Beyond-classical computation in quantum simulation

TL;DR

The paper demonstrates beyond-classical computation by using superconducting quantum annealers to sample Schrödinger dynamics in disordered spin systems, revealing area-law entanglement and universal quantum critical scaling. It compares QPU results with leading classical tensor-network and neural-network methods, finding MPS can match QA quality for some regimes but scales unfavorably with system size, while PEPS and NQS struggle for slower quenches. The work combines experimental QA across multiple topologies with detailed classical simulations (TDVP-MPS, PEPS, NQS) and finite-size scaling analyses to argue that large-scale quantum simulation of nonequilibrium dynamics can surpass classical capabilities, suggesting potential quantum advantages for optimization and AI. It also discusses spatial decomposition strategies and their limits, highlighting the significance of entanglement area laws and correlation lengths in determining classical simulability and guiding future quantum-simulation efforts.

Abstract

Quantum computers hold the promise of solving certain problems that lie beyond the reach of conventional computers. However, establishing this capability, especially for impactful and meaningful problems, remains a central challenge. Here, we show that superconducting quantum annealing processors can rapidly generate samples in close agreement with solutions of the Schrödinger equation. We demonstrate area-law scaling of entanglement in the model quench dynamics of two-, three-, and infinite-dimensional spin glasses, supporting the observed stretched-exponential scaling of effort for matrix-product-state approaches. We show that several leading approximate methods based on tensor networks and neural networks cannot achieve the same accuracy as the quantum annealer within a reasonable time frame. Thus, quantum annealers can answer questions of practical importance that may remain out of reach for classical computation.

Figures (43)

• Figure 1: Sampling post-quench states in Ising spin glasses. Random inputs are generated in topologies of varying dimension. For each input, we consider the task of sampling from the distribution of states following a quantum quench, i.e., rapid change of transverse field $\Gamma(t/t_a)$ and longitudinal field $\mathcal{J}(t/t_a)$ within time $t_a$. This task is performed using quantum annealing and classical methods based on tensor networks (MPS and PEPS) and neural networks (NQS).
• Figure 2: QPU and MPS on quenched 2D spin glasses. ( A-- C) Data for a typical realization of a $6{\times} 6$ cylindrical spin glass. ( A) Squared order parameter (top) and residual energy per spin (bottom) data exhibit close agreement between QPU and the ground truth across nearly three decades of $t_a$. In the fast-quench limit $t_a\rightarrow 0$, correlations and therefore $\langle q^2\rangle$ are small; in the slow-quench limit $t_a\rightarrow \infty$, the system evolves adiabatically and produces classical ground states with vanishing residual energy. Approaching this limit, the problem crosses over into the much easier problem of classical ground-state sampling. ( B) Spin-spin correlations $c_{ij}$ compared against ground-truth values $\tilde{c}_{ij}$ for $t_a{=}7ns$. Left, QPU. Right, MPS at a bond dimension producing output of similar quality. ( C) Comparison of individual state probabilities for the same simulations as B; classical fidelities $\mathcal{F}$ estimated from $10^6$ samples are comparable. ( D-- E) Ensembles of $20$$L{\times} L$ instances (disorder realizations). ( D) Median correlation error $\epsilon_c$ remains roughly constant across system sizes for QPU (left), but matching the QPU error level in MPS requires an increasing QPU-equivalent bond dimension $\chi_{\text{Q}}$. ( E) $\chi_{\text{Q}}$ exhibits exponential growth in $L$ for two quench rates. For $t_a{=}20ns$, the superior processor, ADV2, shows lower errors, and therefore steeper scaling of $\chi_{\text{Q}}$. All error bars represent 95% bootstrap confidence intervals of the ensemble median.
• Figure 3: Breakdown of PEPS in 8${\times}$8 systems. We show correlation error as a function of PEPS bond dimension $D$ for three annealing times. The best-performing PEPS variant, of many evaluatedSM, was "plqt-NN+". Computation time increases sharply with bond dimension $D$. PEPS easily beats QPU error at $t_a{=}2ns$, but degrades for slower quenches. At $20ns$, all PEPS experiments have higher error than QPU, and PEPS appears to be neighborhood-limited: increasing $D$ does not significantly reduce error.
• Figure 4: Entanglement and QPU-equivalent bond dimension. ( A) QPU median correlation error for the four topologies as color coded in the right panels. ( B) For all four problem sets, the bond dimension required by MPS to match QPU simulation quality, $\chi_Q$, exhibits exponential dependence on the bipartition area (asymptotically $N^{(d-1)/d}$, see inset example), consistent with the area-law scaling of entanglement entropy (two-point fit is trivial for diamond at $t_a{=}7ns$). ( C) Median $\log \chi_Q$ against maximum entanglement entropy $S_\text{max}(\chi_Q)$ for MPS with bond dimension $\chi_Q$. All data points for the four topologies and two annealing times fall roughly on the same line. Inset: $S_\text{max}(\chi_Q)$ versus ground-truth $\tilde{S}_\text{max}$ estimates for each input.
• Figure 5: Dynamical scaling and resource estimates for large-scale quantum simulation. ( A) We compare $\langle q^2\rangle$ in QPU output and the ground truth for 20 instances of each topology at $t_a=7ns$ and $20ns$, establishing close agreement (median relative error $<1\%$) for the largest MPS ground truths. ( B) Using a range of system sizes that we cannot simulate classically, we collapse Binder cumulants $U$ using a dynamic finite-size scaling ansatz, yielding best-fit KZ exponents $\mu$. Curve in collapsed data indicates crossover from power-law (KZ) to system-spanning correlations. Data points are computed from ensembles of 300 to 20,400 simulated spin-glass realizations (collapsed sizes $N$, indicated by different shades and markers, are 16--324 for square, 54--432 for cubic, 72--567 for diamond, and 32--96 for biclique). Inset: Estimates of $U$ using simulations shown in A for $t_a=7ns$ and $20ns$ (dark and light stars respectively, also in main plot) show close agreement with the ground truth at simulable sizes, bridging the gap between the classically-simulable and large-scale regimes. ( C) Extracted KZ exponents for one-, two-, three-, and infinite-dimensional systems agree with previous estimates for the corresponding QPT universality classes. ( D) Area-law scaling of $\chi_Q$ (Fig. \ref{['fig:4']}B) allows extrapolation of required classical time based on $N\chi_Q^3$ scaling of MPS methods; all other methods failed at smaller sizes. Filled markers indicate classically-simulated 20-instance ensembles; larger problems are extrapolations.