{ "cells": [ { "attachments": {}, "cell_type": "markdown", "id": "a94f5c51-348c-4fb0-9488-70ad4a19ace0", "metadata": {}, "source": [ "---\n", "title: \"量子シミュレーション\"\n", "description: \"このコースでは、横磁場アイジングハミルトニアンのトロッター化を含む、量子シミュレーションについて扱います\"\n", "---\n", "\n", "{/* cspell:ignore sharex */}\n", "\n", "\n", "\n", "# 量子シミュレーション\n", "\n", "\n", " 川島幸男（2024年5月30日）\n", "\n", " 講演の原文の [PDFをダウンロードする](https://ibm.ent.box.com/s/kzzsmxhw38vph1ioohczaet53euwi310)。 これらは静的画像なので、いくつかのコード・スニペットは非推奨になるかもしれないことに注意してください。\n", "\n", " *この実験の実行にかかるQPUのおおよその時間は7秒である。*\n", "\n", " (このノートブックの大部分は、Qiskit Algorithmsの[チュートリアルノートブックから](https://github.com/qiskit-community/qiskit-algorithms/blob/main/docs/tutorials/13_trotterQRTE.ipynb)引用しています)\n", "\n", "\n" ] }, { "cell_type": "markdown", "id": "aef7b02a-2b09-4409-85a0-8cf8346df430", "metadata": {}, "source": [ "\n", "\n", "## 1. はじめに\n", "\n", "実時間進化技術として、トロッター化は、ある時間スライスのシステムの時間進化を近似するように選択された量子ゲートの連続的な適用で構成される。 シュレーディンガー方程式に従うと、初期状態（ $\\vert\\psi(0)\\rangle$ ）にある系の時間発展は次のようになる：\n", "\n", "$$\n", "\\vert \\psi(t) \\rangle = e^{-i H t} \\vert \\psi(0) \\rangle \\text{,}\n", "$$\n", "\n", "ここで、 $H$ は系を支配する時間非依存ハミルトニアンである。 $H=\\sum_j a_j P_j$ $P_j$ は の量子ビットに作用するパウリ項のテンソル積を表している。 $n$ 特に、これらのパウリ項は互いに交換するかもしれないし、交換しないかもしれない。 ある時刻（ $t=0$ ）の状態が与えられたとき、量子コンピュータを使って、後の時刻（ $|\\psi(t)\\rangle$ ）のシステムの状態をどのように求めるか？ 演算子の指数は、そのテイラー級数を通して最も簡単に理解することができる：\n", "\n", "$$\n", "e^{-i H t} = 1-iHt-\\frac{1}{2}H^2t^2+...\n", "$$\n", "\n", "$e^{iZ}$ のような非常に基本的な指数関数は、コンパクトな量子ゲートのセットを使って量子コンピュータで簡単に実装できる。 関心のあるハミルトニアンのほとんどは、単一の項を持つのではなく、多くの項を持つ。 $H = H_1+H_2$ ：\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", "$H_1$ と $H_2$ が交わる場合、おなじみのケースになる（これは数字にも当てはまり、以下の変数 $a$ と $b$ にも当てはまる）：\n", "\n", "$$\n", "e^{-i (a+b) t} = e^{-i a t}e^{-i b t}\n", "$$\n", "\n", "しかし、演算子が交わらない場合、テイラー級数で項を並べ替えてこのように単純化することはできない。 このように、複雑なハミルトニアンを量子ゲートで表現することは難しい。\n", "\n", "一つの解決策は、テイラー展開の一次項が支配的になるような、非常に小さな時間 $t$ を考慮することである。 その仮定の下で：\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", "もちろん、もっと長い間、状態を進化させる必要があるかもしれない。 それは、このような小さなステップを何度も重ねることで達成される。 このプロセスはトロッター化と呼ばれる：\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", "ここで、 $t/r$ は、選択するタイムスライス（進化ステップ）である。 その結果、 $r$。 タイムステップが小さいほど、より正確な近似につながる。 しかし、これは回路を深くすることにもつながり、実際には、より多くのエラー蓄積（近い将来の量子デバイスでは無視できない懸念）につながる。\n", "\n", "今日は、 $N=2$ と $N=6$ サイトの線形格子における[イジング模型の](https://en.wikipedia.org/wiki/Ising_model)時間発展について研究する。 これらの格子は、最も近い隣人とだけ相互作用するスピン（ $\\sigma_i$ ）の配列で構成されている。 これらのスピンは、 $\\uparrow$ と $\\downarrow$ の2つの向きを持つことができ、それぞれ $+1$ と $-1$ の磁化に対応する。\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", "ここで、 $J$ は相互作用エネルギーを表し、 $h$ は外場（上記ではx方向だが、ここではこれを修正する）の大きさを表す。 パウリ行列を使ってこの式を書き、外場が横方向に対して角度 $\\alpha$ を持つことを考えよう、\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", "このハミルトニアンは、外場の影響を簡単に研究できるという点で有用である。 計算基礎では、システムは以下のように符号化される：\n", "\n", "| 量子状態 | スピン表現 |\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", "このような量子系の時間発展について調査を開始する。 より具体的には、磁化のような系の特定の特性の時間発展を可視化する。\n", "\n" ] }, { "cell_type": "markdown", "id": "8d26b146-12b4-4882-a0c8-3f7b5b15f04d", "metadata": {}, "source": [ "\n", "\n", "### 1.1 要件\n", "\n", "このチュートリアルを始める前に、以下のものがインストールされていることを確認してください：\n", "\n", "* Qiskit SDK v1.2 またはそれ以降 ( `pip install qiskit` )\n", "* Qiskit Runtime v0.30 またはそれ以降 ( `pip install qiskit-ibm-runtime` )\n", "* Numpy v1.24.1 以降 \\< 2 ( `pip install numpy` )\n", "\n" ] }, { "cell_type": "markdown", "id": "e73c8a4a-a540-4637-b179-670da6b1a6e6", "metadata": {}, "source": [ "\n", "\n", "### 1.2 ライブラリをインポートする\n", "\n", "このノートブックでは使っていないが、役に立つかもしれないライブラリ（ MatrixExponential, QDrift）も含まれていることに注意。 時間があれば試してみるといい！\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. 問題の可視化\n", "\n", "\n", "\n", "### 2.1 横磁場イジングハミルトニアンの定義\n", "\n", "ここでは、1次元横磁場イジング模型を考える。\n", "\n", "まず、システムパラメーター $N$、 $J$、 $h$、 $\\alpha$ を受け取り、ハミルトニアンを `SparsePauliOp` として返す関数を作成します。 A [SparsePauliOp](/docs/api/qiskit/qiskit.quantum_info.SparsePauliOp) は、加重[パウリ](/docs/api/qiskit/qiskit.quantum_info.Pauli)項による演算子のスパース表現である。\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", "#### ハミルトニアンを定義する\n", "\n", "これから考えるシステムは、例として $N=6$、 $J=0.2$、 $h=1.2$、 $\\alpha=\\frac{\\pi}{8.0}$ のサイズを持っている。\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 時間発展シミュレーションのパラメータを設定する\n", "\n", "ここでは、3つの異なるトロッター化技術を検討する：\n", "\n", "* リー・トロッター（ファーストオーダー）\n", "* 二次のスズキ・トロッター\n", "* 4次のスズキ・トロッター\n", "\n", "後者の2つは、練習と付録で使用する。\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 量子回路1を準備する（初期状態）\n", "\n", "初期状態を作成する。 ここでは、以下のスピン構成から始める。 $\\uparrow\\uparrow\\downarrow\\downarrow\\uparrow\\uparrow$.\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 量子回路2（時間発展用単一回路）を準備する\n", "\n", "ここでは、Lie-Trotterを使って1つの時間ステップの回路を構成する。\n", "\n", "リー積公式（一次）は [LieTrotter](/docs/api/qiskit/qiskit.synthesis.LieTrotter) クラスに実装されています。 一次式は、冒頭で述べた近似式で構成され、和の行列指数関数が行列指数関数の積で近似される：\n", "\n", "$$\n", "e^{H_1+H_2} \\approx e^{H_1} e^{H_2}\n", "$$\n", "\n", "前述したように、非常に深い回路はエラーの蓄積につながり、現代の量子コンピューターに問題を引き起こす。 2量子ビットゲートは1量子ビットゲートよりもエラー率が高いため、2量子ビット回路の深さが特に注目される。 本当に重要なのは、転置後の2量子ビット回路の深さである（それが量子コンピュータが実際に実行する回路だから）。 しかし、シミュレーターを使って、今からでもこの回路の演算を数える習慣をつけよう。\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 オペレータを測定用に設定する\n", "\n", "*磁化演算子* $\\sum_i \\langle Z_i \\rangle / N$ と*平均スピン相関演算子* $\\sum_i \\langle Z_i Z_{i+1} \\rangle/ (N - 1)$ を定義する。\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 時間発展シミュレーションを実行する\n", "\n", "エネルギー（ハミルトニアンの期待値）、磁化（磁化作用素の期待値）、平均スピン相関（平均スピン相関作用素の期待値）をモニターする。 Qiskitの `StatevectorEstimator` ( EstimatorV2 ) プリミティブは、観測値の期待値を推定する $\\langle\\psi\\vert\\hat{O}\\vert\\psi\\rangle$。\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 観測量の時間発展をプロットする\n", "\n", "計測した期待値を時間に対してプロットしてみた。\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. 練習問題1. 二階鈴木・トロッター法を用いたシミュレーションを実行する\n", "\n", "では、上に示したリー・トロッターの例にならって、2次のスズキ・トロッターでシミュレーションを行ってみよう。\n", "\n", "2次のスズキ・トロッターは、 [SuzukiTrotter クラスによって](/docs/api/qiskit/qiskit.synthesis.SuzukiTrotter) Qiskitで使用できる。 この式を用いると、2次分解は次のようになる：\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 単一時間ステップ用の回路を構築する\n", "\n", "product\\_formula\\_st2 ( SuzukiTrotter(order=2 )) を使用し、2次のスズキ・トロッターを用いて1つの時間ステップの回路を構成する。 また、ゲートの数と回路の深さを数え、Lie-Trotterと比較する。\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 時間発展シミュレーションを実行する\n", "\n", "2次のSuzuki-Trotterを用いて時間発展を行う。\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 二次の鈴木・トロッター結果をプロットする\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 正確な結果と比較する\n", "\n", "以下のデータは、古典的なコンピューターで事前に計算された正確な結果である。\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. 量子ハードウェアの実行\n", "\n", "次に、量子ハードウェア上で時間発展シミュレーションを実行する。 より小さな問題、格子サイズ N=2 を扱う。 パラメータ（ $\\alpha$ ）を変化させ、波動関数のダイナミクスの違いを見る。\n", "\n" ] }, { "cell_type": "markdown", "id": "9a5bb8bf-9a96-4464-8bcc-eeaee193c23a", "metadata": {}, "source": [ "\n", "\n", "### 4.1 ステップ1. 古典的な入力を量子問題にマッピングする\n", "\n", "シミュレーションの初期設定を選ぶ：\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": [ "それから初期回路を設定する：\n", "\n", "スピンの初期設定は \"ダウンアップ\"\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": [ "ここで、理想的な状態ベクトル・シミュレーターを使って基準値を計算する。\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": [ "$\\vert\\psi(0)\\rangle = \\vert10\\rangle$ に対応するスピン配列 $\\downarrow\\uparrow$ を持つ系を最初に用意した。横磁場( $\\alpha=0^\\circ$ )のもとで $t=1.6$ まで進化させた後、 $\\uparrow\\downarrow$ の測定、つまりスピンの入れ替えがほぼ保証される。 (ラベルは右から左に解釈されることに注意）。 磁場が縦方向( $\\alpha=\\pm90^\\circ$ )であれば、進化はないため、最初に準備された状態( $\\downarrow\\uparrow$ )で測定することになる。中間の角度（ $\\alpha=\\pm45^\\circ$ ）では、すべての組み合わせを異なる確率で測定することができる。\n", "\n" ] }, { "cell_type": "markdown", "id": "36ef1a7e-1ab7-48ed-89ad-efbef819c871", "metadata": {}, "source": [ "\n", "\n", "#### HW実験用の回路を構築する\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 ステップ2. ターゲット・ハードウェア用に最適化する\n", "\n", "バックエンドを指定する。\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": [ "次に、選択したバックエンドの回路をトランスパイルする。\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": [ "サーキットをチェックしよう。\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 ステップ3。 IBM Quantum プリミティブを使用して実行する\n", "\n", "Qiskitの `Sampler` ( V2 ) プリミティブは、測定されたビット列のカウントを提供します。\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": [ "結果を保存する\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 ステップ4. ポスト・プロセス結果\n", "\n", "波動関数の分析に相当するビット列のヒストグラムを作成し、上に示した理想値と比較する。\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": [ "ここでは、高次（4次）スズキ・トロッターを用いた回路構成例を示す。\n", "では、上記の例に従って、4次のスズキ・トロッターで回路シミュレーションを組んでみよう。\n", "\n", "4次のスズキ・トロッターは、 [SuzukiTrotter クラスによって](/docs/api/qiskit/qiskit.synthesis.SuzukiTrotter) Qiskitで使用できる。 4次は以下の再帰関係を使って評価できる。 なお、以下の式ではスズキ・トロッターの次数を\" 2k \"と表記している。\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", "#### 単一時間ステップ用の回路を構築する\n", "\n", "product\\_formula\\_st4 ( SuzukiTrotter(order=4 )) を使用し、4次のスズキ・トロッターを用いて1つの時間ステップの回路を構成する。 また、ゲートの数と回路の深さを数え、Lie-Trotterと2次のSuzuki-Trotterと比較する。\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 }