Algorithmic aspects of gauged Gaussian fermionic projected entangled pair states

TL;DR

This paper explores the algorithmic performance of Monte Carlo methods using gauged Gaussian fermionic PEPS for lattice gauge theories, aiming to improve simulations of complex quantum systems beyond traditional approaches.

Contribution

It investigates the numerical behavior of non-action-based Monte Carlo within GGFPEPS, identifying optimal update sizes and effects of gauge fixing and translation invariance on convergence.

Findings

Optimal update size for GGFPEPS-based MC identified

Gauge fixing generally slows convergence

Not exploiting translation-invariance can improve error convergence

Abstract

Lattice gauge theories (LGTs) provide a powerful framework for studying non-perturbative phenomena in gauge theories. However, conventional approaches such as Monte Carlo (MC) simulations in imaginary time are limited, as they do not allow real time evolution and suffer from a sign problem in many important cases. Using Gauged Gaussian fermionic projected entangled pair states (GGFPEPS) as a variational ground state ansatz offers an alternative for studying LGTs through a sign-problem-free variational MC. As this method is extended to larger and more complex systems, understanding its numerical behavior becomes essential. While conventional action based MC has been extensively studied, the performance and characteristics of non-action-based MC within the GGFPEPS framework are far less explored. In this work, we investigate these algorithmic aspects, identifying an optimal update size for GGFPEPS-based MC simulations for $\mathbb{Z}_2$ in $2+1$ dimensions. We show that gauge fixing generally slows convergence, and demonstrate that not exploiting the translation-invariance can, in some cases, improve the computational time scaling of error convergence. We expect that these improvements will allow advancing the simulation to larger and more complex systems.