Operator backpropagation (OBP) para sa pagtatantya ng expectation values

Tantya sa paggamit: 16 minuto sa isang Eagle r3 processor (PAALALA: Ito ay tantya lamang. Maaaring mag-iba ang iyong runtime.)

# Added by doQumentation — required packages for this notebook
!pip install -q matplotlib numpy qiskit qiskit-addon-obp qiskit-addon-utils qiskit-ibm-runtime rustworkx

# This cell is hidden from users;
# it disables linting rules.
# ruff: noqa

Background

Ang operator backpropagation ay isang teknikang nagsasangkot ng pag-absorb ng mga operasyon mula sa dulo ng isang quantum circuit tungo sa sinusukat na observable, na karaniwang nagpapababa ng lalim ng circuit sa halaga ng karagdagang mga term sa observable. Ang layunin ay mag-backpropagate ng kasing dami ng circuit hangga't maaari nang hindi pinapayagang lumaki nang labis ang observable. Ang isang Qiskit-based na implementasyon ay available sa OBP Qiskit addon, at mas maraming detalye ay matatagpuan sa kaukulang docs kasama ang isang simpleng halimbawa para makapagsimula.

Isaalang-alang ang isang halimbawang circuit kung saan ang isang observable $O = \sum_P c_P P$ ay susukatiin, kung saan ang $P$ ay mga Pauli at ang $c_P$ ay mga coefficient. Itakda natin ang circuit bilang isang unitary $U$ na maaaring lohikal na hatiin sa $U = U_C U_Q$ tulad ng ipinapakita sa figure sa ibaba.

Circuit diagram showing Uq followed by Uc

Ang operator backpropagation ay nag-aabsorb ng unitary $U_C$ sa observable sa pamamagitan ng pag-evolve nito bilang $O' = U_C^{\dagger}OU_C = \sum_P c_P U_C^{\dagger}PU_C$ . Sa madaling salita, ang bahagi ng computation ay ginagawa nang classical sa pamamagitan ng evolusyon ng observable mula sa $O$ tungo sa $O'$ . Ang orihinal na problema ay maaari ngayong muling iformula bilang pagsukat ng observable $O'$ para sa bagong mas mababang lalim na circuit na ang unitary ay $U_Q$ .

Ang unitary $U_C$ ay kinakatawan bilang isang bilang ng mga slice $U_C = U_S U_{S-1}...U_2U_1$ . May maraming paraan para tukuyin ang isang slice. Halimbawa, sa halimbawang circuit sa itaas, ang bawat layer ng $R_{zz}$ at bawat layer ng $R_x$ gates ay maaaring ituring bilang indibidwal na slice. Ang backpropagation ay nagsasangkot ng pagkalkula ng $O' = \Pi_{s=1}^S \sum_P c_P U_s^{\dagger} P U_s$ nang classical. Ang bawat slice $U_s$ ay maaaring katawanin bilang $U_s = exp(\frac{-i\theta_s P_s}{2})$ , kung saan ang $P_s$ ay isang $n$ -qubit Pauli at ang $\theta_s$ ay isang scalar. Madaling ma-verify na

U_s^{\dagger} P U_s = P \qquad \text{if} ~[P,P_s] = 0,

U_s^{\dagger} P U_s = \qquad cos(\theta_s)P + i sin(\theta_s)P_sP \qquad \text{if} ~\{P,P_s\} = 0

Sa halimbawa sa itaas, kung $\{P,P_s\} = 0$ , kailangan nating magsagawa ng dalawang quantum circuit, sa halip na isa, upang kalkulahin ang expectation value. Samakatuwid, ang backpropagation ay maaaring magdagdag ng bilang ng mga term sa observable, na nagreresulta sa mas mataas na bilang ng circuit execution. Isang paraan upang payagan ang mas malalim na backpropagation sa circuit, habang pinipigilan ang operator mula sa paglaki nang labis, ay ang mag-truncate ng mga term na may maliliit na coefficient, sa halip na idagdag ang mga ito sa operator. Halimbawa, sa halimbawa sa itaas, maaari tayong pumili na mag-truncate ng term na nagsasangkot ng $P_sP$ kung sakaling ang $\theta_s$ ay sapat na maliit. Ang pag-truncate ng mga term ay maaaring magresulta sa mas kakaunting quantum circuit na isasagawa, ngunit ang paggawa nito ay nagreresulta ng ilang error sa panghuling pagkalkula ng expectation value na proporsyonal sa magnitude ng mga coefficient ng mga na-truncate na term.

Ang tutorial na ito ay nagsasagawa ng isang Qiskit pattern para sa pagsisimula ng quantum dynamics ng isang Heisenberg spin chain gamit ang qiskit-addon-obp.

Requirements

Bago simulan ang tutorial na ito, siguraduhing mayroon ka ng mga sumusunod na naka-install:

Qiskit SDK v1.2 o mas bago (pip install qiskit)
Qiskit Runtime v0.28 o mas bago (pip install qiskit-ibm-runtime)
OBP Qiskit addon (pip install qiskit-addon-obp)
Qiskit addon utils (pip install qiskit-addon-utils)

Setup

import numpy as np
import matplotlib.pyplot as plt

from qiskit.primitives import StatevectorEstimator as Estimator
from qiskit.transpiler.preset_passmanagers import generate_preset_pass_manager
from qiskit.quantum_info import SparsePauliOp
from qiskit.transpiler import CouplingMap
from qiskit.synthesis import LieTrotter

from qiskit_addon_utils.problem_generators import generate_xyz_hamiltonian
from qiskit_addon_utils.problem_generators import (
    generate_time_evolution_circuit,
)
from qiskit_addon_utils.slicing import slice_by_gate_types, combine_slices
from qiskit_addon_obp.utils.simplify import OperatorBudget
from qiskit_addon_obp import backpropagate
from qiskit_addon_obp.utils.truncating import setup_budget

from rustworkx.visualization import graphviz_draw

from qiskit_ibm_runtime import QiskitRuntimeService
from qiskit_ibm_runtime import EstimatorV2, EstimatorOptions

Part I: Small-scale Heisenberg spin chain

Step 1: Map classical inputs to a quantum problem

Map the time-evolution of a quantum Heisenberg model to a quantum experiment.

Ang package na qiskit_addon_utils ay nagbibigay ng ilang muling magagamit na functionality para sa iba't ibang layunin.

Ang module nitong qiskit_addon_utils.problem_generators ay nagbibigay ng mga function upang lumikha ng mga Heisenberg-like Hamiltonian sa isang ibinigay na connectivity graph. Ang graph na ito ay maaaring rustworkx.PyGraph o CouplingMap na nagpapadalì ng paggamit sa mga Qiskit-centric na workflow.

Sa sumusunod, bubuo tayo ng isang linear chain na CouplingMap ng 10 qubit.

num_qubits = 10
layout = [(i - 1, i) for i in range(1, num_qubits)]

# Instantiate a CouplingMap object
coupling_map = CouplingMap(layout)
graphviz_draw(coupling_map.graph, method="circo")

Output of the previous code cell

Susunod, bubuo tayo ng isang Pauli operator na nagmomodelo ng Heisenberg XYZ Hamiltonian.

{\hat{\mathcal{H}}_{XYZ} = \sum_{(j,k)\in E} (J_{x} \sigma_j^{x} \sigma_{k}^{x} + J_{y} \sigma_j^{y} \sigma_{k}^{y} + J_{z} \sigma_j^{z} \sigma_{k}^{z}) + \sum_{j\in V} (h_{x} \sigma_j^{x} + h_{y} \sigma_j^{y} + h_{z} \sigma_j^{z})}

Kung saan ang $G(V,E)$ ay ang graph ng ibinigay na coupling map.

# Get a qubit operator describing the Heisenberg XYZ model
hamiltonian = generate_xyz_hamiltonian(
    coupling_map,
    coupling_constants=(np.pi / 8, np.pi / 4, np.pi / 2),
    ext_magnetic_field=(np.pi / 3, np.pi / 6, np.pi / 9),
)
print(hamiltonian)

SparsePauliOp(['IIIIIIIXXI', 'IIIIIIIYYI', 'IIIIIIIZZI', 'IIIIIXXIII', 'IIIIIYYIII', 'IIIIIZZIII', 'IIIXXIIIII', 'IIIYYIIIII', 'IIIZZIIIII', 'IXXIIIIIII', 'IYYIIIIIII', 'IZZIIIIIII', 'IIIIIIIIXX', 'IIIIIIIIYY', 'IIIIIIIIZZ', 'IIIIIIXXII', 'IIIIIIYYII', 'IIIIIIZZII', 'IIIIXXIIII', 'IIIIYYIIII', 'IIIIZZIIII', 'IIXXIIIIII', 'IIYYIIIIII', 'IIZZIIIIII', 'XXIIIIIIII', 'YYIIIIIIII', 'ZZIIIIIIII', 'IIIIIIIIIX', 'IIIIIIIIIY', 'IIIIIIIIIZ', 'IIIIIIIIXI', 'IIIIIIIIYI', 'IIIIIIIIZI', 'IIIIIIIXII', 'IIIIIIIYII', 'IIIIIIIZII', 'IIIIIIXIII', 'IIIIIIYIII', 'IIIIIIZIII', 'IIIIIXIIII', 'IIIIIYIIII', 'IIIIIZIIII', 'IIIIXIIIII', 'IIIIYIIIII', 'IIIIZIIIII', 'IIIXIIIIII', 'IIIYIIIIII', 'IIIZIIIIII', 'IIXIIIIIII', 'IIYIIIIIII', 'IIZIIIIIII', 'IXIIIIIIII', 'IYIIIIIIII', 'IZIIIIIIII', 'XIIIIIIIII', 'YIIIIIIIII', 'ZIIIIIIIII'],
              coeffs=[0.39269908+0.j, 0.78539816+0.j, 1.57079633+0.j, 0.39269908+0.j,
78539816+0.j, 1.57079633+0.j, 0.39269908+0.j, 0.78539816+0.j,
57079633+0.j, 0.39269908+0.j, 0.78539816+0.j, 1.57079633+0.j,
39269908+0.j, 0.78539816+0.j, 1.57079633+0.j, 0.39269908+0.j,
78539816+0.j, 1.57079633+0.j, 0.39269908+0.j, 0.78539816+0.j,
57079633+0.j, 0.39269908+0.j, 0.78539816+0.j, 1.57079633+0.j,
39269908+0.j, 0.78539816+0.j, 1.57079633+0.j, 1.04719755+0.j,
52359878+0.j, 0.34906585+0.j, 1.04719755+0.j, 0.52359878+0.j,
34906585+0.j, 1.04719755+0.j, 0.52359878+0.j, 0.34906585+0.j,
04719755+0.j, 0.52359878+0.j, 0.34906585+0.j, 1.04719755+0.j,
52359878+0.j, 0.34906585+0.j, 1.04719755+0.j, 0.52359878+0.j,
34906585+0.j, 1.04719755+0.j, 0.52359878+0.j, 0.34906585+0.j,
04719755+0.j, 0.52359878+0.j, 0.34906585+0.j, 1.04719755+0.j,
52359878+0.j, 0.34906585+0.j, 1.04719755+0.j, 0.52359878+0.j,
34906585+0.j])

Mula sa qubit operator, maaari tayong lumikha ng quantum circuit na nagmomodelo ng time evolution nito. Muli, ang module na qiskit_addon_utils.problem_generators ay dumarating sa pagliligtas na may kapaki-pakinabang na function upang gawin iyon:

circuit = generate_time_evolution_circuit(
    hamiltonian,
    time=0.2,
    synthesis=LieTrotter(reps=2),
)
circuit.draw("mpl", style="iqp", scale=0.6)

Output of the previous code cell

Step 2: Optimize problem for quantum hardware execution

Create circuit slices to backpropagate

Tandaan, ang function na backpropagate ay mag-backpropagate ng buong circuit slice nang sabay-sabay, kaya ang pagpili kung paano maghiwa ay maaaring magkaroon ng epekto sa kung gaano kahusay gumagana ang backpropagation para sa isang partikular na problema. Dito, pagsasamahin natin ang mga gate ng parehong uri sa mga slice gamit ang function na slice_by_gate_types.

Para sa mas detalyadong talakayan tungkol sa circuit slicing, tingnan ang how-to guide na ito ng package na qiskit-addon-utils.

slices = slice_by_gate_types(circuit)
print(f"Separated the circuit into {len(slices)} slices.")

Separated the circuit into 18 slices.

Constrain how large the operator may grow during backpropagation

Sa panahon ng backpropagation, ang bilang ng mga term sa operator ay karaniwang lalapit sa $4^N$ nang mabilis, kung saan ang $N$ ay ang bilang ng mga qubit. Kapag ang dalawang term sa operator ay hindi nag-commute nang qubit-wise, kailangan natin ng hiwalay na mga circuit upang makuha ang mga expectation value na tumutugma sa kanila. Halimbawa, kung mayroon tayong 2-qubit observable na $O = 0.1 XX + 0.3 IZ - 0.5 IX$ , dahil ang $[XX,IX] = 0$ , ang pagsukat sa isang basis lamang ay sapat upang kalkulahin ang mga expectation value para sa dalawang term na ito. Gayunpaman, ang $IZ$ ay anti-commute sa dalawang term. Kaya kailangan natin ng hiwalay na basis measurement upang kalkulahin ang expectation value ng $IZ$ . Sa madaling salita, kailangan natin ng dalawang circuit, sa halip na isa, upang kalkulahin ang $\langle O \rangle$ . Habang dumarami ang bilang ng mga term sa operator, may posibilidad na dumarami rin ang kinakailangang bilang ng circuit execution.

Ang laki ng operator ay maaaring limitahan sa pamamagitan ng pagtukoy ng operator_budget kwarg ng function na backpropagate, na tumatanggap ng isang instance ng OperatorBudget.

Upang kontrolin ang dami ng karagdagang resources (oras) na inilalaan, pinapaghihigpitan natin ang maximum na bilang ng qubit-wise commuting Pauli groups na pinapayagang magkaroon ang backpropagated observable. Dito, tinukoy natin na ang backpropagation ay dapat tumigil kapag ang bilang ng qubit-wise commuting Pauli groups sa operator ay lumampas sa 8.

op_budget = OperatorBudget(max_qwc_groups=8)

Backpropagate slices from the circuit

Una, tinukoy natin ang observable bilang $M_Z = \frac{1}{N} \sum_{i=1}^N \langle Z_i \rangle$ , kung saan ang $N$ ay ang bilang ng mga qubit. Mag-backpropagate tayo ng mga slice mula sa time-evolution circuit hanggang sa ang mga term sa observable ay hindi na maaaring pagsamahin sa walong o mas kaunting qubit-wise commuting Pauli groups.

observable = SparsePauliOp.from_sparse_list(
    [("Z", [i], 1 / num_qubits) for i in range(num_qubits)],
    num_qubits=num_qubits,
)
observable

SparsePauliOp(['IIIIIIIIIZ', 'IIIIIIIIZI', 'IIIIIIIZII', 'IIIIIIZIII', 'IIIIIZIIII', 'IIIIZIIIII', 'IIIZIIIIII', 'IIZIIIIIII', 'IZIIIIIIII', 'ZIIIIIIIII'],
              coeffs=[0.1+0.j, 0.1+0.j, 0.1+0.j, 0.1+0.j, 0.1+0.j, 0.1+0.j, 0.1+0.j, 0.1+0.j,
 0.1+0.j, 0.1+0.j])

Sa ibaba, makikita mo na nag-backpropagate tayo ng anim na slice, at ang mga term ay pinagsama sa anim at hindi walong grupo. Ito ay nagpapahiwatig na ang pag-backpropagate pa ng isang slice ay magiging sanhi ng paglampas ng bilang ng mga Pauli group sa walong. Maaari nating i-verify na ito ang kaso sa pamamagitan ng pagsusuri sa ibinalik na metadata. Pansinin din na sa bahaging ito, ang circuit transformation ay eksakto. Ibig sabihin, walang mga term ng bagong observable na $O'$ ang na-truncate. Ang backpropagated circuit at ang backpropagated operator ay nagbibigay ng eksaktong kinalabasan tulad ng orihinal na circuit at operator.

# Backpropagate slices onto the observable
bp_obs, remaining_slices, metadata = backpropagate(
    observable, slices, operator_budget=op_budget
)
# Recombine the slices remaining after backpropagation
bp_circuit = combine_slices(remaining_slices)

print(f"Backpropagated {metadata.num_backpropagated_slices} slices.")
print(
    f"New observable has {len(bp_obs.paulis)} terms, which can be combined into {len(bp_obs.group_commuting(qubit_wise=True))} groups."
)
print(
    f"Note that backpropagating one more slice would result in {metadata.backpropagation_history[-1].num_paulis[0]} terms "
    f"across {metadata.backpropagation_history[-1].num_qwc_groups} groups."
)
print("The remaining circuit after backpropagation looks as follows:")
bp_circuit.draw("mpl", fold=-1, scale=0.6)

Backpropagated 6 slices.
New observable has 60 terms, which can be combined into 6 groups.
Note that backpropagating one more slice would result in 114 terms across 12 groups.
The remaining circuit after backpropagation looks as follows:

Output of the previous code cell

Susunod, tutukuyin natin ang parehong problema na may parehong mga hadlang sa laki ng output observable. Gayunpaman, sa pagkakataong ito, maglalaan tayo ng error budget sa bawat slice gamit ang function na setup_budget. Ang mga Pauli term na may maliliit na coefficient ay ita-truncate mula sa bawat slice hanggang mapunan ang error budget, at ang natitirang budget ay idadagdag sa budget ng sumusunod na slice. Tandaan na sa kasong ito, ang transformation dahil sa backpropagation ay approximation dahil ang ilan sa mga term sa operator ay na-truncate.

Upang paganahin ang truncation na ito, kailangan nating i-setup ang ating error budget tulad nito:

truncation_error_budget = setup_budget(max_error_per_slice=0.005)

Tandaan na sa pamamagitan ng paglalaan ng 5e-3 error sa bawat slice para sa truncation, nakakaya nating mag-alis ng 1 pang slice mula sa circuit, habang nananatili sa loob ng orihinal na budget ng walong commuting Pauli groups sa observable. Bilang default, ang backpropagate ay gumagamit ng L1 norm ng mga na-truncate na coefficient upang limitahan ang kabuuang error na natamo mula sa truncation. Para sa iba pang mga opsyon, sumangguni sa how-to guide on specifying the p_norm.

Sa partikular na halimbawa kung saan nag-backpropagate tayo ng pitong slice, ang kabuuang truncation error ay hindi dapat lumampas sa (5e-3 error/slice) * (7 slices) = 3.5e-2. Para sa karagdagang talakayan tungkol sa pamamahagi ng error budget sa iyong mga slice, tingnan ang how-to guide na ito.

# Run the same experiment but truncate observable terms with small coefficients
bp_obs_trunc, remaining_slices_trunc, metadata = backpropagate(
    observable,
    slices,
    operator_budget=op_budget,
    truncation_error_budget=truncation_error_budget,
)

# Recombine the slices remaining after backpropagation
bp_circuit_trunc = combine_slices(
    remaining_slices_trunc, include_barriers=False
)

print(f"Backpropagated {metadata.num_backpropagated_slices} slices.")
print(
    f"New observable has {len(bp_obs_trunc.paulis)} terms, which can be combined into {len(bp_obs_trunc.group_commuting(qubit_wise=True))} groups.\n"
    f"After truncation, the error in our observable is bounded by {metadata.accumulated_error(0):.3e}"
)
print(
    f"Note that backpropagating one more slice would result in {metadata.backpropagation_history[-1].num_paulis[0]} terms "
    f"across {metadata.backpropagation_history[-1].num_qwc_groups} groups."
)
print("The remaining circuit after backpropagation looks as follows:")
bp_circuit_trunc.draw("mpl", scale=0.6)

Backpropagated 7 slices.
New observable has 82 terms, which can be combined into 8 groups.
After truncation, the error in our observable is bounded by 3.266e-02
Note that backpropagating one more slice would result in 114 terms across 12 groups.
The remaining circuit after backpropagation looks as follows:

Output of the previous code cell

Napapansin natin na ang truncation ay nagpapahintulot sa atin na mag-backpropagate pa nang wala pang pagtaas sa bilang ng commuting groups sa observable.

Now that we have our reduced ansatz and expanded observables, we can transpile our experiments to the backend.

Dito, gagamitin natin ang isang 127-qubit na IBM® quantum computer upang ipakita kung paano mag-transpile sa isang QPU backend.

service = QiskitRuntimeService()
backend = service.least_busy(
    operational=True, simulator=False, min_num_qubits=127
)

pm = generate_preset_pass_manager(backend=backend, optimization_level=1)

# Transpile original experiment
circuit_isa = pm.run(circuit)
observable_isa = observable.apply_layout(circuit_isa.layout)

# Transpile backpropagated experiment
bp_circuit_isa = pm.run(bp_circuit)
bp_obs_isa = bp_obs.apply_layout(bp_circuit_isa.layout)

# Transpile the backpropagated experiment with truncated observable terms
bp_circuit_trunc_isa = pm.run(bp_circuit_trunc)
bp_obs_trunc_isa = bp_obs_trunc.apply_layout(bp_circuit_trunc_isa.layout)

Lumilikha tayo ng Primitive Unified Bloc (PUB) para sa bawat isa sa tatlong kaso.

pub = (circuit_isa, observable_isa)
bp_pub = (bp_circuit_isa, bp_obs_isa)
bp_trunc_pub = (bp_circuit_trunc_isa, bp_obs_trunc_isa)

Step 3: Execute using Qiskit primitives

Calculate expectation value

Sa wakas, maaari tayong magpatakbo ng mga backpropagated experiment at ikumpara ang mga ito sa kumpletong experiment gamit ang noiseless na StatevectorEstimator.

ideal_estimator = Estimator()

# Run the experiments using Estimator primitive to obtain the exact outcome
result_exact = (
    ideal_estimator.run([(circuit, observable)]).result()[0].data.evs.item()
)
print(f"Exact expectation value: {result_exact}")

Exact expectation value: 0.8871244838989416

Gagamitin natin ang resilience_level = 2 para sa halimbawang ito.

options = EstimatorOptions()
options.default_precision = 0.011
options.resilience_level = 2

estimator = EstimatorV2(mode=backend, options=options)

job = estimator.run([pub, bp_pub, bp_trunc_pub])

Step 4: Post-process and return result to desired classical format

result_no_bp = job.result()[0].data.evs.item()
result_bp = job.result()[1].data.evs.item()
result_bp_trunc = job.result()[2].data.evs.item()

std_no_bp = job.result()[0].data.stds.item()
std_bp = job.result()[1].data.stds.item()
std_bp_trunc = job.result()[2].data.stds.item()

print(
    f"Expectation value without backpropagation: {result_no_bp} ± {std_no_bp}"
)
print(f"Backpropagated expectation value: {result_bp} ± {std_bp}")
print(
    f"Backpropagated expectation value with truncation: {result_bp_trunc} ± {std_bp_trunc}"
)

Expectation value without backpropagation: 0.8033194665993642
Backpropagated expectation value: 0.8599808781259016
Backpropagated expectation value with truncation: 0.8868736004169483

methods = [
    "No backpropagation",
    "Backpropagation",
    "Backpropagation w/ truncation",
]
values = [result_no_bp, result_bp, result_bp_trunc]
stds = [std_no_bp, std_bp, std_bp_trunc]

ax = plt.gca()
plt.bar(methods, values, color="#a56eff", width=0.4, edgecolor="#8a3ffc")
plt.axhline(result_exact)
ax.set_ylim([0.6, 0.92])
plt.text(0.2, 0.895, "Exact result")
ax.set_ylabel(r"$M_Z$", fontsize=12)

Text(0, 0.5, '$M_Z$')

Output of the previous code cell

Part B: Scale it up!

Gamitin natin ngayon ang Operator Backpropagation upang pag-aralan ang dynamics ng Hamiltonian ng isang 50-qubit na Heisenberg Spin Chain.

Step 1: Map classical inputs to a quantum problem

Isinasaalang-alang natin ang isang 50-qubit Hamiltonian na $\hat{\mathcal{H}}_{XYZ}$ para sa scaled up na problema na may parehong mga halaga para sa mga coefficient na $J$ at $h$ tulad ng sa small-scale na halimbawa. Ang observable na $M_Z = \frac{1}{N} \sum_{i=1}^N \langle Z_i \rangle$ ay pareho rin sa dati. Ang problemang ito ay lampas sa classical brute-force simulation.

num_qubits = 50
layout = [(i - 1, i) for i in range(1, num_qubits)]

# Instantiate a CouplingMap object
coupling_map = CouplingMap(layout)
graphviz_draw(coupling_map.graph, method="circo")

Output of the previous code cell

hamiltonian = generate_xyz_hamiltonian(
    coupling_map,
    coupling_constants=(np.pi / 8, np.pi / 4, np.pi / 2),
    ext_magnetic_field=(np.pi / 3, np.pi / 6, np.pi / 9),
)
print(hamiltonian)

SparsePauliOp(['IIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIXXI', 'IIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIYYI', 'IIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIZZI', 'IIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIXXIII', 'IIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIYYIII', 'IIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIZZIII', 'IIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIXXIIIII', 'IIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIYYIIIII', 'IIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIZZIIIII', 'IIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIXXIIIIIII', 'IIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIYYIIIIIII', 'IIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIZZIIIIIII', 'IIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIXXIIIIIIIII', 'IIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIYYIIIIIIIII', 'IIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIZZIIIIIIIII', 'IIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIXXIIIIIIIIIII', 'IIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIYYIIIIIIIIIII', 'IIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIZZIIIIIIIIIII', 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57079633+0.j, 0.39269908+0.j, 0.78539816+0.j, 1.57079633+0.j,
39269908+0.j, 0.78539816+0.j, 1.57079633+0.j, 0.39269908+0.j,
78539816+0.j, 1.57079633+0.j, 0.39269908+0.j, 0.78539816+0.j,
57079633+0.j, 0.39269908+0.j, 0.78539816+0.j, 1.57079633+0.j,
39269908+0.j, 0.78539816+0.j, 1.57079633+0.j, 0.39269908+0.j,
78539816+0.j, 1.57079633+0.j, 0.39269908+0.j, 0.78539816+0.j,
57079633+0.j, 0.39269908+0.j, 0.78539816+0.j, 1.57079633+0.j,
39269908+0.j, 0.78539816+0.j, 1.57079633+0.j, 0.39269908+0.j,
78539816+0.j, 1.57079633+0.j, 0.39269908+0.j, 0.78539816+0.j,
57079633+0.j, 0.39269908+0.j, 0.78539816+0.j, 1.57079633+0.j,
39269908+0.j, 0.78539816+0.j, 1.57079633+0.j, 0.39269908+0.j,
78539816+0.j, 1.57079633+0.j, 0.39269908+0.j, 0.78539816+0.j,
57079633+0.j, 0.39269908+0.j, 0.78539816+0.j, 1.57079633+0.j,
39269908+0.j, 0.78539816+0.j, 1.57079633+0.j, 0.39269908+0.j,
78539816+0.j, 1.57079633+0.j, 0.39269908+0.j, 0.78539816+0.j,
57079633+0.j, 0.39269908+0.j, 0.78539816+0.j, 1.57079633+0.j,
39269908+0.j, 0.78539816+0.j, 1.57079633+0.j, 0.39269908+0.j,
78539816+0.j, 1.57079633+0.j, 0.39269908+0.j, 0.78539816+0.j,
57079633+0.j, 0.39269908+0.j, 0.78539816+0.j, 1.57079633+0.j,
39269908+0.j, 0.78539816+0.j, 1.57079633+0.j, 0.39269908+0.j,
78539816+0.j, 1.57079633+0.j, 0.39269908+0.j, 0.78539816+0.j,
57079633+0.j, 0.39269908+0.j, 0.78539816+0.j, 1.57079633+0.j,
39269908+0.j, 0.78539816+0.j, 1.57079633+0.j, 0.39269908+0.j,
78539816+0.j, 1.57079633+0.j, 0.39269908+0.j, 0.78539816+0.j,
57079633+0.j, 0.39269908+0.j, 0.78539816+0.j, 1.57079633+0.j,
39269908+0.j, 0.78539816+0.j, 1.57079633+0.j, 0.39269908+0.j,
78539816+0.j, 1.57079633+0.j, 0.39269908+0.j, 0.78539816+0.j,
57079633+0.j, 0.39269908+0.j, 0.78539816+0.j, 1.57079633+0.j,
39269908+0.j, 0.78539816+0.j, 1.57079633+0.j, 0.39269908+0.j,
78539816+0.j, 1.57079633+0.j, 0.39269908+0.j, 0.78539816+0.j,
57079633+0.j, 0.39269908+0.j, 0.78539816+0.j, 1.57079633+0.j,
39269908+0.j, 0.78539816+0.j, 1.57079633+0.j, 0.39269908+0.j,
78539816+0.j, 1.57079633+0.j, 0.39269908+0.j, 0.78539816+0.j,
57079633+0.j, 0.39269908+0.j, 0.78539816+0.j, 1.57079633+0.j,
39269908+0.j, 0.78539816+0.j, 1.57079633+0.j, 1.04719755+0.j,
52359878+0.j, 0.34906585+0.j, 1.04719755+0.j, 0.52359878+0.j,
34906585+0.j, 1.04719755+0.j, 0.52359878+0.j, 0.34906585+0.j,
04719755+0.j, 0.52359878+0.j, 0.34906585+0.j, 1.04719755+0.j,
52359878+0.j, 0.34906585+0.j, 1.04719755+0.j, 0.52359878+0.j,
34906585+0.j, 1.04719755+0.j, 0.52359878+0.j, 0.34906585+0.j,
04719755+0.j, 0.52359878+0.j, 0.34906585+0.j, 1.04719755+0.j,
52359878+0.j, 0.34906585+0.j, 1.04719755+0.j, 0.52359878+0.j,
34906585+0.j, 1.04719755+0.j, 0.52359878+0.j, 0.34906585+0.j,
04719755+0.j, 0.52359878+0.j, 0.34906585+0.j, 1.04719755+0.j,
52359878+0.j, 0.34906585+0.j, 1.04719755+0.j, 0.52359878+0.j,
34906585+0.j, 1.04719755+0.j, 0.52359878+0.j, 0.34906585+0.j,
04719755+0.j, 0.52359878+0.j, 0.34906585+0.j, 1.04719755+0.j,
52359878+0.j, 0.34906585+0.j, 1.04719755+0.j, 0.52359878+0.j,
34906585+0.j, 1.04719755+0.j, 0.52359878+0.j, 0.34906585+0.j,
04719755+0.j, 0.52359878+0.j, 0.34906585+0.j, 1.04719755+0.j,
52359878+0.j, 0.34906585+0.j, 1.04719755+0.j, 0.52359878+0.j,
34906585+0.j, 1.04719755+0.j, 0.52359878+0.j, 0.34906585+0.j,
04719755+0.j, 0.52359878+0.j, 0.34906585+0.j, 1.04719755+0.j,
52359878+0.j, 0.34906585+0.j, 1.04719755+0.j, 0.52359878+0.j,
34906585+0.j, 1.04719755+0.j, 0.52359878+0.j, 0.34906585+0.j,
04719755+0.j, 0.52359878+0.j, 0.34906585+0.j, 1.04719755+0.j,
52359878+0.j, 0.34906585+0.j, 1.04719755+0.j, 0.52359878+0.j,
34906585+0.j, 1.04719755+0.j, 0.52359878+0.j, 0.34906585+0.j,
04719755+0.j, 0.52359878+0.j, 0.34906585+0.j, 1.04719755+0.j,
52359878+0.j, 0.34906585+0.j, 1.04719755+0.j, 0.52359878+0.j,
34906585+0.j, 1.04719755+0.j, 0.52359878+0.j, 0.34906585+0.j,
04719755+0.j, 0.52359878+0.j, 0.34906585+0.j, 1.04719755+0.j,
52359878+0.j, 0.34906585+0.j, 1.04719755+0.j, 0.52359878+0.j,
34906585+0.j, 1.04719755+0.j, 0.52359878+0.j, 0.34906585+0.j,
04719755+0.j, 0.52359878+0.j, 0.34906585+0.j, 1.04719755+0.j,
52359878+0.j, 0.34906585+0.j, 1.04719755+0.j, 0.52359878+0.j,
34906585+0.j, 1.04719755+0.j, 0.52359878+0.j, 0.34906585+0.j,
04719755+0.j, 0.52359878+0.j, 0.34906585+0.j, 1.04719755+0.j,
52359878+0.j, 0.34906585+0.j, 1.04719755+0.j, 0.52359878+0.j,
34906585+0.j, 1.04719755+0.j, 0.52359878+0.j, 0.34906585+0.j,
04719755+0.j, 0.52359878+0.j, 0.34906585+0.j, 1.04719755+0.j,
52359878+0.j, 0.34906585+0.j, 1.04719755+0.j, 0.52359878+0.j,
34906585+0.j])

observable = SparsePauliOp.from_sparse_list(
    [("Z", [i], 1 / num_qubits) for i in range(num_qubits)],
    num_qubits,
)
observable

SparsePauliOp(['IIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIZ', 'IIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIZI', 'IIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIZII', 'IIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIZIII', 'IIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIZIIII', 'IIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIZIIIII', 'IIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIZIIIIII', 'IIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIZIIIIIII', 'IIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIZIIIIIIII', 'IIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIZIIIIIIIII', 'IIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIZIIIIIIIIII', 'IIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIZIIIIIIIIIII', 'IIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIZIIIIIIIIIIII', 'IIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIZIIIIIIIIIIIII', 'IIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIZIIIIIIIIIIIIII', 'IIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIZIIIIIIIIIIIIIII', 'IIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIZIIIIIIIIIIIIIIII', 'IIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIZIIIIIIIIIIIIIIIII', 'IIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIZIIIIIIIIIIIIIIIIII', 'IIIIIIIIIIIIIIIIIIIIIIIIIIIIIIZIIIIIIIIIIIIIIIIIII', 'IIIIIIIIIIIIIIIIIIIIIIIIIIIIIZIIIIIIIIIIIIIIIIIIII', 'IIIIIIIIIIIIIIIIIIIIIIIIIIIIZIIIIIIIIIIIIIIIIIIIII', 'IIIIIIIIIIIIIIIIIIIIIIIIIIIZIIIIIIIIIIIIIIIIIIIIII', 'IIIIIIIIIIIIIIIIIIIIIIIIIIZIIIIIIIIIIIIIIIIIIIIIII', 'IIIIIIIIIIIIIIIIIIIIIIIIIZIIIIIIIIIIIIIIIIIIIIIIII', 'IIIIIIIIIIIIIIIIIIIIIIIIZIIIIIIIIIIIIIIIIIIIIIIIII', 'IIIIIIIIIIIIIIIIIIIIIIIZIIIIIIIIIIIIIIIIIIIIIIIIII', 'IIIIIIIIIIIIIIIIIIIIIIZIIIIIIIIIIIIIIIIIIIIIIIIIII', 'IIIIIIIIIIIIIIIIIIIIIZIIIIIIIIIIIIIIIIIIIIIIIIIIII', 'IIIIIIIIIIIIIIIIIIIIZIIIIIIIIIIIIIIIIIIIIIIIIIIIII', 'IIIIIIIIIIIIIIIIIIIZIIIIIIIIIIIIIIIIIIIIIIIIIIIIII', 'IIIIIIIIIIIIIIIIIIZIIIIIIIIIIIIIIIIIIIIIIIIIIIIIII', 'IIIIIIIIIIIIIIIIIZIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIII', 'IIIIIIIIIIIIIIIIZIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIII', 'IIIIIIIIIIIIIIIZIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIII', 'IIIIIIIIIIIIIIZIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIII', 'IIIIIIIIIIIIIZIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIII', 'IIIIIIIIIIIIZIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIII', 'IIIIIIIIIIIZIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIII', 'IIIIIIIIIIZIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIII', 'IIIIIIIIIZIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIII', 'IIIIIIIIZIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIII', 'IIIIIIIZIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIII', 'IIIIIIZIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIII', 'IIIIIZIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIII', 'IIIIZIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIII', 'IIIZIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIII', 'IIZIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIII', 'IZIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIII', 'ZIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIII'],
              coeffs=[0.02+0.j, 0.02+0.j, 0.02+0.j, 0.02+0.j, 0.02+0.j, 0.02+0.j, 0.02+0.j,
02+0.j, 0.02+0.j, 0.02+0.j, 0.02+0.j, 0.02+0.j, 0.02+0.j, 0.02+0.j,
02+0.j, 0.02+0.j, 0.02+0.j, 0.02+0.j, 0.02+0.j, 0.02+0.j, 0.02+0.j,
02+0.j, 0.02+0.j, 0.02+0.j, 0.02+0.j, 0.02+0.j, 0.02+0.j, 0.02+0.j,
02+0.j, 0.02+0.j, 0.02+0.j, 0.02+0.j, 0.02+0.j, 0.02+0.j, 0.02+0.j,
02+0.j, 0.02+0.j, 0.02+0.j, 0.02+0.j, 0.02+0.j, 0.02+0.j, 0.02+0.j,
02+0.j, 0.02+0.j, 0.02+0.j, 0.02+0.j, 0.02+0.j, 0.02+0.j, 0.02+0.j,
02+0.j])

Para sa scaled up na problema, isinaalang-alang natin ang oras ng evolusyon bilang $0.2$ na may $4$ na trotter step. Ang problema ay pinili upang ito ay lampas sa classical brute-force simulation, ngunit maaaring i-simulate sa pamamagitan ng tensor network method. Ito ay nagpapahintulot sa atin na i-verify ang kinalalabasan na nakuha sa pamamagitan ng backpropagation sa isang quantum computer gamit ang ideal na kinalabasan.

Ang ideal na expectation value para sa problemang ito, na nakuha sa pamamagitan ng tensor network simulation, ay $\simeq 0.89$ .

circuit = generate_time_evolution_circuit(
    hamiltonian,
    time=0.2,
    synthesis=LieTrotter(reps=4),
)
circuit.draw("mpl", style="iqp", fold=-1, scale=0.6)

Output of the previous code cell

Step 2: Optimize problem for quantum hardware execution

slices = slice_by_gate_types(circuit)
print(f"Separated the circuit into {len(slices)} slices.")

Separated the circuit into 36 slices.

Tinukoy natin ang max_error_per_slice bilang 0.005 tulad ng dati. Gayunpaman, dahil ang bilang ng mga slice para sa large-scale na problema ay mas mataas kaysa sa small scale na problema, ang pagpapahintulot ng 0.005 error sa bawat slice ay maaaring magreresulta ng malaking kabuuang backpropagation error. Maaari nating limitahan ito sa pamamagitan ng pagtukoy ng max_error_total na humahadlang sa kabuuang backpropagation error, at itinakda natin ang halaga nito sa 0.03 (na halos pareho sa small-scale na halimbawa).

Para sa large-scale na halimbawang ito, pinapayagan natin ang mas mataas na halaga para sa bilang ng commuting groups, at itinakda natin ito sa 15.

op_budget = OperatorBudget(max_qwc_groups=15)
truncation_error_budget = setup_budget(
    max_error_total=0.03, max_error_per_slice=0.005
)

Kunin muna natin ang backpropagated circuit at observable nang walang anumang truncation.

bp_obs, remaining_slices, metadata = backpropagate(
    observable, slices, operator_budget=op_budget
)
bp_circuit = combine_slices(remaining_slices)

print(f"Backpropagated {metadata.num_backpropagated_slices} slices.")
print(
    f"New observable has {len(bp_obs.paulis)} terms, which can be combined into {len(bp_obs.group_commuting(qubit_wise=True))} groups."
)
print(
    f"Note that backpropagating one more slice would result in {metadata.backpropagation_history[-1].num_paulis[0]} terms "
    f"across {metadata.backpropagation_history[-1].num_qwc_groups} groups."
)
print("The remaining circuit after backpropagation looks as follows:")
bp_circuit.draw("mpl", fold=-1, scale=0.6)

Backpropagated 7 slices.
New observable has 634 terms, which can be combined into 12 groups.
Note that backpropagating one more slice would result in 1246 terms across 27 groups.
The remaining circuit after backpropagation looks as follows:

Output of the previous code cell

Ngayon, pagpapahintulutan natin ang truncation, nakukuha natin:

bp_obs_trunc, remaining_slices_trunc, metadata = backpropagate(
    observable,
    slices,
    operator_budget=op_budget,
    truncation_error_budget=truncation_error_budget,
)

# Recombine the slices remaining after backpropagation
bp_circuit_trunc = combine_slices(
    remaining_slices_trunc, include_barriers=False
)

print(f"Backpropagated {metadata.num_backpropagated_slices} slices.")
print(
    f"New observable has {len(bp_obs_trunc.paulis)} terms, which can be combined into {len(bp_obs_trunc.group_commuting(qubit_wise=True))} groups.\n"
    f"After truncation, the error in our observable is bounded by {metadata.accumulated_error(0):.3e}"
)
print(
    f"Note that backpropagating one more slice would result in {metadata.backpropagation_history[-1].num_paulis[0]} terms "
    f"across {metadata.backpropagation_history[-1].num_qwc_groups} groups."
)
print("The remaining circuit after backpropagation looks as follows:")
bp_circuit_trunc.draw("mpl", fold=-1, scale=0.6)

Backpropagated 10 slices.
New observable has 646 terms, which can be combined into 14 groups.
After truncation, the error in our observable is bounded by 2.998e-02
Note that backpropagating one more slice would result in 1226 terms across 29 groups.
The remaining circuit after backpropagation looks as follows:

Output of the previous code cell

Napapansin natin na ang pagpapahintulot sa truncation ay humahantong sa backpropagation ng tatlong pang slice. Maaari nating i-verify ang 2-qubit depth ng orihinal na circuit, ang backpropagated circuit, at ang backpropagated circuit na may truncation pagkatapos ng transpilation.

# Transpile original experiment
circuit_isa = pm.run(circuit)
observable_isa = observable.apply_layout(circuit_isa.layout)

# Transpile the backpropagated experiment
bp_circuit_isa = pm.run(bp_circuit)
bp_obs_isa = bp_obs_trunc.apply_layout(bp_circuit_isa.layout)

# Transpile the backpropagated experiment with truncated observable terms
bp_circuit_trunc_isa = pm.run(bp_circuit_trunc)
bp_obs_trunc_isa = bp_obs_trunc.apply_layout(bp_circuit_trunc_isa.layout)

print(
    f"2-qubit depth of original circuit: {circuit_isa.depth(lambda x:x.operation.num_qubits==2)}"
)
print(
    f"2-qubit depth of backpropagated circuit: {bp_circuit_isa.depth(lambda x:x.operation.num_qubits==2)}"
)
print(
    f"2-qubit depth of backpropagated circuit with truncation: {bp_circuit_trunc_isa.depth(lambda x:x.operation.num_qubits==2)}"
)

2-qubit depth of original circuit: 48
2-qubit depth of backpropagated circuit: 40
2-qubit depth of backpropagated circuit with truncation: 36

Step 3: Execute using Qiskit primitives

pubs = [
    (circuit_isa, observable_isa),
    (bp_circuit_isa, bp_obs_isa),
    (bp_circuit_trunc_isa, bp_obs_trunc_isa),
]

options = EstimatorOptions()
options.default_precision = 0.01
options.resilience_level = 2
options.resilience.zne.noise_factors = [1, 1.2, 1.4]
options.resilience.zne.extrapolator = ["linear"]

estimator = EstimatorV2(mode=backend, options=options)

job = estimator.run(pubs)

Step 4: Post-process and return result to desired classical format

result_no_bp = job.result()[0].data.evs.item()
result_bp = job.result()[1].data.evs.item()
result_bp_trunc = job.result()[2].data.evs.item()

print(f"Expectation value without backpropagation: {result_no_bp}")
print(f"Backpropagated expectation value: {result_bp}")
print(f"Backpropagated expectation value with truncation: {result_bp_trunc}")

Expectation value without backpropagation: 0.7887194658035515
Backpropagated expectation value: 0.9532818300978584
Backpropagated expectation value with truncation: 0.8913400398926913

methods = [
    "No backpropagation",
    "Backpropagation",
    "Backpropagation w/ truncation",
]
values = [result_no_bp, result_bp, result_bp_trunc]

ax = plt.gca()
plt.bar(methods, values, color="#a56eff", width=0.4, edgecolor="#8a3ffc")
plt.axhline(0.89)
ax.set_ylim([0.6, 0.98])
plt.text(0.2, 0.895, "Exact result")
ax.set_ylabel(r"$M_Z$", fontsize=12)

Text(0, 0.5, '$M_Z$')

Output of the previous code cell

Tutorial survey

Mangyaring sagutan ang maikling survey na ito upang magbigay ng feedback sa tutorial na ito. Ang inyong mga pananaw ay makakatulong sa amin na mapabuti ang aming mga alok na content at user experience.

Link to survey

Background​

Requirements​

Setup​

Part I: Small-scale Heisenberg spin chain​

Step 1: Map classical inputs to a quantum problem​

Map the time-evolution of a quantum Heisenberg model to a quantum experiment.​

Step 2: Optimize problem for quantum hardware execution​

Create circuit slices to backpropagate​

Constrain how large the operator may grow during backpropagation​

Backpropagate slices from the circuit​

Now that we have our reduced ansatz and expanded observables, we can transpile our experiments to the backend.​

Step 3: Execute using Qiskit primitives​

Calculate expectation value​

Step 4: Post-process and return result to desired classical format​

Part B: Scale it up!​

Step 1: Map classical inputs to a quantum problem​

Step 2: Optimize problem for quantum hardware execution​

Step 3: Execute using Qiskit primitives​

Step 4: Post-process and return result to desired classical format​

Tutorial survey​

Background

Requirements

Setup

Part I: Small-scale Heisenberg spin chain

Step 1: Map classical inputs to a quantum problem

Map the time-evolution of a quantum Heisenberg model to a quantum experiment.

Step 2: Optimize problem for quantum hardware execution

Create circuit slices to backpropagate

Constrain how large the operator may grow during backpropagation

Backpropagate slices from the circuit

Now that we have our reduced ansatz and expanded observables, we can transpile our experiments to the backend.

Step 3: Execute using Qiskit primitives

Calculate expectation value

Step 4: Post-process and return result to desired classical format

Part B: Scale it up!

Step 1: Map classical inputs to a quantum problem

Step 2: Optimize problem for quantum hardware execution

Step 3: Execute using Qiskit primitives

Step 4: Post-process and return result to desired classical format

Tutorial survey