{ "cells": [ { "cell_type": "markdown", "id": "b97a7e0a-66a6-46b2-b33f-47feeefad60d", "metadata": {}, "source": [ "---\n", "title: \"ユーティリティII\"\n", "description: \"このノートは第7課の方法と技法に従っています。我々の目標は、時間依存シュレーディンガー方程式を数値的に解くことである。\"\n", "---\n", "\n", "{/* cspell:ignore sharex */}\n", "\n", "\n", "\n", "# ユーティリティ規模の実験II\n", "\n", "\n", " 川島幸夫（2024年7月12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/bipgoms7gr6b6vhkoc1uw6oi4wsanfoq)。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", " *この実験にかかるQPUのおおよその時間は2m30秒である。*\n", "\n", " (このノートブックは、Qiskit Algorithmsの[チュートリアルノートブック(](https://github.com/qiskit-community/qiskit-algorithms/blob/main/docs/tutorials/13_trotterQRTE.ipynb) 現在は廃止)のテキスト、図解、コードを使用していることに注意)\n", "\n", "\n" ] }, { "cell_type": "markdown", "id": "76421ffc-f47f-46a4-8198-af3bc21e3c5d", "metadata": {}, "source": [ "\n", "\n", "## 1. 導入と時間発展の概観\n", "\n", "このノートブックは、レッスン7のメソッドとテクニックを踏襲している。我々の目標は、時間依存のシュレーディンガー方程式を数値的に解くことである。レッスン7で説明したように、トロッター化とは、ある時間スライスにおける系の時間発展を近似するために選択された量子ゲートを連続的に適用することである。ここでは便宜上、その議論を繰り返す。最近レッスン7を復習された方は、以下のコード・セルまで読み飛ばしてください。\n", "\n", "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す。 $n$ 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": "code", "execution_count": 1, "id": "81aff7b3-67e5-453b-9f5b-68baf5209560", "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": "c88eda88-da8a-4e5e-9b8b-53ac0d0ed91b", "metadata": { "scrolled": true, "tags": [] }, "outputs": [], "source": [ "# Import the qiskit library\n", "\n", "import numpy as np\n", "import warnings\n", "\n", "from qiskit import QuantumCircuit, QuantumRegister\n", "from qiskit.circuit.library import PauliEvolutionGate\n", "from qiskit.quantum_info import SparsePauliOp\n", "from qiskit.synthesis import LieTrotter\n", "from qiskit.transpiler.preset_passmanagers import generate_preset_pass_manager\n", "\n", "from qiskit_aer import AerSimulator\n", "from qiskit_ibm_runtime import QiskitRuntimeService, Estimator\n", "\n", "warnings.filterwarnings(\"ignore\")" ] }, { "cell_type": "markdown", "id": "ed6c1602-d994-46d6-9426-d38272a63d56", "metadata": {}, "source": [ "\n", "\n", "## 2. 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 $N$、 $J$、 $h$ 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": "markdown", "id": "b3137f47-640a-4382-95b6-6bb9e760bc96", "metadata": {}, "source": [ "\n", "\n", "### 2.1 アクティビティ1\n", "\n", "量子ビット数\"、\"Jパラメータ\"、\"hパラメータ \"を引数として、横場イジング・ハミルトニアン（上式参照）を構築する関数を構築する。これまでの例を使って、自分でやってみよう。解決策は下にスクロールしてください。\n", "\n", "**解決策：**\n", "\n" ] }, { "cell_type": "code", "execution_count": 3, "id": "3de92af7-7504-4dca-b0b4-0d934f0c2069", "metadata": {}, "outputs": [], "source": [ "def get_hamiltonian(nqubits, J, h):\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", " X_tuples = [(\"X\", [i], -h) 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, *X_tuples], num_qubits=nqubits\n", " )\n", " return hamiltonian.simplify()" ] }, { "cell_type": "markdown", "id": "8b147fb0-6c3e-4eaf-8553-8850b53ff121", "metadata": {}, "source": [ "磁化を追跡しながら、量子系の時間発展を調べ始める。\n", "ここでは、StatevectorとMatrix Product Stateシミュレータの結果を比較する。\n", "\n", "\n", "\n", "#### ハミルトニアンを定義する\n", "\n", "ここで考えるシステムのサイズは $N=20$ である。\n", "\n" ] }, { "cell_type": "code", "execution_count": 4, "id": "56764f93-3851-4b1c-b9e2-ea34161ddd85", "metadata": {}, "outputs": [ { "data": { "text/plain": [ "SparsePauliOp(['IIIIIIIIIIIIIIIIIIZZ', 'IIIIIIIIIIIIIIIIIZZI', 'IIIIIIIIIIIIIIIIZZII', 'IIIIIIIIIIIIIIIZZIII', 'IIIIIIIIIIIIIIZZIIII', 'IIIIIIIIIIIIIZZIIIII', 'IIIIIIIIIIIIZZIIIIII', 'IIIIIIIIIIIZZIIIIIII', 'IIIIIIIIIIZZIIIIIIII', 'IIIIIIIIIZZIIIIIIIII', 'IIIIIIIIZZIIIIIIIIII', 'IIIIIIIZZIIIIIIIIIII', 'IIIIIIZZIIIIIIIIIIII', 'IIIIIZZIIIIIIIIIIIII', 'IIIIZZIIIIIIIIIIIIII', 'IIIZZIIIIIIIIIIIIIII', 'IIZZIIIIIIIIIIIIIIII', 'IZZIIIIIIIIIIIIIIIII', 'ZZIIIIIIIIIIIIIIIIII', 'IIIIIIIIIIIIIIIIIIIX', 'IIIIIIIIIIIIIIIIIIXI', 'IIIIIIIIIIIIIIIIIXII', 'IIIIIIIIIIIIIIIIXIII', 'IIIIIIIIIIIIIIIXIIII', 'IIIIIIIIIIIIIIXIIIII', 'IIIIIIIIIIIIIXIIIIII', 'IIIIIIIIIIIIXIIIIIII', 'IIIIIIIIIIIXIIIIIIII', 'IIIIIIIIIIXIIIIIIIII', 'IIIIIIIIIXIIIIIIIIII', 'IIIIIIIIXIIIIIIIIIII', 'IIIIIIIXIIIIIIIIIIII', 'IIIIIIXIIIIIIIIIIIII', 'IIIIIXIIIIIIIIIIIIII', 'IIIIXIIIIIIIIIIIIIII', 'IIIXIIIIIIIIIIIIIIII', 'IIXIIIIIIIIIIIIIIIII', 'IXIIIIIIIIIIIIIIIIII', 'XIIIIIIIIIIIIIIIIIII'],\n", " coeffs=[-1.+0.j, -1.+0.j, -1.+0.j, -1.+0.j, -1.+0.j, -1.+0.j, -1.+0.j, -1.+0.j,\n", " -1.+0.j, -1.+0.j, -1.+0.j, -1.+0.j, -1.+0.j, -1.+0.j, -1.+0.j, -1.+0.j,\n", " -1.+0.j, -1.+0.j, -1.+0.j, 5.+0.j, 5.+0.j, 5.+0.j, 5.+0.j, 5.+0.j,\n", " 5.+0.j, 5.+0.j, 5.+0.j, 5.+0.j, 5.+0.j, 5.+0.j, 5.+0.j, 5.+0.j,\n", " 5.+0.j, 5.+0.j, 5.+0.j, 5.+0.j, 5.+0.j, 5.+0.j, 5.+0.j])" ] }, "execution_count": 4, "metadata": {}, "output_type": "execute_result" } ], "source": [ "n_qubits = 20\n", "hamiltonian = get_hamiltonian(nqubits=n_qubits, J=1.0, h=-5.0)\n", "hamiltonian" ] }, { "cell_type": "markdown", "id": "eb075fee-fdb5-4016-bea2-643b444062b8", "metadata": {}, "source": [ "\n", "\n", "#### 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、リー・トロッター（一次）について考える。\n", "\n" ] }, { "cell_type": "code", "execution_count": 5, "id": "3a5af3d3-015b-4a5c-86cd-52830067a1e2", "metadata": {}, "outputs": [], "source": [ "num_timesteps = 20\n", "evolution_time = 2.0\n", "dt = evolution_time / num_timesteps\n", "product_formula_lt = LieTrotter()" ] }, { "cell_type": "markdown", "id": "e5fc52a2-15af-464f-aac2-890417745c82", "metadata": {}, "source": [ "\n", "\n", "#### 量子回路を準備する（初期状態）\n", "\n", "初期状態を作成する。我々は、強磁性状態（オールアップまたはオールダウン）である基底状態から始める。ここでは、オールアップ（すべて「0」）を例にしている。\n", "\n" ] }, { "cell_type": "code", "execution_count": 6, "id": "64f622f2-6e1c-4b42-b548-dd3e5aa32786", "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(\"00000000000000000000\")\n", "# Change reps and see the difference when you decompose the circuit\n", "initial_circuit.decompose(reps=1).draw(\"mpl\")" ] }, { "cell_type": "markdown", "id": "dab0957f-186f-475f-bdee-907469b54174", "metadata": {}, "source": [ "\n", "\n", "#### 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", "リー積の公式（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", "この回路の動作を数えてみよう。\n", "\n" ] }, { "cell_type": "code", "execution_count": 7, "id": "b812f4b6-fb83-4c89-a8bc-ac0d85784720", "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "\n", "Trotter step with Lie-Trotter\n", "-----------------------------\n", "Depth: 58\n", "Gate count: 77\n", "Nonlocal gate count: 38\n", "Gate breakdown: CX: 38, U3: 20, U1: 19\n", "\n" ] }, { "data": { "text/plain": [ " $\"Output$ " ] }, "execution_count": 7, "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": "474e7a29-82bc-470e-8556-87e88195f213", "metadata": {}, "source": [ "\n", "\n", "#### 測定対象の演算子を設定する\n", "\n", "*磁化演算子* $\\sum_i Z_i / N$ を定義しよう。\n", "\n" ] }, { "cell_type": "code", "execution_count": 8, "id": "2488d18d-70f0-43a2-9675-4d34562e47ce", "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "magnetization : SparsePauliOp(['IIIIIIIIIIIIIIIIIIIZ', 'IIIIIIIIIIIIIIIIIIZI', 'IIIIIIIIIIIIIIIIIZII', 'IIIIIIIIIIIIIIIIZIII', 'IIIIIIIIIIIIIIIZIIII', 'IIIIIIIIIIIIIIZIIIII', 'IIIIIIIIIIIIIZIIIIII', 'IIIIIIIIIIIIZIIIIIII', 'IIIIIIIIIIIZIIIIIIII', 'IIIIIIIIIIZIIIIIIIII', 'IIIIIIIIIZIIIIIIIIII', 'IIIIIIIIZIIIIIIIIIII', 'IIIIIIIZIIIIIIIIIIII', 'IIIIIIZIIIIIIIIIIIII', 'IIIIIZIIIIIIIIIIIIII', 'IIIIZIIIIIIIIIIIIIII', 'IIIZIIIIIIIIIIIIIIII', 'IIZIIIIIIIIIIIIIIIII', 'IZIIIIIIIIIIIIIIIIII', 'ZIIIIIIIIIIIIIIIIIII'],\n", " coeffs=[0.05+0.j, 0.05+0.j, 0.05+0.j, 0.05+0.j, 0.05+0.j, 0.05+0.j, 0.05+0.j,\n", " 0.05+0.j, 0.05+0.j, 0.05+0.j, 0.05+0.j, 0.05+0.j, 0.05+0.j, 0.05+0.j,\n", " 0.05+0.j, 0.05+0.j, 0.05+0.j, 0.05+0.j, 0.05+0.j, 0.05+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", "print(\"magnetization : \", magnetization)" ] }, { "cell_type": "markdown", "id": "308ed03d-432d-4113-9c06-2ce10c02ecd5", "metadata": {}, "source": [ "\n", "\n", "#### 時間発展シミュレーションを実行する\n", "\n", "磁化（磁化作用素の期待値）をモニターする。 StatevectorとMPSのシミュレーターを使い、結果を比較する。\n", "\n" ] }, { "cell_type": "code", "execution_count": 9, "id": "25bd2824-009e-4d76-be83-a906bf8b0d44", "metadata": { "scrolled": true }, "outputs": [], "source": [ "# Step 1. Map the problem\n", "# 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", "\n", "# Define backend (simulator)\n", "# MPS\n", "backend_mps = AerSimulator(method=\"matrix_product_state\")\n", "# Statevector\n", "backend_sv = AerSimulator(method=\"statevector\")\n", "\n", "# Set Runtime Estimator\n", "# MPS\n", "estimator_mps = Estimator(mode=backend_mps)\n", "# Statevector\n", "estimator_sv = Estimator(mode=backend_sv)\n", "\n", "# Step 2. Optimize\n", "# Set pass manager\n", "# MPS\n", "pm_mps = generate_preset_pass_manager(optimization_level=3, backend=backend_mps)\n", "# Statevector\n", "pm_sv = generate_preset_pass_manager(optimization_level=3, backend=backend_sv)\n", "\n", "# Transpile initial circuit\n", "# MPS\n", "evolved_state_mps = pm_mps.run(evolved_state)\n", "# Statevector\n", "evolved_state_sv = pm_sv.run(evolved_state)\n", "\n", "# Apply layout to the operator\n", "# MPS\n", "magnetization_mps = magnetization.apply_layout(evolved_state_mps.layout)\n", "# Statevector\n", "magnetization_sv = magnetization.apply_layout(evolved_state_sv.layout)\n", "\n", "mag_mps_list = []\n", "mag_sv_list = []\n", "\n", "# Step 3. Run the circuit\n", "# Estimate expectation values for t=0.0: MPS\n", "job = estimator_mps.run([(evolved_state_mps, [magnetization_mps])])\n", "# Get estimated expectation values: MPS\n", "evs = job.result()[0].data.evs\n", "# Collect data: MPS\n", "mag_mps_list.append(evs[0])\n", "\n", "# Estimate expectation values for t=0.0: Statevector\n", "job = estimator_sv.run([(evolved_state_sv, [magnetization_sv])])\n", "# Get estimated expectation values: Statevector\n", "evs = job.result()[0].data.evs\n", "# Collect data: Statevector\n", "mag_sv_list.append(evs[0])\n", "\n", "# Start time evolution\n", "for n in range(num_timesteps):\n", " # Step 1. Map the problem\n", " # Expand the circuit to describe delta-t\n", " evolved_state.append(single_step_evolution_lt, evolved_state.qubits)\n", " # Step 2. Optimize\n", " # Transpile the circuit: MPS\n", " evolved_state_mps = pm_mps.run(evolved_state)\n", " # Apply the physical layout of the qubits to the operator: MPS\n", " magnetization_mps = magnetization.apply_layout(evolved_state_mps.layout)\n", " # Step 3. Run the circuit\n", " # Estimate expectation values at delta-t: MPS\n", " job = estimator_mps.run([(evolved_state_mps, [magnetization_mps])])\n", " # Get estimated expectation values: MPS\n", " evs = job.result()[0].data.evs\n", " # Collect data: MPS\n", " mag_mps_list.append(evs[0])\n", "\n", " # Step 2. Optimize\n", " # Transpile the circuit: Statevector\n", " evolved_state_sv = pm_sv.run(evolved_state)\n", " # Apply the physical layout of the qubits to the operator: Statevector\n", " magnetization_sv = magnetization.apply_layout(evolved_state_sv.layout)\n", " # Step 3. Run the circuit\n", " # Estimate expectation values at delta-t: Statevector\n", " job = estimator_sv.run([(evolved_state_sv, [magnetization_sv])])\n", " # Get estimated expectation values: Statevector\n", " evs = job.result()[0].data.evs\n", " # Collect data: Statevector\n", " mag_sv_list.append(evs[0])\n", "\n", "# Transform the list of expectation values (at each time step) to arrays\n", "mag_mps_array = np.array(mag_mps_list)\n", "mag_sv_array = np.array(mag_sv_list)" ] }, { "cell_type": "markdown", "id": "046df34e-4cc8-455a-a3e8-f6b345c9249a", "metadata": {}, "source": [ "\n", "\n", "#### 観測量の時間発展をプロットする\n", "\n", "測定した期待値を時間に対してプロットしてみた。状態ベクトル空間シミュレータと行列積空間シミュレータの結果が一致することを確認する。\n", "\n" ] }, { "cell_type": "code", "execution_count": null, "id": "1762b191-c234-4acc-a6ea-8433a59ef3f8", "metadata": {}, "outputs": [ { "data": { "text/plain": [ "Text(0.5, 0.98, 'Observable evolution')" ] }, "execution_count": 10, "metadata": {}, "output_type": "execute_result" }, { "data": { "text/plain": [ " $\"Output$ " ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "import matplotlib.pyplot as plt\n", "\n", "# Step 4. Post-processing\n", "fig, axes = plt.subplots(2, sharex=True)\n", "times = np.linspace(0, evolution_time, num_timesteps + 1) # includes initial state\n", "axes[0].plot(\n", " times, mag_mps_array, label=\"MPS\", marker=\"x\", c=\"darkmagenta\", ls=\"-\", lw=0.8\n", ")\n", "axes[1].plot(\n", " times, mag_sv_array, label=\"SV\", marker=\"x\", c=\"darkmagenta\", ls=\"-\", lw=0.8\n", ")\n", "\n", "axes[0].set_ylabel(\"MPS\")\n", "axes[1].set_ylabel(\"Statevector\")\n", "axes[1].set_xlabel(\"Time\")\n", "fig.suptitle(\"Observable evolution\")" ] }, { "cell_type": "markdown", "id": "4a4a4efa-de95-470e-8030-1b842e63f904", "metadata": {}, "source": [ "量子系の時間発展を、性質を追跡しながら調べ始める。\n", "ここでは、行列積状態シミュレータと実際の量子デバイスの結果を比較する。\n", "\n", "\n", "\n", "### 2.2 アクティビティ2\n", "\n", "\n", "\n", "#### ハミルトニアンを定義する\n", "\n", "$N=70$ 他の条件は20量子ビットの問題と同じである。解答は下にスクロールしてください。\n", "\n", "**解決策：**\n", "\n" ] }, { "cell_type": "code", "execution_count": 11, "id": "b8ee438b-69ac-438d-a8c9-a2055c347f0e", "metadata": {}, "outputs": [ { "data": { "text/plain": [ "SparsePauliOp(['IIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIZZ', 'IIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIZZI', 'IIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIZZII', 'IIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIZZIII', 'IIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIZZIIII', 'IIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIZZIIIII', 'IIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIZZIIIIII', 'IIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIZZIIIIIII', 'IIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIZZIIIIIIII', 'IIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIZZIIIIIIIII', 'IIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIZZIIIIIIIIII', 'IIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIZZIIIIIIIIIII', 'IIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIZZIIIIIIIIIIII', 'IIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIZZIIIIIIIIIIIII', 'IIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIZZIIIIIIIIIIIIII', 'IIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIZZIIIIIIIIIIIIIII', 'IIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIZZIIIIIIIIIIIIIIII', 'IIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIZZIIIIIIIIIIIIIIIII', 'IIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIZZIIIIIIIIIIIIIIIIII', 'IIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIZZIIIIIIIIIIIIIIIIIII', 'IIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIZZIIIIIIIIIIIIIIIIIIII', 'IIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIZZIIIIIIIIIIIIIIIIIIIII', 'IIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIZZIIIIIIIIIIIIIIIIIIIIII', 'IIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIZZIIIIIIIIIIIIIIIIIIIIIII', 'IIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIZZIIIIIIIIIIIIIIIIIIIIIIII', 'IIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIZZIIIIIIIIIIIIIIIIIIIIIIIII', 'IIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIZZIIIIIIIIIIIIIIIIIIIIIIIIII', 'IIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIZZIIIIIIIIIIIIIIIIIIIIIIIIIII', 'IIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIZZIIIIIIIIIIIIIIIIIIIIIIIIIIII', 'IIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIZZIIIIIIIIIIIIIIIIIIIIIIIIIIIII', 'IIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIZZIIIIIIIIIIIIIIIIIIIIIIIIIIIIII', 'IIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIZZIIIIIIIIIIIIIIIIIIIIIIIIIIIIIII', 'IIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIZZIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIII', 'IIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIZZIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIII', 'IIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIZZIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIII', 'IIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIZZIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIII', 'IIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIZZIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIII', 'IIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIZZIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIII', 'IIIIIIIIIIIIIIIIIIIIIIIIIIIIIIZZIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIII', 'IIIIIIIIIIIIIIIIIIIIIIIIIIIIIZZIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIII', 'IIIIIIIIIIIIIIIIIIIIIIIIIIIIZZIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIII', 'IIIIIIIIIIIIIIIIIIIIIIIIIIIZZIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIII', 'IIIIIIIIIIIIIIIIIIIIIIIIIIZZIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIII', 'IIIIIIIIIIIIIIIIIIIIIIIIIZZIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIII', 'IIIIIIIIIIIIIIIIIIIIIIIIZZIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIII', 'IIIIIIIIIIIIIIIIIIIIIIIZZIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIII', 'IIIIIIIIIIIIIIIIIIIIIIZZIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIII', 'IIIIIIIIIIIIIIIIIIIIIZZIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIII', 'IIIIIIIIIIIIIIIIIIIIZZIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIII', 'IIIIIIIIIIIIIIIIIIIZZIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIII', 'IIIIIIIIIIIIIIIIIIZZIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIII', 'IIIIIIIIIIIIIIIIIZZIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIII', 'IIIIIIIIIIIIIIIIZZIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIII', 'IIIIIIIIIIIIIIIZZIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIII', 'IIIIIIIIIIIIIIZZIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIII', 'IIIIIIIIIIIIIZZIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIII', 'IIIIIIIIIIIIZZIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIII', 'IIIIIIIIIIIZZIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIII', 'IIIIIIIIIIZZIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIII', 'IIIIIIIIIZZIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIII', 'IIIIIIIIZZIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIII', 'IIIIIIIZZIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIII', 'IIIIIIZZIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIII', 'IIIIIZZIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIII', 'IIIIZZIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIII', 'IIIZZIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIII', 'IIZZIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIII', 'IZZIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIII', 'ZZIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIII', 'IIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIX', 'IIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIXI', 'IIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIXII', 'IIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIXIII', 'IIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIXIIII', 'IIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIXIIIII', 'IIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIXIIIIII', 'IIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIXIIIIIII', 'IIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIXIIIIIIII', 'IIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIXIIIIIIIII', 'IIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIXIIIIIIIIII', 'IIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIXIIIIIIIIIII', 'IIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIXIIIIIIIIIIII', 'IIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIXIIIIIIIIIIIII', 'IIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIXIIIIIIIIIIIIII', 'IIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIXIIIIIIIIIIIIIII', 'IIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIXIIIIIIIIIIIIIIII', 'IIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIXIIIIIIIIIIIIIIIII', 'IIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIXIIIIIIIIIIIIIIIIII', 'IIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIXIIIIIIIIIIIIIIIIIII', 'IIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIXIIIIIIIIIIIIIIIIIIII', 'IIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIXIIIIIIIIIIIIIIIIIIIII', 'IIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIXIIIIIIIIIIIIIIIIIIIIII', 'IIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIXIIIIIIIIIIIIIIIIIIIIIII', 'IIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIXIIIIIIIIIIIIIIIIIIIIIIII', 'IIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIXIIIIIIIIIIIIIIIIIIIIIIIII', 'IIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIXIIIIIIIIIIIIIIIIIIIIIIIIII', 'IIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIXIIIIIIIIIIIIIIIIIIIIIIIIIII', 'IIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIXIIIIIIIIIIIIIIIIIIIIIIIIIIII', 'IIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIXIIIIIIIIIIIIIIIIIIIIIIIIIIIII', 'IIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIXIIIIIIIIIIIIIIIIIIIIIIIIIIIIII', 'IIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIXIIIIIIIIIIIIIIIIIIIIIIIIIIIIIII', 'IIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIXIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIII', 'IIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIXIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIII', 'IIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIXIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIII', 'IIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIXIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIII', 'IIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIXIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIII', 'IIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIXIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIII', 'IIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIXIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIII', 'IIIIIIIIIIIIIIIIIIIIIIIIIIIIIIXIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIII', 'IIIIIIIIIIIIIIIIIIIIIIIIIIIIIXIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIII', 'IIIIIIIIIIIIIIIIIIIIIIIIIIIIXIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIII', 'IIIIIIIIIIIIIIIIIIIIIIIIIIIXIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIII', 'IIIIIIIIIIIIIIIIIIIIIIIIIIXIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIII', 'IIIIIIIIIIIIIIIIIIIIIIIIIXIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIII', 'IIIIIIIIIIIIIIIIIIIIIIIIXIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIII', 'IIIIIIIIIIIIIIIIIIIIIIIXIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIII', 'IIIIIIIIIIIIIIIIIIIIIIXIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIII', 'IIIIIIIIIIIIIIIIIIIIIXIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIII', 'IIIIIIIIIIIIIIIIIIIIXIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIII', 'IIIIIIIIIIIIIIIIIIIXIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIII', 'IIIIIIIIIIIIIIIIIIXIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIII', 'IIIIIIIIIIIIIIIIIXIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIII', 'IIIIIIIIIIIIIIIIXIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIII', 'IIIIIIIIIIIIIIIXIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIII', 'IIIIIIIIIIIIIIXIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIII', 'IIIIIIIIIIIIIXIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIII', 'IIIIIIIIIIIIXIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIII', 'IIIIIIIIIIIXIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIII', 'IIIIIIIIIIXIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIII', 'IIIIIIIIIXIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIII', 'IIIIIIIIXIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIII', 'IIIIIIIXIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIII', 'IIIIIIXIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIII', 'IIIIIXIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIII', 'IIIIXIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIII', 'IIIXIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIII', 'IIXIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIII', 'IXIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIII', 'XIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIII'],\n", " coeffs=[-1.+0.j, -1.+0.j, -1.+0.j, -1.+0.j, -1.+0.j, -1.+0.j, -1.+0.j, -1.+0.j,\n", " -1.+0.j, -1.+0.j, -1.+0.j, -1.+0.j, -1.+0.j, -1.+0.j, -1.+0.j, -1.+0.j,\n", " -1.+0.j, -1.+0.j, -1.+0.j, -1.+0.j, -1.+0.j, -1.+0.j, -1.+0.j, -1.+0.j,\n", " -1.+0.j, -1.+0.j, -1.+0.j, -1.+0.j, -1.+0.j, -1.+0.j, -1.+0.j, -1.+0.j,\n", " -1.+0.j, -1.+0.j, -1.+0.j, -1.+0.j, -1.+0.j, -1.+0.j, -1.+0.j, -1.+0.j,\n", " -1.+0.j, -1.+0.j, -1.+0.j, -1.+0.j, -1.+0.j, -1.+0.j, -1.+0.j, -1.+0.j,\n", " -1.+0.j, -1.+0.j, -1.+0.j, -1.+0.j, -1.+0.j, -1.+0.j, -1.+0.j, -1.+0.j,\n", " -1.+0.j, -1.+0.j, -1.+0.j, -1.+0.j, -1.+0.j, -1.+0.j, -1.+0.j, -1.+0.j,\n", " -1.+0.j, -1.+0.j, -1.+0.j, -1.+0.j, -1.+0.j, 5.+0.j, 5.+0.j, 5.+0.j,\n", " 5.+0.j, 5.+0.j, 5.+0.j, 5.+0.j, 5.+0.j, 5.+0.j, 5.+0.j, 5.+0.j,\n", " 5.+0.j, 5.+0.j, 5.+0.j, 5.+0.j, 5.+0.j, 5.+0.j, 5.+0.j, 5.+0.j,\n", " 5.+0.j, 5.+0.j, 5.+0.j, 5.+0.j, 5.+0.j, 5.+0.j, 5.+0.j, 5.+0.j,\n", " 5.+0.j, 5.+0.j, 5.+0.j, 5.+0.j, 5.+0.j, 5.+0.j, 5.+0.j, 5.+0.j,\n", " 5.+0.j, 5.+0.j, 5.+0.j, 5.+0.j, 5.+0.j, 5.+0.j, 5.+0.j, 5.+0.j,\n", " 5.+0.j, 5.+0.j, 5.+0.j, 5.+0.j, 5.+0.j, 5.+0.j, 5.+0.j, 5.+0.j,\n", " 5.+0.j, 5.+0.j, 5.+0.j, 5.+0.j, 5.+0.j, 5.+0.j, 5.+0.j, 5.+0.j,\n", " 5.+0.j, 5.+0.j, 5.+0.j, 5.+0.j, 5.+0.j, 5.+0.j, 5.+0.j, 5.+0.j,\n", " 5.+0.j, 5.+0.j, 5.+0.j])" ] }, "execution_count": 11, "metadata": {}, "output_type": "execute_result" } ], "source": [ "# Set the number of qubits\n", "n_qubits2 = 70\n", "# Construct the Hamiltonian by calling the function you made in Activity 1\n", "hamiltonian2 = get_hamiltonian(nqubits=n_qubits2, J=1.0, h=-5.0)\n", "hamiltonian2" ] }, { "cell_type": "markdown", "id": "8bcad10c-1ad8-4d62-ba05-6c3d4792e4b9", "metadata": {}, "source": [ "\n", "\n", "### 2.3 アクティビティ３\n", "\n", "初期状態を作成する。我々は、強磁性状態（オールアップまたはオールダウン）である基底状態から始める。ここでは、オールアップ（すべて「0」）を例にしている。解答は下にスクロールしてください。\n", "\n", "**解決策：**\n", "\n" ] }, { "cell_type": "code", "execution_count": 12, "id": "b7bfecd3-cc23-4005-89d5-61754b72ac7d", "metadata": {}, "outputs": [ { "data": { "text/plain": [ " $\"Output$ " ] }, "execution_count": 12, "metadata": {}, "output_type": "execute_result" } ], "source": [ "# Initiate the (quantum)circuit\n", "initial_circuit2 = QuantumCircuit(n_qubits2)\n", "# Use QuantumCircuit.prepare_state() to define the initial state\n", "initial_circuit2.prepare_state(\n", " \"0000000000000000000000000000000000000000000000000000000000000000000000\"\n", ")\n", "# Change reps and see the difference when you decompose the circuit\n", "initial_circuit2.decompose(reps=1).draw(\"mpl\")" ] }, { "cell_type": "markdown", "id": "d3699dfb-5f77-4c2b-b4c7-13ca873b032b", "metadata": {}, "source": [ "\n", "\n", "### 2.4 アクティビティ4\n", "\n", "\n", "\n", "#### 70量子ビット問題のための量子回路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", "20量子ビットの場合と同様に、リー積の公式（一次）が [LieTrotter](/docs/api/qiskit/qiskit.synthesis.LieTrotter) クラスに実装される。ここでも、一次式は上記の近似式で構成される：\n", "\n", "$$\n", "e^{H_1+H_2} \\approx e^{H_1} e^{H_2}\n", "$$\n", "\n", "20qubitの場合を例にして、自分で試してみよう。前回と同様に、この回路の動作を数える。\n", "\n", "**解決策：**\n", "\n" ] }, { "cell_type": "code", "execution_count": 13, "id": "adb8395b-478a-4db0-b85e-456be61c9e69", "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "\n", "Trotter step with Lie-Trotter\n", "-----------------------------\n", "Depth: 208\n", "Gate count: 277\n", "Nonlocal gate count: 138\n", "Gate breakdown: CX: 138, U3: 70, U1: 69\n", "\n" ] }, { "data": { "text/plain": [ " $\"Output$ " ] }, "execution_count": 13, "metadata": {}, "output_type": "execute_result" } ], "source": [ "# Construct the gates using PauliEvolutionGate()\n", "single_step_evolution_gates_lt2 = PauliEvolutionGate(\n", " hamiltonian2, dt, synthesis=LieTrotter()\n", ")\n", "# Initiate the quantum circuit\n", "single_step_evolution_lt2 = QuantumCircuit(n_qubits2)\n", "# Append the gates defined above\n", "single_step_evolution_lt2.append(\n", " single_step_evolution_gates_lt2, single_step_evolution_lt2.qubits\n", ")\n", "\n", "print(\n", " f\"\"\"\n", "Trotter step with Lie-Trotter\n", "-----------------------------\n", "Depth: {single_step_evolution_lt2.decompose(reps=3).depth()}\n", "Gate count: {len(single_step_evolution_lt2.decompose(reps=3))}\n", "Nonlocal gate count: {single_step_evolution_lt2.decompose(reps=3).num_nonlocal_gates()}\n", "Gate breakdown: {\", \".join([f\"{k.upper()}: {v}\" for k, v in single_step_evolution_lt2.decompose(reps=3).count_ops().items()])}\n", "\"\"\"\n", ")\n", "single_step_evolution_lt2.decompose(reps=3).draw(\"mpl\", fold=-1)" ] }, { "cell_type": "markdown", "id": "acede2cd-d3d8-45b8-8bc5-b307fc1f32fe", "metadata": {}, "source": [ "\n", "\n", "### 2.5 アクティビティ5\n", "\n", "\n", "\n", "#### 測定対象の演算子を設定する\n", "\n", "20量子ビットの場合と全く同じ*磁化作用素を*定義します。 $\\sum_i Z_i / N$。20量子ビットの解を修正することで、自分で試してみてください。\n", "\n", "**解決策：**\n", "\n" ] }, { "cell_type": "code", "execution_count": 14, "id": "9cac351a-b15c-4aff-87f1-c877946664d3", "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "magnetization : SparsePauliOp(['IIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIZ', 'IIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIZI', 'IIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIZII', 'IIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIZIII', 'IIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIZIIII', 'IIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIZIIIII', 'IIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIZIIIIII', 'IIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIZIIIIIII', 'IIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIZIIIIIIII', 'IIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIZIIIIIIIII', 'IIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIZIIIIIIIIII', 'IIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIZIIIIIIIIIII', 'IIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIZIIIIIIIIIIII', 'IIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIZIIIIIIIIIIIII', 'IIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIZIIIIIIIIIIIIII', 'IIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIZIIIIIIIIIIIIIII', 'IIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIZIIIIIIIIIIIIIIII', 'IIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIZIIIIIIIIIIIIIIIII', 'IIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIZIIIIIIIIIIIIIIIIII', 'IIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIZIIIIIIIIIIIIIIIIIII', 'IIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIZIIIIIIIIIIIIIIIIIIII', 'IIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIZIIIIIIIIIIIIIIIIIIIII', 'IIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIZIIIIIIIIIIIIIIIIIIIIII', 'IIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIZIIIIIIIIIIIIIIIIIIIIIII', 'IIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIZIIIIIIIIIIIIIIIIIIIIIIII', 'IIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIZIIIIIIIIIIIIIIIIIIIIIIIII', 'IIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIZIIIIIIIIIIIIIIIIIIIIIIIIII', 'IIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIZIIIIIIIIIIIIIIIIIIIIIIIIIII', 'IIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIZIIIIIIIIIIIIIIIIIIIIIIIIIIII', 'IIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIZIIIIIIIIIIIIIIIIIIIIIIIIIIIII', 'IIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIZIIIIIIIIIIIIIIIIIIIIIIIIIIIIII', 'IIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIZIIIIIIIIIIIIIIIIIIIIIIIIIIIIIII', 'IIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIZIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIII', 'IIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIZIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIII', 'IIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIZIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIII', 'IIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIZIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIII', 'IIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIZIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIII', 'IIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIZIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIII', 'IIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIZIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIII', 'IIIIIIIIIIIIIIIIIIIIIIIIIIIIIIZIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIII', 'IIIIIIIIIIIIIIIIIIIIIIIIIIIIIZIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIII', 'IIIIIIIIIIIIIIIIIIIIIIIIIIIIZIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIII', 'IIIIIIIIIIIIIIIIIIIIIIIIIIIZIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIII', 'IIIIIIIIIIIIIIIIIIIIIIIIIIZIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIII', 'IIIIIIIIIIIIIIIIIIIIIIIIIZIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIII', 'IIIIIIIIIIIIIIIIIIIIIIIIZIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIII', 'IIIIIIIIIIIIIIIIIIIIIIIZIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIII', 'IIIIIIIIIIIIIIIIIIIIIIZIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIII', 'IIIIIIIIIIIIIIIIIIIIIZIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIII', 'IIIIIIIIIIIIIIIIIIIIZIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIII', 'IIIIIIIIIIIIIIIIIIIZIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIII', 'IIIIIIIIIIIIIIIIIIZIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIII', 'IIIIIIIIIIIIIIIIIZIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIII', 'IIIIIIIIIIIIIIIIZIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIII', 'IIIIIIIIIIIIIIIZIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIII', 'IIIIIIIIIIIIIIZIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIII', 'IIIIIIIIIIIIIZIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIII', 'IIIIIIIIIIIIZIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIII', 'IIIIIIIIIIIZIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIII', 'IIIIIIIIIIZIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIII', 'IIIIIIIIIZIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIII', 'IIIIIIIIZIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIII', 'IIIIIIIZIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIII', 'IIIIIIZIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIII', 'IIIIIZIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIII', 'IIIIZIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIII', 'IIIZIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIII', 'IIZIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIII', 'IZIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIII', 'ZIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIII'],\n", " coeffs=[0.01428571+0.j, 0.01428571+0.j, 0.01428571+0.j, 0.01428571+0.j,\n", " 0.01428571+0.j, 0.01428571+0.j, 0.01428571+0.j, 0.01428571+0.j,\n", " 0.01428571+0.j, 0.01428571+0.j, 0.01428571+0.j, 0.01428571+0.j,\n", " 0.01428571+0.j, 0.01428571+0.j, 0.01428571+0.j, 0.01428571+0.j,\n", " 0.01428571+0.j, 0.01428571+0.j, 0.01428571+0.j, 0.01428571+0.j,\n", " 0.01428571+0.j, 0.01428571+0.j, 0.01428571+0.j, 0.01428571+0.j,\n", " 0.01428571+0.j, 0.01428571+0.j, 0.01428571+0.j, 0.01428571+0.j,\n", " 0.01428571+0.j, 0.01428571+0.j, 0.01428571+0.j, 0.01428571+0.j,\n", " 0.01428571+0.j, 0.01428571+0.j, 0.01428571+0.j, 0.01428571+0.j,\n", " 0.01428571+0.j, 0.01428571+0.j, 0.01428571+0.j, 0.01428571+0.j,\n", " 0.01428571+0.j, 0.01428571+0.j, 0.01428571+0.j, 0.01428571+0.j,\n", " 0.01428571+0.j, 0.01428571+0.j, 0.01428571+0.j, 0.01428571+0.j,\n", " 0.01428571+0.j, 0.01428571+0.j, 0.01428571+0.j, 0.01428571+0.j,\n", " 0.01428571+0.j, 0.01428571+0.j, 0.01428571+0.j, 0.01428571+0.j,\n", " 0.01428571+0.j, 0.01428571+0.j, 0.01428571+0.j, 0.01428571+0.j,\n", " 0.01428571+0.j, 0.01428571+0.j, 0.01428571+0.j, 0.01428571+0.j,\n", " 0.01428571+0.j, 0.01428571+0.j, 0.01428571+0.j, 0.01428571+0.j,\n", " 0.01428571+0.j, 0.01428571+0.j])\n" ] } ], "source": [ "# Define the magnetization operator in SparsePauliOp\n", "magnetization2 = (\n", " SparsePauliOp.from_sparse_list(\n", " [(\"Z\", [i], 1.0) for i in range(0, n_qubits2)], num_qubits=n_qubits2\n", " )\n", " / n_qubits2\n", ")\n", "print(\"magnetization : \", magnetization2)" ] }, { "cell_type": "markdown", "id": "91f92286-acb7-4946-9f5f-150587e7f4d0", "metadata": {}, "source": [ "\n", "\n", "### 2.6 アクティビティ6\n", "\n", "\n", "\n", "#### 時間発展シミュレーションを実行する\n", "\n", "磁化（磁化作用素の期待値）をモニターする。ハードウェアから計算された結果を比較するための基準値を得るために、MPSシミュレータを使用します。このチュートリアルでMPSシミュレーターを使ったことがあると思います。この新しい計算に合うように、必要な場合はその例を修正する。\n", "\n", "**解決策：**\n", "\n" ] }, { "cell_type": "code", "execution_count": 15, "id": "7e7b754c-deff-40fd-9250-8b4deafa18d6", "metadata": {}, "outputs": [], "source": [ "# Step 1. Map the problem\n", "# Initiate the circuit\n", "evolved_state2 = QuantumCircuit(initial_circuit2.num_qubits)\n", "# Start from the initial spin configuration\n", "evolved_state2.append(initial_circuit2, evolved_state2.qubits)\n", "# Define backend (MPs simulator)\n", "backend_mps2 = AerSimulator(method=\"matrix_product_state\")\n", "# Initiate Runtime Estimator\n", "estimator_mps2 = Estimator(mode=backend_mps2)\n", "# Step 2. Optimize\n", "# Initiate pass manager\n", "pm_mps2 = generate_preset_pass_manager(optimization_level=3, backend=backend_mps2)\n", "# Transpile\n", "evolved_state_mps2 = pm_mps2.run(evolved_state2)\n", "# Apply qubit layout to the observable to measure\n", "magnetization_mps2 = magnetization2.apply_layout(evolved_state_mps2.layout)\n", "# Initiate list\n", "mag_mps_list2 = []\n", "# Step 3. Run the circuit\n", "# Estimate expectation values for t=0.0\n", "job = estimator_mps2.run([(evolved_state_mps2, [magnetization_mps2])])\n", "# Get estimated expectation values\n", "evs = job.result()[0].data.evs\n", "# Append to list\n", "mag_mps_list2.append(evs[0])\n", "\n", "# Start time evolution\n", "for n in range(num_timesteps):\n", " # Step 1. Map the problem\n", " # Expand the circuit to describe delta-t\n", " evolved_state2.append(single_step_evolution_lt2, evolved_state2.qubits)\n", " # Step 2. Optimize\n", " # Transpile the circuit\n", " evolved_state_mps2 = pm_mps2.run(evolved_state2)\n", " # Apply the physical layout of the qubits to the operator\n", " magnetization_mps2 = magnetization2.apply_layout(evolved_state_mps2.layout)\n", " # Step 3. Run the circuit\n", " # Estimate expectation values at delta-t\n", " job = estimator_mps2.run([(evolved_state_mps2, [magnetization_mps2])])\n", " # Get estimated expectation values\n", " evs = job.result()[0].data.evs\n", " # Append to list\n", " mag_mps_list2.append(evs[0])\n", "# Transform the list of expectation values (at each time step) to arrays\n", "mag_mps_array2 = np.array(mag_mps_list2)" ] }, { "cell_type": "markdown", "id": "c8612280-d29b-4952-826f-16a686a051dd", "metadata": {}, "source": [ "これまでのレッスンと同様に、Qiskitパターンフレームワークを実装します。ここまでのレッスンでは、問題を説明するための正しい量子回路を作ることに焦点を絞ってきた。これが事実上のステップ1だ。\n", "\n", "\n", "\n", "#### ステップ2: 対象ハードウェア向けに最適化\n", "\n", "まず、対象となるバックエンドを定義する。\n", "\n" ] }, { "cell_type": "code", "execution_count": null, "id": "610ecd2c-9a47-43c1-872b-a4038fb83817", "metadata": {}, "outputs": [ { "data": { "text/plain": [ "'ibm_kingston'" ] }, "execution_count": 19, "metadata": {}, "output_type": "execute_result" } ], "source": [ "service = QiskitRuntimeService()\n", "backend = service.least_busy(operational=True, simulator=False)\n", "backend.name" ] }, { "cell_type": "markdown", "id": "ba36b8f4-9fae-4151-b3fa-8f848eb77a0d", "metadata": {}, "source": [ "回路をトランスパイルし、リストにまとめる。これには数分かかることがあります。\n", "\n" ] }, { "cell_type": "code", "execution_count": 20, "id": "116e914e-8fa4-4e60-9980-af71d9bcc2f9", "metadata": {}, "outputs": [], "source": [ "pm_hw = generate_preset_pass_manager(optimization_level=3, backend=backend)\n", "circuit_isa = []\n", "# Step 1. Map the problem\n", "evolved_state_hw = QuantumCircuit(initial_circuit2.num_qubits)\n", "evolved_state_hw.append(initial_circuit2, evolved_state_hw.qubits)\n", "# Step 2. Optimize\n", "circuit_isa.append(pm_hw.run(evolved_state_hw))\n", "\n", "for n in range(num_timesteps):\n", " # Step 1. Map the problem\n", " evolved_state_hw.append(single_step_evolution_lt2, evolved_state_hw.qubits)\n", " # Step 2. Optimize\n", " circuit_isa.append(pm_hw.run(evolved_state_hw))" ] }, { "cell_type": "markdown", "id": "60c95c4c-9eab-4c14-b17d-c710beecb2ce", "metadata": {}, "source": [ "\n", "\n", "#### ステップ3: 対象ハードウェア上で実行する\n", "\n", "Runtime Estimatorを定義し、PUBのリストを作成する。また、測定するオペレーターにもレイアウトを適用しなければならない。\n", "\n" ] }, { "cell_type": "code", "execution_count": 21, "id": "3da4db25-5e78-4a86-b740-75fdacbdb831", "metadata": {}, "outputs": [], "source": [ "# Step 2. Optimize\n", "estimator_hw = Estimator(mode=backend)\n", "pub_list = []\n", "for circuit in circuit_isa:\n", " temp = (circuit, magnetization2.apply_layout(circuit.layout))\n", " pub_list.append(temp)" ] }, { "cell_type": "markdown", "id": "2433f80e-5ed8-4e73-8adb-c4c688a79775", "metadata": {}, "source": [ "これでジョブを実行する準備ができた。\n", "\n" ] }, { "cell_type": "code", "execution_count": 22, "id": "ba037ca8-dd6f-4962-a7f4-65f0bde66f8b", "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "d147hfdqf56g0081sxs0\n" ] } ], "source": [ "job = estimator_hw.run(pub_list)\n", "job_id = job.job_id()\n", "print(job_id)" ] }, { "cell_type": "code", "execution_count": 28, "id": "efcb2851-3529-4e76-8a0f-1b65be8f4818", "metadata": {}, "outputs": [ { "data": { "text/plain": [ "'DONE'" ] }, "execution_count": 28, "metadata": {}, "output_type": "execute_result" } ], "source": [ "# check job status\n", "job.status()" ] }, { "cell_type": "markdown", "id": "f5dd6d18-08e9-49eb-90b4-8b6037fd6d34", "metadata": {}, "source": [ "\n", "\n", "#### ステップ4：結果の後処理\n", "\n", "まずは結果を出す。\n", "\n" ] }, { "cell_type": "code", "execution_count": 20, "id": "eb661ec3-b8fd-4e2b-a525-547685cc6864", "metadata": {}, "outputs": [], "source": [ "job = service.job(job_id)\n", "pub_result = job.result()" ] }, { "cell_type": "markdown", "id": "1840130b-3c40-4a10-ac83-1bc4350a1f58", "metadata": {}, "source": [ "さて、これらの結果から期待値を抽出しなければならない。\n", "\n" ] }, { "cell_type": "code", "execution_count": 21, "id": "be186c8e-0408-4d2d-b843-5d27c82ba99c", "metadata": {}, "outputs": [], "source": [ "mag_hw_list = []\n", "for res in pub_result:\n", " evs = res.data.evs\n", " mag_hw_list.append(evs)" ] }, { "cell_type": "markdown", "id": "0a03e25f-d447-4c52-9595-c776c9b7a23c", "metadata": {}, "source": [ "以下、比較のためにこれを使用する。まず、回路をさらに最適化できるかどうかを見てみよう。\n", "\n", "\n", "\n", "## 3. 実際の量子コンピュータを用いた解決策 II\n", "\n", "Qiskitのパターン・ステップ1に戻り、回路の奥行きを短くできるか見てみよう。\n", "\n", "\n", "\n", "### 3.1 ステップ1. 問題を量子回路と演算子に写像する\n", "\n", "\n", "\n", "#### アクティビティ7\n", "\n", "時間発展回路を構築する。これまでのレッスンで得た知識を活かして、回路の深さを浅くしてみよう。\n", "\n", "**解決策：**\n", "\n" ] }, { "cell_type": "code", "execution_count": 22, "id": "52a28d34-0852-4099-bcb8-ea9487701d43", "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "\n", "Trotter step with Lie-Trotter\n", "-----------------------------\n", "Depth: 7\n", "Gate count: 277\n", "Nonlocal gate count: 138\n", "Gate breakdown: CX: 138, U3: 70, U1: 69\n", "\n" ] }, { "data": { "text/plain": [ " $\"Output$ " ] }, "execution_count": 22, "metadata": {}, "output_type": "execute_result" } ], "source": [ "# Define J\n", "J = 1.0\n", "# Define h\n", "h = -5.0\n", "# Create instruction for rotation around ZZ:\n", "# Initiate the circuit (use 2 qubits)\n", "Rzz_circ = QuantumCircuit(2)\n", "# Add Rzz gate (do not forget to multiply the angle by 2.0)\n", "Rzz_circ.rzz(-J * dt * 2.0, 0, 1)\n", "# Transform the QuantumCircuit to instruction (QuantumCircuit.to_instruction())\n", "Rzz_instr = Rzz_circ.to_instruction(label=\"RZZ\")\n", "\n", "# Create instruction for rotation around X:\n", "# Initiate the circuit (use 1 qubit)\n", "Rx_circ = QuantumCircuit(1)\n", "# Add Rx gate (do not forget to multiply the angle by 2.0)\n", "Rx_circ.rx(-h * dt * 2.0, 0)\n", "# Transform the QuantumCircuit to instruction (QuantumCircuit.to_instruction())\n", "Rx_instr = Rx_circ.to_instruction(label=\"RX\")\n", "\n", "# Define the interaction list\n", "interaction_list = [\n", " [[i, i + 1] for i in range(0, n_qubits2 - 1, 2)],\n", " [[i, i + 1] for i in range(1, n_qubits2 - 1, 2)],\n", "] # linear chain\n", "\n", "# Define the registers\n", "qr = QuantumRegister(n_qubits2)\n", "# Initiate the circuit\n", "single_step_evolution_sh = QuantumCircuit(qr)\n", "# Construct the Rzz gates\n", "for i, color in enumerate(interaction_list):\n", " for interaction in color:\n", " single_step_evolution_sh.append(Rzz_instr, interaction)\n", "\n", "# Construct the Rx gates\n", "for i in range(0, n_qubits2):\n", " single_step_evolution_sh.append(Rx_instr, [i])\n", "\n", "print(\n", " f\"\"\"\n", "Trotter step with Lie-Trotter\n", "-----------------------------\n", "Depth: {single_step_evolution_sh.decompose(reps=3).depth()}\n", "Gate count: {len(single_step_evolution_sh.decompose(reps=3))}\n", "Nonlocal gate count: {single_step_evolution_sh.decompose(reps=3).num_nonlocal_gates()}\n", "Gate breakdown: {\", \".join([f\"{k.upper()}: {v}\" for k, v in single_step_evolution_sh.decompose(reps=3).count_ops().items()])}\n", "\"\"\"\n", ")\n", "\n", "single_step_evolution_sh.decompose(reps=2).draw(\"mpl\")" ] }, { "cell_type": "markdown", "id": "5cde0bf4-37e8-469a-a88f-bb9c37168cea", "metadata": {}, "source": [ "これは大成功だった。これで残りのQiskitパターンのステップに進むことができる。\n", "\n", "\n", "\n", "### 3.2 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", "回路をトランスパイルし、リストにまとめる。もう一度言うが、これには数分かかるかもしれない。\n", "\n" ] }, { "cell_type": "code", "execution_count": 32, "id": "958e2251-2a1e-4caf-b501-946f32b4fe9e", "metadata": {}, "outputs": [], "source": [ "pm_hw2 = generate_preset_pass_manager(backend=backend, optimization_level=3)\n", "circuit_isa2 = []\n", "# Step 1. Map the problem\n", "evolved_state_hw2 = QuantumCircuit(initial_circuit2.num_qubits)\n", "evolved_state_hw2.append(initial_circuit2, evolved_state_hw2.qubits)\n", "# Step 2. Optimize\n", "circuit_isa2.append(pm_hw2.run(evolved_state_hw2))\n", "for n in range(num_timesteps):\n", " # Step 1. Map the problem\n", " evolved_state_hw2.append(single_step_evolution_sh, evolved_state_hw2.qubits)\n", " # Step 2. Optimize\n", " circuit_isa2.append(pm_hw2.run(evolved_state_hw2))" ] }, { "cell_type": "markdown", "id": "8c74b53d-abac-48ac-883c-029adcd4894c", "metadata": {}, "source": [ "Runtime Estimatorを定義し、PUBのリストを作成する。\n", "\n" ] }, { "cell_type": "code", "execution_count": 33, "id": "754e4c6d-04cc-4cc1-af81-c6fc13c353e0", "metadata": {}, "outputs": [], "source": [ "estimator_hw2 = Estimator(mode=backend)\n", "pub_list2 = []\n", "for circuit in circuit_isa2:\n", " temp = (circuit, magnetization2.apply_layout(circuit.layout))\n", " pub_list2.append(temp)" ] }, { "cell_type": "markdown", "id": "e2fa2986-701d-4f68-ab38-d6bf24a599c6", "metadata": {}, "source": [ "\n", "\n", "### 3.3 ステップ３．ターゲット・ハードウェア上で実行する\n", "\n", "ジョブを実行します。\n", "\n" ] }, { "cell_type": "code", "execution_count": 34, "id": "5288e0f3-e12c-4e8c-80f8-6e2535ffb0d1", "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "d147qqeqf56g0081sye0\n" ] } ], "source": [ "job2 = estimator_hw2.run(pub_list2)\n", "job2_id = job2.job_id()\n", "print(job2_id)" ] }, { "cell_type": "code", "execution_count": 36, "id": "37dfc4b9-4757-4a87-a1a4-972a372b931a", "metadata": {}, "outputs": [ { "data": { "text/plain": [ "'DONE'" ] }, "execution_count": 36, "metadata": {}, "output_type": "execute_result" } ], "source": [ "# check job status\n", "job2.status()" ] }, { "cell_type": "markdown", "id": "4d08139b-71af-4578-b3b5-45f9a3741c0b", "metadata": {}, "source": [ "結果を出す。\n", "\n" ] }, { "cell_type": "code", "execution_count": 24, "id": "83d2d003-2f97-4294-a745-2e0a1c7ae80c", "metadata": {}, "outputs": [], "source": [ "job2 = service.job(job2_id)\n", "pub_result2 = job2.result()" ] }, { "cell_type": "markdown", "id": "7f32a06a-2974-4f22-9dbe-d2ec6143f3cf", "metadata": {}, "source": [ "\n", "\n", "### 3.4 ステップ4. 後処理\n", "\n", "結果から期待値を抽出する。\n", "\n" ] }, { "cell_type": "code", "execution_count": 25, "id": "402c9208-9a1d-4424-8ef2-636407c398d7", "metadata": {}, "outputs": [], "source": [ "mag_hw_list2 = []\n", "for res in pub_result2:\n", " evs = res.data.evs\n", " mag_hw_list2.append(evs)" ] }, { "cell_type": "markdown", "id": "0c255702-0ef3-46ff-bb6e-4675af2676a3", "metadata": {}, "source": [ "プロットするために、リストをnumpy配列に変換する。\n", "\n" ] }, { "cell_type": "code", "execution_count": 26, "id": "0d28a88f-990c-4330-9506-4cf23ae57415", "metadata": {}, "outputs": [], "source": [ "mag_hw_array = np.array(mag_hw_list)\n", "mag_hw_array2 = np.array(mag_hw_list2)" ] }, { "cell_type": "markdown", "id": "128662bd-44dc-4730-bdaa-fd686d1dbf04", "metadata": {}, "source": [ "では、結果をプロットし、ハードウェア（デフォルトと浅い回路）の結果とMPSシミュレータを比較してみよう。実際のハードウェアの誤差は結果にどう影響するのか？\n", "\n" ] }, { "cell_type": "code", "execution_count": 27, "id": "3aa52fc2-1685-482c-8673-c25096ddb44f", "metadata": {}, "outputs": [ { "data": { "text/plain": [ "Text(0.5, 0.98, 'Observable evolution')" ] }, "execution_count": 27, "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, mag_mps_array2, label=\"MPS\", marker=\"x\", c=\"darkmagenta\", ls=\"-\", lw=0.8\n", ")\n", "axes[1].plot(\n", " times, mag_hw_array, label=\"HW\", marker=\"x\", c=\"darkmagenta\", ls=\"-\", lw=0.8\n", ")\n", "axes[2].plot(\n", " times, mag_hw_array2, label=\"HW2\", marker=\"x\", c=\"darkmagenta\", ls=\"-\", lw=0.8\n", ")\n", "axes[0].set_ylabel(\"MPS\")\n", "axes[1].set_ylabel(\"HW\")\n", "axes[2].set_ylabel(\"HW2\")\n", "axes[2].set_xlabel(\"Time\")\n", "fig.suptitle(\"Observable evolution\")" ] }, { "cell_type": "markdown", "id": "ec7c30e2-eea0-4fb7-80c8-e0dfb826c95e", "metadata": {}, "source": [ "おめでとうございます! ユーティリティスケールの量子の旅が一歩前進した。残されたレッスンは1つだけだ！\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" }, "widgets": { "application/vnd.jupyter.widget-state+json": { "state": {}, "version_major": 2, "version_minor": 0 } } }, "nbformat": 4, "nbformat_minor": 4 }