Backpropagating Hybrid Monte Carlo algorithm for fast Lefschetz thimble calculations

TL;DR

This paper introduces a fast Hybrid Monte Carlo algorithm that leverages backpropagation of forces to efficiently evaluate complex integrals using Lefschetz thimbles, improving computational cost and enabling better saddle point identification.

Contribution

The paper proposes a novel backpropagation-based Hybrid Monte Carlo method that reduces computational costs and extends to flow time integration for complex integral evaluation.

Findings

Reduces computational cost by a factor of the system size.

Enables efficient identification of all dominant saddle points.

Successfully applied to real-time wave function evolution.

Abstract

The Picard-Lefschetz theory has been attracting much attention as a tool to evaluate a multi-variable integral with a complex weight, which appears in various important problems in theoretical physics. The idea is to deform the integration contour based on Cauchy's theorem using the so-called gradient flow equation. In this paper, we propose a fast Hybrid Monte Carlo algorithm for evaluating the integral, where we "backpropagate" the force of the fictitious Hamilton dynamics on the deformed contour to that on the original contour, thereby reducing the required computational cost by a factor of the system size. Our algorithm can be readily extended to the case in which one integrates over the flow time in order to solve not only the sign problem but also the ergodicity problem that occurs when there are more than one thimbles contributing to the integral. This enables, in particular, efficient identification of all the dominant saddle points and the associated thimbles. We test our algorithm by calculating the real-time evolution of the wave function using the path integral formalism.