Benchmarking quantum simulation with neutron-scattering experiments

Abstract

A central goal of quantum computation is the realistic simulation of quantum materials. Although quantum processors have advanced rapidly in scale and fidelity, it has remained unclear whether pre-fault-tolerant devices can perform quantitatively reliable material simulations within their limited gate budgets. Here, we demonstrate that a superconducting quantum processor operating on up to 50 qubits can already produce meaningful, quantitative comparisons with inelastic neutron-scattering measurements of KCuF$_3$, a canonical realization of a gapless Luttinger liquid system with a strongly correlated ground state and a spectrum of emergent spinons. The quantum simulation is enabled by a quantum-classical workflow for computing dynamical structure factors (DSFs). The resulting spectra are benchmarked against experimental measurements using multiple metrics, highlighting the impact of circuit depth and circuit fidelity on simulation accuracy. Finally, we extend our simulations to 1D XXZ Heisenberg model with next-nearest neighbor interactions and a strong anisotropy, producing a gapped excitation spectrum, which could be used to describe the CsCoX$_3$ compounds above the Néel temperature. Our results establish a framework for computing DSFs for quantum materials in classically challenging regimes of strong entanglement and long-range interactions, enabling quantum simulations that are directly testable against laboratory measurements.

Figures (17)

• Figure 1: Quantum Simulation of the Inelastic Neutron Scattering (INS) Spectra: We use spin Hamiltonians to model the materials of interest and compute their retarded Green’s functions. The ground state is prepared using a variational ansatz, applied to an appropriate initial state yu2023simulating. Next, we perturb the $j$-th qubit with a rotation gate $U^\beta$ as defined in equation \ref{['perturb']}, evolve the system in time, and finally measure $\sigma^\alpha_i$ to obtain the retarded Green’s function (RGF) $G_{\alpha,\beta}^R(i,j,t)$. Fourier transforming the RGF yields the dynamical structure factor (DSF) in energy-momentum space, a quantity that can be directly compared with the INS spectrum. Crystal structure of (B) KCuF$_3$ and CsCoX$_3$. KCuF$_3$ is well described by a nearest neighbor XXZ model at the isotropic point, whereas CsCoX$_3$ requires the inclusion of next-nearest-neighbor interactions for realistic modelling.
• Figure 2: Inelastic neutron scattering (INS) spectrum for XX model and KCuF$_3$. (A) Illustration of the spread of fractionalized excitation (spinon) after interacting with a neutron. Comparison of the spatio-temporal retarded Green's function and DSF obtained by 50-qubit MPS simulation and quantum simulation for (B,C) XX model and (D,E) KCuF$_3$. The experimental data (first figure in (D)) was replotted from Ref. scheie2022quantum, and all spectral intensities are normalized. Both MPS and quantum simulation results display pixelated features arising from finite-size and finite-time effects, while the latter also shows smearing due to noise. The spread of the RGF illustrates the light-cone property of the measured observable. We used a time-step length $\Delta t = 0.6$ for both MPS simulation and quantum experiments, with 30 total time steps for the XX model and 20 for KCuF$_3$. Benchmarking experimental, MPS, and quantum results of KCuF$_3$ with line-scan comparisons at (F) $q=\pi$ and (G) $q = \pi/2$, (H) quantum Fisher information, and (I) spin–spin correlations.
• Figure 3: Analysis of the spectrum with finite-size effects, finite-time effects, and depolarization errors. (A) MPS simulation of finite-size effects (first row), finite-time effects (second row), and noisy circuit-MPS simulation of noisy results with depolarization errors (last row). The baseline setup consists of 50 qubits, 20 time steps, and a fixed time step length of 0.6; in each row, we vary either the system size or the number of time steps. The noisy circuit-MPS results are simulated using a depolarization model applied only to two-qubit gates in the light-cone region of the quantum circuit. (B) The cumulative distributions of gate errors for 50-qubit layout on $ibm\_kingston$ and $ibm\_boston$. The median error rates of single-qubit gates are $2.130 \times 10^{-4}$ and $1.449 \times 10^{-4}$; those of two-qubit gates are $1.877 \times 10^{-3}$ and $1.080 \times 10^{-3}$; and those of readout are $7.876 \times 10^{-3}$ and $5.123\times 10^{-3}$. Benchmarking MPS, noisy circuit-MPS, and quantum results for KCuF$_3$ using (C) MSE, (D) SSIM, and (E) nQFI. The markers correspond to the cases considered in (A).
• Figure 4: Inelastic neutron scattering (INS) spectrum for two-soliton continuum and CsCoX$_3$ model. Comparison of the spatio-temporal retarded Green’s functions and DSF obtained from 50-qubit MPS and quantum simulation with (A,C) only nearest-neighbor (NN) interaction and (B,D) additional NNN interaction. With only NN interactions, the spectrum exhibits a bow-tie structure consistent with the theoretical two-soliton continuum. Upon including NNN interactions, this bow-tie feature breaks down and evolves into a lower-band dispersion. For both the MPS simulation and the quantum experiments, we used a time-step length $\Delta t = 0.8$, and 30 first-order trotter steps. (E) Comparison of various metrics for MPS and quantum spectra.
• Figure S1: Quantum circuit for measuring the dynamical structural factor. A Quantum circuit for XX model and KCuF$_3$, where $U$ refers to second-order trotterization. B Quantum circuit for Ising-like XXZ model with next-nearest-neighbor interactions, where the initial state is the Néel state instead of the singlet product state.