유틸리티 규모 실험 II
참고
Yukio Kawashima (2024년 7월 12일)
원본 강의의 PDF를 다운로드하세요. 일부 코드 스니펫은 정적 이미지이므로 더 이상 사용되지 않을 수 있습니다.
이 실험을 실행하는 데 필요한 예상 QPU 시간은 2분 30초입니다.
(이 노트북은 Qiskit Algorithms의 현재 더 이상 사용되지 않는 튜토리얼 노트북의 텍스트, 그림, 코드를 참고하였습니다.)
1. 시간 진화 소개 및 복습
이 노트북은 레슨 7의 방법과 기법을 따릅니다. 목표는 시간 의존 슈뢰딩거 방정식을 수치적으로 풀어내는 것입니다. 레슨 7에서 논의한 바와 같이, Trotterization은 특정 시간 슬라이스 동안 시스템의 시간 진화를 근사하도록 선택된 양자 Gate 또는 Gate들을 연속적으로 적용하는 방법입니다. 여기서는 편의를 위해 그 논의를 다시 소개합니다. 레슨 7을 최근에 복습하셨다면 아래의 코드 셀로 바로 넘어가셔도 됩니다.
슈뢰딩거 방정식에 따르면, 초기 상태 에 있는 시스템의 시간 진화는 다음과 같은 형태를 취합니다:
여기서 는 시스템을 지배하는 시간 독립 해밀토니안입니다. 개의 Qubit에 작용하는 파울리 항들의 텐서 곱을 나타내는 를 사용하여, 해밀토니안을 파울리 항의 가중 합 으로 표현할 수 있습니다. 특히, 이 파울리 항들은 서로 교환 가능할 수도 있고 그렇지 않을 수도 있습니다. 에서의 상태가 주어졌을 때, 양자 컴퓨터를 사용하여 이후 시간 에서의 시스템 상태를 어떻게 구할 수 있을까요? 연산자의 지수 함수는 테일러 급수를 통해 가장 쉽게 이해할 수 있습니다:
와 같은 매우 기본적인 지수 함수는 간결한 양자 Gate 집합을 사용하여 양자 컴퓨터에서 쉽게 구현할 수 있습니다. 관심 있는 대부분의 해밀토니안은 단일 항만 있는 것이 아니라 많은 항을 포함합니다. 인 경우 어떤 일이 일어나는지 살펴보겠습니다:
과 가 교환 가능한 경우, 우리가 잘 아는 다음의 공식이 성립합니다(숫자 및 아래의 변수 , 에 대해서도 마찬가지입니다):
그러나 연산자가 교환 불가능한 경우, 테일러 급수에서 항들을 이와 같이 재배열하여 단순화할 수 없습니다. 따라서 복잡한 해밀토니안을 양자 Gate로 표현하는 것은 어려운 과제입니다.
한 가지 해결책은 매우 짧은 시간 를 고려하여 테일러 전개의 1차 항이 지배적인 경우를 활용하는 것입니다. 이 가정 하에서:
물론, 더 긴 시간 동안 상태를 진화시켜야 할 수도 있습니다. 이는 이러한 작은 시간 스텝을 여러 번 반복함으로써 달성할 수 있습니다. 이 과정을 Trotterization이라고 합니다:
여기서 은 우리가 선택하는 시간 슬라이스(진화 스텝)입니다. 결과적으로 번 적용되는 Gate가 생성됩니다. 시간 스텝이 작을수록 더 정확한 근사가 가능합니다. 그러나 이는 더 깊은 Circuit으로 이어지며, 실제로는 더 많은 오류가 누적됩니다(근미래 양자 장치에서는 무시할 수 없는 우려 사항입니다).
오늘은 및 격자점을 가진 선형 격자에서 이징 모델의 시간 진화를 연구합니다. 이 격자들은 오직 최근접 이웃과만 상호작용하는 스핀 의 배열로 구성됩니다. 이 스핀들 은 두 가지 방향을 가질 수 있습니다: 와 , 각각 자화값 과 에 해당합니다.
여기서 는 상호작용 에너지를 나타내고, 는 외부 자기장의 세기를 나타냅니다(위에서는 x 방향이지만, 이를 수정할 것입니다). 파울리 행렬을 사용하여 이 표현식을 작성하고, 외부 자기장이 횡방향과 각도 를 이루는 경우를 고려하면:
이 해밀토니안은 외부 자기장의 영향을 쉽게 연구할 수 있다는 점에서 유용합니다. 계산 기저에서 시스템은 다음과 같이 인코딩됩니다:
| 양자 상태 | 스핀 표현 |
|---|---|
이제 이러한 양자 시스템의 시간 진화를 조사해 보겠습니다. 구체적으로, 자화와 같은 시스템의 특정 속성의 시간 진화를 시각화할 것입니다.
# Added by doQumentation — required packages for this notebook
!pip install -q matplotlib numpy qiskit qiskit-aer qiskit-ibm-runtime
# Check the version of Qiskit
import qiskit
qiskit.__version__
'2.0.2'
# Import the qiskit library
import numpy as np
import warnings
from qiskit import QuantumCircuit, QuantumRegister
from qiskit.circuit.library import PauliEvolutionGate
from qiskit.quantum_info import SparsePauliOp
from qiskit.synthesis import LieTrotter
from qiskit.transpiler.preset_passmanagers import generate_preset_pass_manager
from qiskit_aer import AerSimulator
from qiskit_ibm_runtime import QiskitRuntimeService, Estimator
warnings.filterwarnings("ignore")
2. 횡자기장 이징 해밀토니안 정의하기
여기서는 1차원 횡자기장 이징 모델을 다룹니다.
먼저, 시스템 매개변수 , , 를 입력받아 해밀토니안을 SparsePauliOp으로 반환하는 함수를 만들겠습니다. SparsePauliOp은 가중치가 부여된 Pauli 항들의 조합으로 연산자를 희소하게 표현하는 방식입니다.
2.1 Activity 1
"Qubit 수", "J 매개변수", "h 매개변수"를 인수로 받아 횡자기장 이징 해밀토니안(위의 방정식 참고)을 구성하는 함수를 만들어 보세요. 앞서 살펴본 예제를 참고하여 직접 작성해 보세요. 정답은 아래로 스크롤하면 확인할 수 있습니다.
정답:
def get_hamiltonian(nqubits, J, h):
# List of Hamiltonian terms as 3-tuples containing
# (1) the Pauli string,
# (2) the qubit indices corresponding to the Pauli string,
# (3) the coefficient.
ZZ_tuples = [("ZZ", [i, i + 1], -J) for i in range(0, nqubits - 1)]
X_tuples = [("X", [i], -h) for i in range(0, nqubits)]
# We create the Hamiltonian as a SparsePauliOp, via the method
# `from_sparse_list`, and multiply by the interaction term.
hamiltonian = SparsePauliOp.from_sparse_list(
[*ZZ_tuples, *X_tuples], num_qubits=nqubits
)
return hamiltonian.simplify()
이제 자기화(magnetization)를 추적하면서 양자 시스템의 시간 진화를 살펴보겠습니다. 여기서는 Statevector 시뮬레이터와 행렬 곱 상태(Matrix Product State) 시뮬레이터의 결과를 비교합니다.
해밀토니안 정의하기
이제 다룰 시스템의 크기는 입니다.
n_qubits = 20
hamiltonian = get_hamiltonian(nqubits=n_qubits, J=1.0, h=-5.0)
hamiltonian
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'],
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,
-1.+0.j, -1.+0.j, -1.+0.j, -1.+0.j, -1.+0.j, -1.+0.j, -1.+0.j, -1.+0.j,
-1.+0.j, -1.+0.j, -1.+0.j, 5.+0.j, 5.+0.j, 5.+0.j, 5.+0.j, 5.+0.j,
5.+0.j, 5.+0.j, 5.+0.j, 5.+0.j, 5.+0.j, 5.+0.j, 5.+0.j, 5.+0.j,
5.+0.j, 5.+0.j, 5.+0.j, 5.+0.j, 5.+0.j, 5.+0.j, 5.+0.j])
시간 진화 시뮬레이션 매개변수 설정하기
여기서는 Lie–Trotter (1차) 방법을 사용합니다.
num_timesteps = 20
evolution_time = 2.0
dt = evolution_time / num_timesteps
product_formula_lt = LieTrotter()
양자 Circuit 준비하기 (초기 상태)
초기 상태를 생성합니다. 강자성 상태(모두 위 또는 모두 아래)인 바닥 상태에서 시작하겠습니다. 여기서는 모두 위('0') 상태를 예시로 사용합니다.
initial_circuit = QuantumCircuit(n_qubits)
initial_circuit.prepare_state("00000000000000000000")
# Change reps and see the difference when you decompose the circuit
initial_circuit.decompose(reps=1).draw("mpl")
양자 Circuit 준비하기 2 (시간 진화를 위한 단일 Circuit)
여기서는 Lie–Trotter를 사용하여 단일 시간 스텝에 대한 Circuit을 구성합니다. Lie 곱 공식(1차)은 LieTrotter 클래스에 구현되어 있습니다. 1차 공식은 합의 행렬 지수를 행렬 지수들의 곱으로 근사하는 서론의 근사식으로 구성됩니다:
이 Circuit의 연산 수를 세어 봅시다.
single_step_evolution_gates_lt = PauliEvolutionGate(
hamiltonian, dt, synthesis=product_formula_lt
)
single_step_evolution_lt = QuantumCircuit(n_qubits)
single_step_evolution_lt.append(
single_step_evolution_gates_lt, single_step_evolution_lt.qubits
)
print(
f"""
Trotter step with Lie-Trotter
-----------------------------
Depth: {single_step_evolution_lt.decompose(reps=3).depth()}
Gate count: {len(single_step_evolution_lt.decompose(reps=3))}
Nonlocal gate count: {single_step_evolution_lt.decompose(reps=3).num_nonlocal_gates()}
Gate breakdown: {", ".join([f"{k.upper()}: {v}" for k, v in single_step_evolution_lt.decompose(reps=3).count_ops().items()])}
"""
)
single_step_evolution_lt.decompose(reps=3).draw("mpl", fold=-1)
Trotter step with Lie-Trotter
-----------------------------
Depth: 58
Gate count: 77
Nonlocal gate count: 38
Gate breakdown: CX: 38, U3: 20, U1: 19
측정할 연산자 설정하기
자기화 연산자 를 정의합니다.
magnetization = (
SparsePauliOp.from_sparse_list(
[("Z", [i], 1.0) for i in range(0, n_qubits)], num_qubits=n_qubits
)
/ n_qubits
)
print("magnetization : ", magnetization)
magnetization : SparsePauliOp(['IIIIIIIIIIIIIIIIIIIZ', 'IIIIIIIIIIIIIIIIIIZI', 'IIIIIIIIIIIIIIIIIZII', 'IIIIIIIIIIIIIIIIZIII', 'IIIIIIIIIIIIIIIZIIII', 'IIIIIIIIIIIIIIZIIIII', 'IIIIIIIIIIIIIZIIIIII', 'IIIIIIIIIIIIZIIIIIII', 'IIIIIIIIIIIZIIIIIIII', 'IIIIIIIIIIZIIIIIIIII', 'IIIIIIIIIZIIIIIIIIII', 'IIIIIIIIZIIIIIIIIIII', 'IIIIIIIZIIIIIIIIIIII', 'IIIIIIZIIIIIIIIIIIII', 'IIIIIZIIIIIIIIIIIIII', 'IIIIZIIIIIIIIIIIIIII', 'IIIZIIIIIIIIIIIIIIII', 'IIZIIIIIIIIIIIIIIIII', 'IZIIIIIIIIIIIIIIIIII', 'ZIIIIIIIIIIIIIIIIIII'],
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,
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,
0.05+0.j, 0.05+0.j, 0.05+0.j, 0.05+0.j, 0.05+0.j, 0.05+0.j])
시간 진화 시뮬레이션 수행하기
자기화(자기화 연산자의 기댓값)를 모니터링합니다. Statevector 시뮬레이터와 MPS 시뮬레이터를 사용하여 결과를 비교합니다.
# Step 1. Map the problem
# Initiate the circuit
evolved_state = QuantumCircuit(initial_circuit.num_qubits)
# Start from the initial spin configuration
evolved_state.append(initial_circuit, evolved_state.qubits)
# Define backend (simulator)
# MPS
backend_mps = AerSimulator(method="matrix_product_state")
# Statevector
backend_sv = AerSimulator(method="statevector")
# Set Runtime Estimator
# MPS
estimator_mps = Estimator(mode=backend_mps)
# Statevector
estimator_sv = Estimator(mode=backend_sv)
# Step 2. Optimize
# Set pass manager
# MPS
pm_mps = generate_preset_pass_manager(optimization_level=3, backend=backend_mps)
# Statevector
pm_sv = generate_preset_pass_manager(optimization_level=3, backend=backend_sv)
# Transpile initial circuit
# MPS
evolved_state_mps = pm_mps.run(evolved_state)
# Statevector
evolved_state_sv = pm_sv.run(evolved_state)
# Apply layout to the operator
# MPS
magnetization_mps = magnetization.apply_layout(evolved_state_mps.layout)
# Statevector
magnetization_sv = magnetization.apply_layout(evolved_state_sv.layout)
mag_mps_list = []
mag_sv_list = []
# Step 3. Run the circuit
# Estimate expectation values for t=0.0: MPS
job = estimator_mps.run([(evolved_state_mps, [magnetization_mps])])
# Get estimated expectation values: MPS
evs = job.result()[0].data.evs
# Collect data: MPS
mag_mps_list.append(evs[0])
# Estimate expectation values for t=0.0: Statevector
job = estimator_sv.run([(evolved_state_sv, [magnetization_sv])])
# Get estimated expectation values: Statevector
evs = job.result()[0].data.evs
# Collect data: Statevector
mag_sv_list.append(evs[0])
# Start time evolution
for n in range(num_timesteps):
# Step 1. Map the problem
# Expand the circuit to describe delta-t
evolved_state.append(single_step_evolution_lt, evolved_state.qubits)
# Step 2. Optimize
# Transpile the circuit: MPS
evolved_state_mps = pm_mps.run(evolved_state)
# Apply the physical layout of the qubits to the operator: MPS
magnetization_mps = magnetization.apply_layout(evolved_state_mps.layout)
# Step 3. Run the circuit
# Estimate expectation values at delta-t: MPS
job = estimator_mps.run([(evolved_state_mps, [magnetization_mps])])
# Get estimated expectation values: MPS
evs = job.result()[0].data.evs
# Collect data: MPS
mag_mps_list.append(evs[0])
# Step 2. Optimize
# Transpile the circuit: Statevector
evolved_state_sv = pm_sv.run(evolved_state)
# Apply the physical layout of the qubits to the operator: Statevector
magnetization_sv = magnetization.apply_layout(evolved_state_sv.layout)
# Step 3. Run the circuit
# Estimate expectation values at delta-t: Statevector
job = estimator_sv.run([(evolved_state_sv, [magnetization_sv])])
# Get estimated expectation values: Statevector
evs = job.result()[0].data.evs
# Collect data: Statevector
mag_sv_list.append(evs[0])
# Transform the list of expectation values (at each time step) to arrays
mag_mps_array = np.array(mag_mps_list)
mag_sv_array = np.array(mag_sv_list)
관측량의 시간 진화 플롯하기
측정한 기댓값을 시간에 대해 플롯합니다. Statevector 시뮬레이터와 행렬 곱 공간 시뮬레이터의 결과가 일치하는지 확인하세요.
import matplotlib.pyplot as plt
# Step 4. Post-processing
fig, axes = plt.subplots(2, sharex=True)
times = np.linspace(0, evolution_time, num_timesteps + 1) # includes initial state
axes[0].plot(
times, mag_mps_array, label="MPS", marker="x", c="darkmagenta", ls="-", lw=0.8
)
axes[1].plot(
times, mag_sv_array, label="SV", marker="x", c="darkmagenta", ls="-", lw=0.8
)
axes[0].set_ylabel("MPS")
axes[1].set_ylabel("Statevector")
axes[1].set_xlabel("Time")
fig.suptitle("Observable evolution")
Text(0.5, 0.98, 'Observable evolution')
이제 특성을 추적하면서 양자 시스템의 시간 진화를 살펴보겠습니다. 여기서는 행렬 곱 상태(Matrix Product State) 시뮬레이터와 실제 양자 디바이스의 결과를 비교합니다.
2.2 Activity 2
해밀토니안 정의하기
이제 고려할 시스템의 크기는 입니다. 다른 조건들은 20-큐비트 문제와 동일합니다. 직접 풀어 보세요. 아래로 스크롤하면 풀이를 확인할 수 있습니다.
풀이:
# Set the number of qubits
n_qubits2 = 70
# Construct the Hamiltonian by calling the function you made in Activity 1
hamiltonian2 = get_hamiltonian(nqubits=n_qubits2, J=1.0, h=-5.0)
hamiltonian2
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'],
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,
-1.+0.j, -1.+0.j, -1.+0.j, -1.+0.j, -1.+0.j, -1.+0.j, -1.+0.j, -1.+0.j,
-1.+0.j, -1.+0.j, -1.+0.j, -1.+0.j, -1.+0.j, -1.+0.j, -1.+0.j, -1.+0.j,
-1.+0.j, -1.+0.j, -1.+0.j, -1.+0.j, -1.+0.j, -1.+0.j, -1.+0.j, -1.+0.j,
-1.+0.j, -1.+0.j, -1.+0.j, -1.+0.j, -1.+0.j, -1.+0.j, -1.+0.j, -1.+0.j,
-1.+0.j, -1.+0.j, -1.+0.j, -1.+0.j, -1.+0.j, -1.+0.j, -1.+0.j, -1.+0.j,
-1.+0.j, -1.+0.j, -1.+0.j, -1.+0.j, -1.+0.j, -1.+0.j, -1.+0.j, -1.+0.j,
-1.+0.j, -1.+0.j, -1.+0.j, -1.+0.j, -1.+0.j, -1.+0.j, -1.+0.j, -1.+0.j,
-1.+0.j, -1.+0.j, -1.+0.j, -1.+0.j, -1.+0.j, 5.+0.j, 5.+0.j, 5.+0.j,
5.+0.j, 5.+0.j, 5.+0.j, 5.+0.j, 5.+0.j, 5.+0.j, 5.+0.j, 5.+0.j,
5.+0.j, 5.+0.j, 5.+0.j, 5.+0.j, 5.+0.j, 5.+0.j, 5.+0.j, 5.+0.j,
5.+0.j, 5.+0.j, 5.+0.j, 5.+0.j, 5.+0.j, 5.+0.j, 5.+0.j, 5.+0.j,
5.+0.j, 5.+0.j, 5.+0.j, 5.+0.j, 5.+0.j, 5.+0.j, 5.+0.j, 5.+0.j,
5.+0.j, 5.+0.j, 5.+0.j, 5.+0.j, 5.+0.j, 5.+0.j, 5.+0.j, 5.+0.j,
5.+0.j, 5.+0.j, 5.+0.j, 5.+0.j, 5.+0.j, 5.+0.j, 5.+0.j, 5.+0.j,
5.+0.j, 5.+0.j, 5.+0.j, 5.+0.j, 5.+0.j, 5.+0.j, 5.+0.j, 5.+0.j,
5.+0.j, 5.+0.j, 5.+0.j, 5.+0.j, 5.+0.j, 5.+0.j, 5.+0.j, 5.+0.j,
5.+0.j, 5.+0.j, 5.+0.j])
2.3 Activity 3
초기 상태를 만들어 봅니다. 기저 상태(ground state)인 강자성 상태(모두 위 또는 모두 아래)에서 시작할 것입니다. 여기서는 모두 위('0') 상태를 예시로 사용합니다. 직접 풀어 보세요. 아래로 스크롤하면 풀이를 확인할 수 있습니다.
풀이:
# Initiate the (quantum)circuit
initial_circuit2 = QuantumCircuit(n_qubits2)
# Use QuantumCircuit.prepare_state() to define the initial state
initial_circuit2.prepare_state(
"0000000000000000000000000000000000000000000000000000000000000000000000"
)
# Change reps and see the difference when you decompose the circuit
initial_circuit2.decompose(reps=1).draw("mpl")
2.4 Activity 4
양자 회로 2 준비하기 (70-큐비트 문제의 단일 시간 진화 Circuit)
여기서는 Lie–Trotter 방법을 이용하여 단일 시간 스텝에 대한 Circuit을 구성합니다. 20-큐비트 경우와 동일하게, Lie 곱 공식(1차)은 LieTrotter 클래스에 구현되어 있습니다. 1차 공식은 위에서 설명한 근사, 즉 합의 행렬 지수함수를 행렬 지수함수의 곱으로 근사하는 방식으로 구성됩니다.
20-큐비트 예시를 기반으로 직접 풀어 보세요. 앞서와 마찬가지로 이 Circuit의 연산 횟수를 세어 보세요.
풀이:
# Construct the gates using PauliEvolutionGate()
single_step_evolution_gates_lt2 = PauliEvolutionGate(
hamiltonian2, dt, synthesis=LieTrotter()
)
# Initiate the quantum circuit
single_step_evolution_lt2 = QuantumCircuit(n_qubits2)
# Append the gates defined above
single_step_evolution_lt2.append(
single_step_evolution_gates_lt2, single_step_evolution_lt2.qubits
)
print(
f"""
Trotter step with Lie-Trotter
-----------------------------
Depth: {single_step_evolution_lt2.decompose(reps=3).depth()}
Gate count: {len(single_step_evolution_lt2.decompose(reps=3))}
Nonlocal gate count: {single_step_evolution_lt2.decompose(reps=3).num_nonlocal_gates()}
Gate breakdown: {", ".join([f"{k.upper()}: {v}" for k, v in single_step_evolution_lt2.decompose(reps=3).count_ops().items()])}
"""
)
single_step_evolution_lt2.decompose(reps=3).draw("mpl", fold=-1)
Trotter step with Lie-Trotter
-----------------------------
Depth: 208
Gate count: 277
Nonlocal gate count: 138
Gate breakdown: CX: 138, U3: 70, U1: 69
2.5 Activity 5
측정할 연산자 설정
20-Qubit 경우와 정확히 동일한 방식으로 자기화 연산자를 정의합니다: . 20-Qubit 풀이를 수정하여 직접 시도해 보세요.
풀이:
# Define the magnetization operator in SparsePauliOp
magnetization2 = (
SparsePauliOp.from_sparse_list(
[("Z", [i], 1.0) for i in range(0, n_qubits2)], num_qubits=n_qubits2
)
/ n_qubits2
)
print("magnetization : ", magnetization2)
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'],
coeffs=[0.01428571+0.j, 0.01428571+0.j, 0.01428571+0.j, 0.01428571+0.j,
0.01428571+0.j, 0.01428571+0.j, 0.01428571+0.j, 0.01428571+0.j,
0.01428571+0.j, 0.01428571+0.j, 0.01428571+0.j, 0.01428571+0.j,
0.01428571+0.j, 0.01428571+0.j, 0.01428571+0.j, 0.01428571+0.j,
0.01428571+0.j, 0.01428571+0.j, 0.01428571+0.j, 0.01428571+0.j,
0.01428571+0.j, 0.01428571+0.j, 0.01428571+0.j, 0.01428571+0.j,
0.01428571+0.j, 0.01428571+0.j, 0.01428571+0.j, 0.01428571+0.j,
0.01428571+0.j, 0.01428571+0.j, 0.01428571+0.j, 0.01428571+0.j,
0.01428571+0.j, 0.01428571+0.j, 0.01428571+0.j, 0.01428571+0.j,
0.01428571+0.j, 0.01428571+0.j, 0.01428571+0.j, 0.01428571+0.j,
0.01428571+0.j, 0.01428571+0.j, 0.01428571+0.j, 0.01428571+0.j,
0.01428571+0.j, 0.01428571+0.j, 0.01428571+0.j, 0.01428571+0.j,
0.01428571+0.j, 0.01428571+0.j, 0.01428571+0.j, 0.01428571+0.j,
0.01428571+0.j, 0.01428571+0.j, 0.01428571+0.j, 0.01428571+0.j,
0.01428571+0.j, 0.01428571+0.j, 0.01428571+0.j, 0.01428571+0.j,
0.01428571+0.j, 0.01428571+0.j, 0.01428571+0.j, 0.01428571+0.j,
0.01428571+0.j, 0.01428571+0.j, 0.01428571+0.j, 0.01428571+0.j,
0.01428571+0.j, 0.01428571+0.j])
2.6 Activity 6
시간 진화 시뮬레이션 수행
자기화(자기화 연산자의 기댓값)를 모니터링할 것입니다. 하드웨어에서 계산된 결과와 비교하기 위한 기준값을 얻기 위해 MPS 시뮬레이터를 사용할 것입니다. 이 튜토리얼에서 MPS 시뮬레이터를 이전에 사용한 적이 있습니다. 이 새로운 계산에 맞게 해당 예시를 필요한 부분만 수정해 보세요.
풀이:
# Step 1. Map the problem
# Initiate the circuit
evolved_state2 = QuantumCircuit(initial_circuit2.num_qubits)
# Start from the initial spin configuration
evolved_state2.append(initial_circuit2, evolved_state2.qubits)
# Define backend (MPs simulator)
backend_mps2 = AerSimulator(method="matrix_product_state")
# Initiate Runtime Estimator
estimator_mps2 = Estimator(mode=backend_mps2)
# Step 2. Optimize
# Initiate pass manager
pm_mps2 = generate_preset_pass_manager(optimization_level=3, backend=backend_mps2)
# Transpile
evolved_state_mps2 = pm_mps2.run(evolved_state2)
# Apply qubit layout to the observable to measure
magnetization_mps2 = magnetization2.apply_layout(evolved_state_mps2.layout)
# Initiate list
mag_mps_list2 = []
# Step 3. Run the circuit
# Estimate expectation values for t=0.0
job = estimator_mps2.run([(evolved_state_mps2, [magnetization_mps2])])
# Get estimated expectation values
evs = job.result()[0].data.evs
# Append to list
mag_mps_list2.append(evs[0])
# Start time evolution
for n in range(num_timesteps):
# Step 1. Map the problem
# Expand the circuit to describe delta-t
evolved_state2.append(single_step_evolution_lt2, evolved_state2.qubits)
# Step 2. Optimize
# Transpile the circuit
evolved_state_mps2 = pm_mps2.run(evolved_state2)
# Apply the physical layout of the qubits to the operator
magnetization_mps2 = magnetization2.apply_layout(evolved_state_mps2.layout)
# Step 3. Run the circuit
# Estimate expectation values at delta-t
job = estimator_mps2.run([(evolved_state_mps2, [magnetization_mps2])])
# Get estimated expectation values
evs = job.result()[0].data.evs
# Append to list
mag_mps_list2.append(evs[0])
# Transform the list of expectation values (at each time step) to arrays
mag_mps_array2 = np.array(mag_mps_list2)
이전 모든 레슨과 마찬가지로 Qiskit 패턴 프레임워크를 구현할 것입니다. 이 레슨에서 지금까지의 내용은 문제를 설명하는 올바른 양자 Circuit을 만드는 데 초점을 맞추었습니다. 이것이 사실상 1단계에 해당합니다.
2단계: 대상 하드웨어에 맞게 최적화
대상 Backend를 정의하는 것부터 시작합니다.
service = QiskitRuntimeService()
backend = service.least_busy(operational=True, simulator=False)
backend.name
'ibm_kingston'
Circuit을 트랜스파일하고 리스트에 수집합니다. 몇 분 정도 걸릴 수 있습니다.
pm_hw = generate_preset_pass_manager(optimization_level=3, backend=backend)
circuit_isa = []
# Step 1. Map the problem
evolved_state_hw = QuantumCircuit(initial_circuit2.num_qubits)
evolved_state_hw.append(initial_circuit2, evolved_state_hw.qubits)
# Step 2. Optimize
circuit_isa.append(pm_hw.run(evolved_state_hw))
for n in range(num_timesteps):
# Step 1. Map the problem
evolved_state_hw.append(single_step_evolution_lt2, evolved_state_hw.qubits)
# Step 2. Optimize
circuit_isa.append(pm_hw.run(evolved_state_hw))
3단계: 대상 하드웨어에서 실행
Runtime Estimator를 정의하고 PUB 리스트를 구성할 것입니다. 또한 측정할 연산자에 레이아웃을 적용해야 합니다.
# Step 2. Optimize
estimator_hw = Estimator(mode=backend)
pub_list = []
for circuit in circuit_isa:
temp = (circuit, magnetization2.apply_layout(circuit.layout))
pub_list.append(temp)
이제 작업을 실행할 준비가 되었습니다.
job = estimator_hw.run(pub_list)
job_id = job.job_id()
print(job_id)
d147hfdqf56g0081sxs0
# check job status
job.status()
'DONE'
4단계: 결과 후처리
먼저 결과를 가져옵니다.
job = service.job(job_id)
pub_result = job.result()
이제 이 결과에서 기댓값을 추출해야 합니다.
mag_hw_list = []
for res in pub_result:
evs = res.data.evs
mag_hw_list.append(evs)
이것은 아래 비교에 사용할 것입니다. 먼저 Circuit을 더욱 최적화할 수 있는지 살펴보겠습니다.
3. 실제 양자 컴퓨터를 사용한 풀이 II
Qiskit 패턴 1단계로 돌아가서, Circuit의 깊이를 줄일 수 있는지 살펴보겠습니다.
3.1 1단계. 문제를 양자 Circuit 및 연산자로 매핑하기
활동 7
시간 진화 Circuit을 구성하세요. 이전 학습에서 익힌 지식을 활용하여 Circuit의 깊이를 줄여보세요.
풀이:
# Define J
J = 1.0
# Define h
h = -5.0
# Create instruction for rotation around ZZ:
# Initiate the circuit (use 2 qubits)
Rzz_circ = QuantumCircuit(2)
# Add Rzz gate (do not forget to multiply the angle by 2.0)
Rzz_circ.rzz(-J * dt * 2.0, 0, 1)
# Transform the QuantumCircuit to instruction (QuantumCircuit.to_instruction())
Rzz_instr = Rzz_circ.to_instruction(label="RZZ")
# Create instruction for rotation around X:
# Initiate the circuit (use 1 qubit)
Rx_circ = QuantumCircuit(1)
# Add Rx gate (do not forget to multiply the angle by 2.0)
Rx_circ.rx(-h * dt * 2.0, 0)
# Transform the QuantumCircuit to instruction (QuantumCircuit.to_instruction())
Rx_instr = Rx_circ.to_instruction(label="RX")
# Define the interaction list
interaction_list = [
[[i, i + 1] for i in range(0, n_qubits2 - 1, 2)],
[[i, i + 1] for i in range(1, n_qubits2 - 1, 2)],
] # linear chain
# Define the registers
qr = QuantumRegister(n_qubits2)
# Initiate the circuit
single_step_evolution_sh = QuantumCircuit(qr)
# Construct the Rzz gates
for i, color in enumerate(interaction_list):
for interaction in color:
single_step_evolution_sh.append(Rzz_instr, interaction)
# Construct the Rx gates
for i in range(0, n_qubits2):
single_step_evolution_sh.append(Rx_instr, [i])
print(
f"""
Trotter step with Lie-Trotter
-----------------------------
Depth: {single_step_evolution_sh.decompose(reps=3).depth()}
Gate count: {len(single_step_evolution_sh.decompose(reps=3))}
Nonlocal gate count: {single_step_evolution_sh.decompose(reps=3).num_nonlocal_gates()}
Gate breakdown: {", ".join([f"{k.upper()}: {v}" for k, v in single_step_evolution_sh.decompose(reps=3).count_ops().items()])}
"""
)
single_step_evolution_sh.decompose(reps=2).draw("mpl")
Trotter step with Lie-Trotter
-----------------------------
Depth: 7
Gate count: 277
Nonlocal gate count: 138
Gate breakdown: CX: 138, U3: 70, U1: 69
매우 성공적인 결과입니다. 이제 나머지 Qiskit 패턴 단계를 진행할 수 있습니다.
3.2 2단계. 대상 하드웨어에 맞게 최적화하기
Circuit을 Transpile하고 목록에 모아둡니다. 이번에도 몇 분 정도 소요될 수 있습니다.
pm_hw2 = generate_preset_pass_manager(backend=backend, optimization_level=3)
circuit_isa2 = []
# Step 1. Map the problem
evolved_state_hw2 = QuantumCircuit(initial_circuit2.num_qubits)
evolved_state_hw2.append(initial_circuit2, evolved_state_hw2.qubits)
# Step 2. Optimize
circuit_isa2.append(pm_hw2.run(evolved_state_hw2))
for n in range(num_timesteps):
# Step 1. Map the problem
evolved_state_hw2.append(single_step_evolution_sh, evolved_state_hw2.qubits)
# Step 2. Optimize
circuit_isa2.append(pm_hw2.run(evolved_state_hw2))
Runtime Estimator를 정의하고 PUB 목록을 구성합니다.
estimator_hw2 = Estimator(mode=backend)
pub_list2 = []
for circuit in circuit_isa2:
temp = (circuit, magnetization2.apply_layout(circuit.layout))
pub_list2.append(temp)
3.3 3단계. 대상 하드웨어에서 실행하기
작업을 실행합니다.
job2 = estimator_hw2.run(pub_list2)
job2_id = job2.job_id()
print(job2_id)
d147qqeqf56g0081sye0
# check job status
job2.status()
'DONE'
결과를 가져옵니다.
job2 = service.job(job2_id)
pub_result2 = job2.result()
3.4 4단계. 후처리
결과에서 기댓값을 추출합니다.
mag_hw_list2 = []
for res in pub_result2:
evs = res.data.evs
mag_hw_list2.append(evs)
목록을 플로팅을 위한 numpy 배열로 변환합니다.
mag_hw_array = np.array(mag_hw_list)
mag_hw_array2 = np.array(mag_hw_list2)
이제 결과를 플로팅하고, 하드웨어 결과(기본 Circuit과 얕은 Circuit 모두)를 MPS 시뮬레이터 결과와 비교해 보겠습니다. 실제 하드웨어의 오류가 결과에 어떤 영향을 미치나요?
fig, axes = plt.subplots(3, sharex=True)
times = np.linspace(0, evolution_time, num_timesteps + 1) # includes initial state
axes[0].plot(
times, mag_mps_array2, label="MPS", marker="x", c="darkmagenta", ls="-", lw=0.8
)
axes[1].plot(
times, mag_hw_array, label="HW", marker="x", c="darkmagenta", ls="-", lw=0.8
)
axes[2].plot(
times, mag_hw_array2, label="HW2", marker="x", c="darkmagenta", ls="-", lw=0.8
)
axes[0].set_ylabel("MPS")
axes[1].set_ylabel("HW")
axes[2].set_ylabel("HW2")
axes[2].set_xlabel("Time")
fig.suptitle("Observable evolution")
Text(0.5, 0.98, 'Observable evolution')
