Neural Importance Resampling: A Practical Sampling Strategy for Neural Quantum States

TL;DR

Neural Importance Resampling (NIR) addresses sampling bottlenecks in neural quantum states by decoupling the sampling process from the NQS and introducing a dedicated autoregressive proposal network learned through forward KL objectives. Sampling is performed via importance resampling, with an adaptive retraining loop guided by effective sample size to maintain high overlap between the target and proposal distributions. Empirical results on the 2D transverse-field Ising model and the J1-J2 Heisenberg model show NIR outperforms Metropolis-Hastings in challenging regimes and yields energies competitive with DMRG, while enabling robust multi-state NQS training. Overall, NIR offers a practical, scalable alternative for variational NQS that preserves flexibility in network design and facilitates multi-state determinant constructions for larger quantum systems.

Abstract

Neural quantum states (NQS) have emerged as powerful tools for simulating many-body quantum systems, but their practical use is often hindered by limitations of current sampling techniques. Markov chain Monte Carlo (MCMC) methods suffer from slow mixing and require manual tuning, while autoregressive NQS impose restrictive architectural constraints that complicate the enforcement of symmetries and the construction of determinant-based multi-state wave functions. In this work, we introduce Neural Importance Resampling (NIR), a new sampling algorithm that combines importance resampling with a separately trained autoregressive proposal network. This approach enables efficient and unbiased sampling without constraining the NQS architecture. We demonstrate that NIR supports stable and scalable training, including for multi-state NQS, and mitigates issues faced by MCMC and autoregressive approaches. Numerical experiments on the 2D transverse-field Ising and $J_1$-$J_2$ Heisenberg models show that NIR outperforms MCMC in challenging regimes and yields results competitive with state of the art methods. Our results establish NIR as a robust alternative for sampling in variational NQS algorithms.

Figures (4)

• Figure 1: Scheme for the proposed NQS training algorithm.
• Figure 2: Left: Dependence of the trace of the final variational energy matrix on transverse field strength $g$ for three lattice sizes ($4\times4$, $6\times6$, $8\times8$). For every combination, two data points are shown, differing only in the sampler used. Right: Example of the error evolution in the sum of the three lowest energies during a training run of NQS using NIR (blue) and MCMC (red) for TFI on an $8 \times 8$ lattice and $g = 0.1$. The transparent curves show the values obtained at each iteration from the batch used in the optimization step, and the solid lines represent their moving averages.
• Figure 3: Evolution of the JSD between empirical configuration probabilities from the MCMC and NIR samplers and the exact probabilities during NQS optimization for a $2\times3$ lattice and $g=0.01$.
• Figure 4: Left: Dependence of the three lowest energy levels (relative to the ground state found with NQS) on $g$ for TFI on an $8\times 8$ lattice. Computed using NQS (blue) and DMRG (red). Right: Dependence of the difference between energies obtained with NQS and DMRG on $g$.