Improving efficiency of the path optimization method for a gauge theory

TL;DR

This paper explores improving the efficiency of path optimization in lattice gauge theories by using gauge-covariant neural networks and Jacobian approximations, significantly reducing computational costs while maintaining effectiveness.

Contribution

It introduces a gauge-covariant neural network with smearing for better path optimization and demonstrates a Jacobian approximation that drastically cuts computational costs.

Findings

Gauge-covariant neural network improves phase factor.

Jacobian approximation reduces computational complexity.

Path optimization remains effective with Jacobian approximation.

Abstract

We investigate efficiency of a gauge-covariant neural network and an approximation of the Jacobian in optimizing the complexified integration path toward evading the sign problem in lattice field theories. For the construction of the complexified integration path, we employ the path optimization method. The $2$-dimensional $\text{U}(1)$ gauge theory with the complex gauge coupling constant is used as a laboratory to evaluate the efficiency. It is found that the gauge-covariant neural network, which is composed of the Stout-like smearing, can enhance the average phase factor, as the gauge-invariant input does. For the approximation of the Jacobian, we test the most drastic case in which we perfectly drop the Jacobian during the learning process. It reduces the numerical cost of the Jacobian calculation from ${\cal O}(N^3)$ to ${\cal O}(1)$, where $N$ means the number of degrees of freedom of the theory. The path optimization using this Jacobian approximation still enhances the average phase factor at expense of a slight increase of the statistical error.