Reducing the Sign Problem with Line Integrals

TL;DR

This paper introduces a new method using line integrals along paths of changing imaginary action to significantly mitigate the sign problem, enabling standard Monte Carlo sampling in complex quantum systems.

Contribution

It proposes a novel approach to reduce the sign problem via line integrals, providing a practical alternative to complex analysis methods like Lefschetz-thimbles.

Findings

Effective reduction of sign problem in 1D quantum anharmonic oscillator

Enables Monte Carlo sampling in previously intractable cases

Provides a differential equation framework for line integrals

Abstract

We present a novel strategy to strongly reduce the severity of the sign problem, using line integrals along paths of changing imaginary action. Highly oscillating regions along these paths cancel out, decreasing their contributions. As a result, sampling with standard Monte-Carlo techniques becomes possible in cases that otherwise require methods taking advantage of complex analysis, such as Lefschetz-thimbles or Complex Langevin. We lay out how to write down an ordinary differential equation for the line integrals. As an example of its usage, we apply the results to a 1d quantum mechanical anharmonic oscillator with a $x^4$ potential in real time, finite temperature.