Title: Mitigating the Quantum Monte Carlo Sign Problem via Neural Autoregressive Control Variates

Abstract:

To address the sign problem inherent in quantum Monte Carlo simulations, we introduce a methodology that employs a pair of trained autoregressive models to generate zero-mean control variates. These distinct autoregressive networks are restricted to mutually exclusive positive- and negative-sign sectors, ensuring that each model is precisely normalized within its respective domain. Consequently, the subtraction of these models yields a structural zero-mean value, functioning as an unbiased auxiliary observable. The efficacy of variance reduction is governed by the correlation between this auxiliary observable and the sign estimator.

We integrated this approach into the stochastic series expansion framework, for which we developed an incremental loop-topology update to accommodate frustrated lattices. To ensure sign-ergodic sampling on non-bipartite structures, we utilized a twist channel, identified as the sole sign-changing mechanism in such contexts. The control variates were realized as autoregressive transformers, featuring an end-of-sequence parity mask to guarantee exact sign-sector classification. Additionally, topological features, including cumulative frustration parity and incremental loop-count changes, were embedded into the model.

We validated the method on the triangular-lattice Heisenberg antiferromagnet within the small-$N$ limit. The results demonstrate that the control variate decreases the standard error of the average sign by as much as tenfold and reduces the standard error of the energy estimator by a factor of three to five. Notably, the technique maintains its effectiveness even when the average sign falls below $10^{-3}$. This study establishes the theoretical framework and offers a proof-of-principle demonstration that autoregressive control variates can successfully alleviate the sign problem. Future research will focus on scaling the method to larger systems using architectures informed by physical principles.

Source: arXiv Generated at: 2026-06-04 00:00:00 UTC