Path optimization method for the sign problem

TL;DR

This paper introduces a path optimization method (POM) to mitigate the sign problem in Monte Carlo simulations of complex actions, demonstrating its effectiveness in a toy model and exploring neural network-based path optimization.

Contribution

The paper proposes a novel path optimization approach that avoids singular points and improves sign problem mitigation, including the potential use of neural networks for optimization.

Findings

POM successfully avoids the sign problem in a toy model.

Optimizing the path enhances the average phase factor.

Neural networks may be utilized for path optimization.

Abstract

We propose a path optimization method (POM) to evade the sign problem in the Monte-Carlo calculations for complex actions. Among many approaches to the sign problem, the Lefschetz-thimble path-integral method and the complex Langevin method are promising and extensively discussed. In these methods, real field variables are complexified and the integration manifold is determined by the flow equations or stochastically sampled. When we have singular points of the action or multiple critical points near the original integral surface, however, we have a risk to encounter the residual and global sign problems or the singular drift term problem. One of the ways to avoid the singular points is to optimize the integration path which is designed not to hit the singular points of the Boltzmann weight. By specifying the one-dimensional integration-path as $z=t+if(t) (f\in \mathbb{R})$ and by optimizing $f(t)$ to enhance the average phase factor, we demonstrate that we can avoid the sign problem in a one-variable toy model for which the complex Langevin method is found to fail. In this proceedings, we propose POM and discuss how we can avoid the sign problem in a toy model. We also discuss the possibility to utilize the neural network to optimize the path.