FFT-Accelerated Auxiliary Variable MCMC for Fermionic Lattice Models: A Determinant-Free Approach with $O(N\log N)$ Complexity

• URL: http://arxiv.org/abs/2510.13866v1
• Date: Mon, 13 Oct 2025 07:57:21 GMT
• Title: FFT-Accelerated Auxiliary Variable MCMC for Fermionic Lattice Models: A Determinant-Free Approach with $O(N\log N)$ Complexity
• Abstract summary: We introduce a Markov Chain Monte Carlo (MCMC) algorithm that dramatically accelerates the simulation of quantum many-body systems.<n>We validate our algorithm on benchmark quantum physics problems, accurately reproducing known theoretical results.<n>Our work provides a powerful tool for large-scale probabilistic inference and opens avenues for physics-inspired generative models.
• Score: 52.3171766248012
• License: http://creativecommons.org/licenses/by/4.0/
• Abstract: We introduce a Markov Chain Monte Carlo (MCMC) algorithm that dramatically accelerates the simulation of quantum many-body systems, a grand challenge in computational science. State-of-the-art methods for these problems are severely limited by $O(N^3)$ computational complexity. Our method avoids this bottleneck, achieving near-linear $O(N \log N)$ scaling per sweep. Our approach samples a joint probability measure over two coupled variable sets: (1) particle trajectories of the fundamental fermions, and (2) auxiliary variables that decouple fermion interactions. The key innovation is a novel transition kernel for particle trajectories formulated in the Fourier domain, revealing the transition probability as a convolution that enables massive acceleration via the Fast Fourier Transform (FFT). The auxiliary variables admit closed-form, factorized conditional distributions, enabling efficient exact Gibbs sampling update. We validate our algorithm on benchmark quantum physics problems, accurately reproducing known theoretical results and matching traditional $O(N^3)$ algorithms on $32\times 32$ lattice simulations at a fraction of the wall-clock time, empirically demonstrating $N \log N$ scaling. By reformulating a long-standing physics simulation problem in machine learning language, our work provides a powerful tool for large-scale probabilistic inference and opens avenues for physics-inspired generative models.

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