", "id": "d21490c0-e86c-4fe0-af90-c9625813e0c0", "metadata": {}, "source": [ "\n", "\n", "## Utilize a norma “ L1 ”\n", "\n", "Por padrão, e como você já viu no guia [sobre termos truncados de Pauli](/docs/addons/qiskit-addon-obp/guides/truncate-operator-terms), `p_norm=1`, o que significa que o erro é estimado da seguinte forma:\n", "\n", "$$\n", "|\\langle\\psi|\\Delta|\\psi\\rangle| \\leq \\sum_{P\\in\\mathcal{T}} |c_P|\n", "$$\n", "\n", "onde $\\psi$ é o estado quântico, $\\Delta$ é a diferença real entre a observável exata e a truncada (que é desconhecida), $\\mathcal{T}$ é o conjunto de termos de Pauli que foram truncados e $c_P$ é o coeficiente dos termos de Pauli.\n", "Essa desigualdade é um limite superior rigoroso, mas bastante aproximado, para a maioria dos cenários.\n", "\n" ] }, { "cell_type": "markdown", "id": "425293ce-49fb-4bd0-8d2f-44a3a43db967", "metadata": {}, "source": [ "Este guia realiza a retropropagação em seis fatias do circuito de exemplo, utilizando um erro constante por fatia de `0.001`. Esse valor deve ser entendido como o orçamento dentro do `p_norm`.\n", "\n" ] }, { "cell_type": "code", "execution_count": 4, "id": "b46531b3-055b-4112-903a-11f2dd52a9ac", "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "TruncationErrorBudget(per_slice_budget=[0.001], max_error_total=inf, p_norm=1)\n" ] } ], "source": [ "from qiskit_addon_obp.utils.truncating import setup_budget\n", "\n", "l1_truncation_error_budget = setup_budget(max_error_per_slice=0.001, p_norm=1)\n", "print(l1_truncation_error_budget)" ] }, { "cell_type": "code", "execution_count": 5, "id": "bc252678-58a0-407e-91fe-b7c545d57ae8", "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "Backpropagated 6 circuit slices.\n", "New observable contains 116 terms and 10 commuting groups.\n" ] } ], "source": [ "from qiskit_addon_obp import backpropagate\n", "\n", "max_slices = 6\n", "l1_bp_obs, l1_remaining_slices, l1_metadata = backpropagate(\n", " obs,\n", " slices[-max_slices:],\n", " truncation_error_budget=l1_truncation_error_budget,\n", ")\n", "l1_reduced_circuit = combine_slices(\n", " slices[:-max_slices] + l1_remaining_slices\n", ")\n", "print(\n", " f\"Backpropagated {max_slices - len(l1_remaining_slices)} circuit slices.\"\n", ")\n", "print(\n", " f\"New observable contains {len(l1_bp_obs)} terms and {len(l1_bp_obs.group_commuting(qubit_wise=True))} commuting groups.\"\n", ")" ] }, { "cell_type": "markdown", "id": "9f2782f8-1e93-4319-af26-dc467d7ed459", "metadata": {}, "source": [ "Agora podemos calcular o valor esperado do observável retropropagado, bem como o erro em relação à referência exata:\n", "\n" ] }, { "cell_type": "code", "execution_count": 6, "id": "5be8c457-6569-452d-8502-0fc5e39bf918", "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "9.317869899338842 0.00032796052330397174\n" ] } ], "source": [ "estimator = StatevectorEstimator()\n", "job = estimator.run([(l1_reduced_circuit, l1_bp_obs)])\n", "res = job.result()\n", "l1_exp = res[0].data.evs\n", "l1_error = exact_exp - l1_exp\n", "print(l1_exp, l1_error)" ] }, { "cell_type": "markdown", "id": "c0eccc5b-dc71-45b3-ad79-3d3dcb491f2d", "metadata": {}, "source": [ "Por fim, podemos representar graficamente o erro incorrido durante a retropropagação de cada fatia, bem como o erro acumulado.\n", "O erro acumulado é a soma dos erros das fatias. Podemos observar que o erro acumulado é um limite superior bastante aproximado do erro real.\n", "\n" ] }, { "cell_type": "code", "execution_count": 7, "id": "ff72e77f-e2c0-4cce-8575-c5aa587f78fb", "metadata": {}, "outputs": [ { "data": { "text/plain": [ "\"Output" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "from matplotlib import pyplot as plt\n", "from qiskit_addon_obp.utils.visualization import (\n", " plot_accumulated_error,\n", " plot_slice_errors,\n", ")\n", "\n", "fig, axes = plt.subplots(1, 2, figsize=(14, 5))\n", "axes[1].plot([6], [l1_error], \"x\", color=\"red\", label=\"actual error\")\n", "plot_slice_errors(l1_metadata, axes[0])\n", "plot_accumulated_error(l1_metadata, axes[1])" ] }, { "cell_type": "markdown", "id": "10dd1139-c597-43c2-9f58-0a0988de768f", "metadata": {}, "source": [ "\n", "\n", "## Utilize a norma “ L2 ”\n", "\n", "Pode-se argumentar que a norma “ L2 ” é uma aproximação melhor do erro incorrido do que a norma “ L1 ”.\n", "Isso ocorre porque podemos supor que o estado quântico $|\\psi\\rangle$ seja extraído de um conjunto aleatório de Haar; nesse caso, o erro incorrido segue uma distribuição com média nula e uma variância que pode ser aproximadamente limitada pela norma L2 :\n", "\n", "$$\n", "|\\langle\\psi|\\Delta|\\psi\\rangle| \\lesssim \\left( \\sum_{P\\in\\mathcal{T}} |c_P|^2 \\right)^{1/2}\n", "$$\n", "\n", "Embora o limite não seja rigoroso, ele só será violado em casos patológicos.\n", "\n" ] }, { "cell_type": "markdown", "id": "f302d808-b341-49cb-8bbd-b03f2bfc6dd6", "metadata": {}, "source": [ "Mais uma vez, realizamos a retropropagação em seis fatias do circuito de exemplo, utilizando um erro máximo por fatia de `0.001` (desta vez, interpretado na norma L2 ).\n", "\n" ] }, { "cell_type": "code", "execution_count": 8, "id": "c7b7e21f-22e4-479d-954c-39400974f8c1", "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "TruncationErrorBudget(per_slice_budget=[0.001], max_error_total=inf, p_norm=2)\n" ] } ], "source": [ "l2_truncation_error_budget = setup_budget(max_error_per_slice=0.001, p_norm=2)\n", "print(l2_truncation_error_budget)" ] }, { "cell_type": "code", "execution_count": 9, "id": "e92680c5-497e-47e0-ae0f-69465028dd66", "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "Backpropagated 6 circuit slices.\n", "New observable contains 84 terms and 6 commuting groups.\n" ] } ], "source": [ "max_slices = 6\n", "l2_bp_obs, l2_remaining_slices, l2_metadata = backpropagate(\n", " obs,\n", " slices[-max_slices:],\n", " truncation_error_budget=l2_truncation_error_budget,\n", ")\n", "l2_reduced_circuit = combine_slices(\n", " slices[:-max_slices] + l2_remaining_slices\n", ")\n", "print(\n", " f\"Backpropagated {max_slices - len(l2_remaining_slices)} circuit slices.\"\n", ")\n", "print(\n", " f\"New observable contains {len(l2_bp_obs)} terms and {len(l2_bp_obs.group_commuting(qubit_wise=True))} commuting groups.\"\n", ")" ] }, { "cell_type": "markdown", "id": "a05f992a-3a38-4912-9e1b-bcd40463c35e", "metadata": {}, "source": [ "Mais uma vez, calculamos o valor esperado do observável retropropagado e do erro em relação à referência exata:\n", "\n" ] }, { "cell_type": "code", "execution_count": 10, "id": "73fe737a-c3b0-4639-83dc-d614521dd77a", "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "9.317829770853422 0.00036808900872387085\n" ] } ], "source": [ "estimator = StatevectorEstimator()\n", "job = estimator.run([(l2_reduced_circuit, l2_bp_obs)])\n", "res = job.result()\n", "l2_exp = res[0].data.evs\n", "l2_error = exact_exp - l2_exp\n", "print(l2_exp, l2_error)" ] }, { "cell_type": "markdown", "id": "81fd099c-f144-4c59-9b76-72fd8b06d8ad", "metadata": {}, "source": [ "Ao traçar os erros ocorridos por fatia e o erro acumulado, obtém-se um quadro semelhante ao anterior.\n", "\n" ] }, { "cell_type": "code", "execution_count": 11, "id": "fe7f454c-7d79-4c61-b1dd-6a98564fb5f1", "metadata": {}, "outputs": [ { "data": { "text/plain": [ "\"Output" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "fig, axes = plt.subplots(1, 2, figsize=(14, 5))\n", "axes[1].plot([6], [l2_error], \"x\", color=\"red\", label=\"actual error\")\n", "plot_slice_errors(l2_metadata, axes[0])\n", "plot_accumulated_error(l2_metadata, axes[1])" ] }, { "cell_type": "markdown", "id": "9ef23a93-5c3b-48be-a02a-e666179c8c65", "metadata": {}, "source": [ "Observe que o erro acumulado é, mais uma vez, a soma dos erros de cada fatia. Este é outro limite aproximado decorrente da desigualdade de Minkowski, já que precisamos calcular esse limite de forma recursiva:\n", "\n", "$$\n", "|\\langle\\psi|\\Delta_{i}|\\psi\\rangle| \\leq |\\langle\\psi|\\tilde{\\Delta}_{i-1}|\\psi\\rangle| + \\left( \\sum_{P\\in\\mathcal{T_i}} |c_P|^2 \\right)^{1/2} = |\\langle\\psi|\\tilde{\\Delta}_{i}|\\psi\\rangle|\n", "$$\n", "\n", "onde o novo índice $i$ indica a iteração atual da fatia, tornando $\\Delta_i$ o erro real na iteração de retropropagação $i$, $\\tilde{\\Delta}_{i-1}$ o erro de truncamento aproximado da iteração $i-1$ e $\\mathcal{T}_i$ o conjunto de termos de Pauli truncados na iteração $i$.\n", "\n" ] }, { "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" } }, "nbformat": 4, "nbformat_minor": 5 }