Predicting sampling advantage of stochastic Ising Machines for Quantum Simulations

TL;DR

The paper addresses whether stochastic Ising machines (sIMs) can provide a computational advantage for simulating quantum magnets via Neural Network Quantum States (NQS) by mapping the NQS wavefunction to an Ising model and comparing sampling on a software-emulated sIM against Metropolis-Hastings sampling. It introduces a hardware-agnostic metric based on autocorrelation times to predict iso-accuracy sampling requirements, and uses pre-trained RBM models for the 2D Heisenberg model to quantify potential speedups. The results indicate that for alpha=2 networks, sIM-based sampling can outperform MH by 2-4 orders of magnitude, while larger networks encounter higher autocorrelation due to energy barriers; sparse models may mitigate this. The study provides a practical framework for predicting sIM advantages without hardware deployment and highlights pathways toward scalable, energy-efficient quantum simulations.

Abstract

Stochastic Ising machines, sIMs, are highly promising accelerators for optimization and sampling of computational problems that can be formulated as an Ising model. Here we investigate the computational advantage of sIM for simulations of quantum magnets with neural-network quantum states (NQS), in which the quantum many-body wave function is mapped onto an Ising model. We study the sampling performance of sIM for NQS by comparing sampling on a software-emulated sIM with standard Metropolis-Hastings sampling for NQS. We quantify the sampling efficiency by the number of steps required to reach iso-accurate stochastic estimation of the variational energy and show that this is entirely determined by the autocorrelation time of the sampling. This enables predications of sampling advantage without direct deployment on hardware. For the quantum Heisenberg models studied and experimental results on the runtime of sIMs, we project a possible speed-up of 100 to 10000, suggesting great opportunities for studying complex quantum systems at larger scales.

Figures (10)

• Figure 1: (a) Restricted Boltzmann Machine with $n$ visible spins and $M$ hidden spins. The visible spins (red) are fully connected to the hidden spins (blue), but there are no intralayer connections. The connections correspond to non-zero weights in the weight matrix $J_{ij}$. (b) Setup of the sampling methods. The MH sampling on the left performs of $N_\mathrm{MH}$ MH sweeps each consisting of $n$ antiparallel spin flips of the visible spins. The stochastic Ising Machine (sIM) on the right performs $N_\mathrm{Ising}$ sIM sweeps corresponding to updating all hidden and then all visible spins.
• Figure 2: The relative error, $\varepsilon_{\mathrm{rel}}(N)$ as a function of $N$, which corresponds to the number of sIM sweeps or MH sweeps. In (a) and (b) the scaling of the relative error is shown for respectively $n_\mathrm{spins} = 36$ and $n_\mathrm{spins} = 484$ spins at $\alpha = 2$. The relative error for sIM and MH sampling is calculated as an average value over $10$ runs. The MH fit is a fit of the MH sampling results. The fit is then scaled by the ratio of the autocorrelation time of sIM and MH sampling which should again overlap with the sIM sampling results.
• Figure 3: Ratio of autocorrelation time $\tau_\mathrm{sIM} / \tau_{MH} = N_\mathrm{sIM}/ N_{MH}$ for different models. Each point is the average of $5$ pre-trained models with the same $\alpha$ and $n_\mathrm{spins} = L^2$.
• Figure 4: Performance comparison for generating the same amount of independent samples as one MH-sweep across five methods as a function of $n_\mathrm{spins} = L^2$ for $\alpha = 2$. The baseline, MH (CPU) obtained using UltraFast fabiani_ultrafastcode_, is a traditional NQS implementation with MH sampling. Furthermore, there are four sIM implementations: CPU-based implementation, FPGA running at 70 MHz patel_isingmodeloptimization_2020, conservative projection based on gallo_64core_2023, and optimistic projection based on sutton_autonomous_2020.
• Figure 5: The relative error, Eq. \ref{['sup:eq:erel:compnqsqmc']}, as a function of the side length $L$ of 2D antiferromagnetic Heisenberg model. The QMC ground state energies are taken from sandvik_finitesizescalinggroundstate_1997 for comparison. Standard refers to the ansatz used in carleo_solving_2017fabiani_investigating_2019. The model id's, $0$ through $4$, correspond to the 5 pre-trained models.