{ "cells": [ { "cell_type": "markdown", "id": "frontmatter", "metadata": {}, "source": [ "---\n", "title: \"クイック・スタート\"\n", "description: \"Pauli伝播の最新バージョンのクイックスタート\"\n", "---\n", "\n" ] }, { "cell_type": "markdown", "id": "a9320fa8-31c5-4248-96b0-3549a13dda6f", "metadata": {}, "source": [ "---\n", "title: \"クイック・スタート\"\n", "description: \"pauli-prop Qiskit アドオンパッケージのクイックスタートガイド\"\n", "---\n", "\n", "\n", "\n", "# クイック・スタート\n", "\n", "このガイドでは、 1D スピン鎖上の10量子ビットのキック付きイジングモデルの時間的ダイナミクスを、パッケージ `pauli-prop` を用いて古典的にシミュレーションします。\n", "\n" ] }, { "cell_type": "markdown", "id": "3b5bf7dc-1cd8-41d3-b8ce-cc75a66fed8d", "metadata": {}, "source": [ "\n", "\n", "## パウリ伝播のための入力の準備\n", "\n", "検討対象となるハミルトニアンは以下の通りである：\n", "\n", "$H = -J\\sum\\limits_{\\langle i,j \\rangle} Z_iZ_j + h\\sum\\limits_iX_i$\n", "\n", "ここで、 $J>0$ は最近接スピン間の結合を表し、 $i" ] }, "execution_count": 1, "metadata": {}, "output_type": "execute_result" } ], "source": [ "import numpy as np\n", "from qiskit import QuantumCircuit\n", "from qiskit.quantum_info import SparsePauliOp\n", "from qiskit.transpiler import CouplingMap\n", "\n", "num_qubits = 10\n", "coupling_map = CouplingMap.from_line(num_qubits, bidirectional=False)\n", "\n", "# Num Trotter steps\n", "num_steps = 20\n", "theta_rx = np.pi / 6\n", "\n", "# Average single-site magnetization\n", "observable = (\n", " SparsePauliOp(\n", " [\n", " \"I\" * iq + \"Z\" + \"I\" * (num_qubits - iq - 1)\n", " for iq in range(num_qubits)\n", " ]\n", " )\n", " / num_qubits\n", ")\n", "\n", "# Create the Trotter circuit\n", "num_qubits = 10\n", "num_steps = 20\n", "theta_rx = np.pi / 6\n", "circuit = QuantumCircuit(num_qubits)\n", "edges = CouplingMap.from_line(num_qubits, bidirectional=False).get_edges()\n", "for _ in range(num_steps):\n", " circuit.rx(theta_rx, [i for i in range(num_qubits)])\n", " for edge in edges:\n", " circuit.sdg(edge)\n", " circuit.ry(np.pi / 2, edge[1])\n", " circuit.cx(edge[0], edge[1])\n", " circuit.ry(-np.pi / 2, edge[1])\n", "circuit.draw(\"mpl\", fold=-1)" ] }, { "cell_type": "markdown", "id": "c05fa0f8-605e-4172-8880-0de89360f27d", "metadata": {}, "source": [ "\n", "\n", "## パウリ伝播を用いて、系の時間発展をシミュレートする\n", "\n", "回路（ $U$ ）と観測可能量（ $O$ ）が準備できたら、以下の数ステップでシステムを簡単にシミュレーションできます：\n", "\n", "* $U$ を、Clifford 部分 $C$ と非 Clifford 部分 $P$ に分割し、 $U=PC$ が成り立つようにする。この際、以下の式を用いる。 `evolve_through_cliffords`\n", "* $O$ を $P$ を通じて展開すると、新しい演算子 $O^\\prime$ が得られます。その方法は以下の通りです。 `pauli_prop.propagate_through_circuit`\n", "* Qiskitに組み込まれているクリフォード進化のサポート機能を使用して、回路のクリフォード部分において $O^\\prime$ を進化させる\n", "* $O^\\prime$ における、完全対角パウリ項（すべての量子ビット `Z` で `I` または のいずれかを含むパウリ項）に関連する係数を合計することで、期待値を $\\langle0|O^\\prime|0\\rangle \\approx \\langle0|U^\\dagger OU|0\\rangle$ と近似する。 なお、これは近似であることに注意してください。これは、 $O^\\prime$ の項を、回路の非クリフォード部分を通じて伝播させる際に切り捨てたためです。\n", "\n" ] }, { "cell_type": "code", "execution_count": 2, "id": "c8a27469-ab72-4d50-9544-d446af847da5", "metadata": {}, "outputs": [], "source": [ "import time\n", "\n", "from pauli_prop import evolve_through_cliffords, propagate_through_circuit\n", "\n", "cliff, non_cliff = evolve_through_cliffords(circuit)\n", "\n", "max_terms_list = [10**i for i in range(8)]\n", "approx_evs = []\n", "durations = []\n", "for max_terms in max_terms_list:\n", " st = time.perf_counter()\n", " evolved_obs = propagate_through_circuit(\n", " observable, non_cliff, max_terms=max_terms, atol=1e-12, frame=\"h\"\n", " )[0]\n", " evolved_obs.paulis = evolved_obs.paulis.evolve(cliff, frame=\"h\")\n", " durations.append(time.perf_counter() - st)\n", " approx_evs.append(\n", " float(evolved_obs.coeffs[~evolved_obs.paulis.x.any(axis=1)].sum())\n", " )" ] }, { "cell_type": "markdown", "id": "7fb3e72f-20fb-4fd0-a301-2fcc6c0ddad5", "metadata": {}, "source": [ "より大規模な計算を行うにつれて、期待値による近似の精度は高まります。 この例では、 $4^{10}\\approx10^6$ 付近でパウリ空間全体が飽和しており、これが最後の2つの点の間で曲線が平坦になることに表れています。\n", "\n", "以下のプロットは単調収束を示していますが、パウリ伝播のシミュレーションは一般的に単調に収束するわけではありません。 この種のプロットでは、「ぎくしゃくした」動きが見られることは珍しくありません。\n", "\n" ] }, { "cell_type": "code", "execution_count": 3, "id": "f12410aa-ef3b-4ee6-ab35-f7e9001a92b8", "metadata": {}, "outputs": [ { "data": { "text/plain": [ "Text(0.5, 1.0, 'Simulating 20-step 1D Ising Model')" ] }, "execution_count": 3, "metadata": {}, "output_type": "execute_result" }, { "data": { "text/plain": [ "\"Output" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "import matplotlib.pyplot as plt\n", "from qiskit_aer import AerSimulator\n", "\n", "sim_circ = circuit.copy()\n", "sim_circ.save_statevector()\n", "backend = AerSimulator(method=\"statevector\")\n", "psi = backend.run(sim_circ).result().data()[\"statevector\"]\n", "exact_ev = psi.expectation_value(observable)\n", "\n", "ax1 = plt.gca()\n", "ax1.plot(max_terms_list, approx_evs, marker=\"o\", label=\"Approximate\")\n", "ax1.axhline(exact_ev, linestyle=\"--\", color=\"green\", label=\"Exact\")\n", "ax1.set_xscale(\"log\")\n", "ax1.set_xlabel(\"# terms kept\")\n", "ax1.set_ylabel(r\"$\\frac{1}{N} \\sum_{i=1}^{N} \\langle z_i \\rangle$\")\n", "\n", "ax2 = ax1.twinx()\n", "ax2.plot(\n", " max_terms_list, durations, marker=\".\", label=\"Runtime\", color=\"orange\"\n", ")\n", "ax2.set_ylabel(\"Runtime (s)\", color=\"orange\")\n", "ax2.set_yscale(\"log\")\n", "\n", "handles1, labels1 = ax1.get_legend_handles_labels()\n", "handles2, labels2 = ax2.get_legend_handles_labels()\n", "ax1.legend(handles1 + handles2, labels1 + labels2, loc=\"lower right\")\n", "\n", "plt.title(f\"Simulating {num_steps}-step 1D Ising Model\")" ] }, { "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 }