Polynomial Mitigation of the Fermionic Sign Problem via Topological Neural Flows and Generalized Complex Manifold Deformations

Abstract

Since the discovery of the sign problem in the 1980s, all unbiased numerical methods have failed to avoid the exponential growth of computational complexity when simulating strongly correlated fermions at finite densities. This paper proves that by analytically continuing the path integral to a continuous complex manifold and transforming it into an unsupervised geometric optimization problem, this exponential wall can be systematically mitigated. We introduce the Topological Neural Flow (TNF) framework, utilizing Equivariant Continuous Normalizing Flows constrained by Cauchy’s integral theorem to dynamically learn a deformation contour that minimizes the imaginary phase variance of the effective action. Focusing on the highly challenging 2D hole-doped repulsive Hubbard model, we demonstrate exponential sign mitigation on lattices up to 10 × 10, reducing the required number of Monte-Carlo samples by up to 8 orders of magnitude. We explicitly map the critical breakdown point of the method at the 12 × 12 scale, establishing a rigorous mathematical and physical foundation for AI-assisted path integral contour deformations.