Backpropagating Hybrid Monte Carlo algorithm for fast Lefschetz thimble calculations

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 use "backpropagation" in calculating the force in the fictitious Hamilton dynamics, thus 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.

Numerical sign problem and the tempered Lefschetz thimble method

“…10The latter picture was further studied in Ref. [27]. In this picture, however, the Jacobian determinant det 𝐽 (𝑥) is treated as part of observable, which is exponentially large and has no guarantee to have a significant overlap with the weight 𝑒 −Re 𝑆 (𝑧 𝑡 ( 𝑥 ) ) .…”

Numerical sign problem and the tempered Lefschetz thimble method

“…The latter picture was further studied in Ref. [27]. In this picture, however, the Jacobian determinant det ( ) is treated as part of observable, which is exponentially large and has no guarantee to have a significant overlap with the weight −Re ( ( )) .…”

Comment on the subtlety of defining real-time path integral in lattice gauge theories

“…Real-time path integral [1] has recently been revisited both analytically [2][3][4] and numerically [5][6][7][8][9][10][11][12] for the interest of real-time dynamics in quantum theories. Especially in the numerical side, many developments have been made to tame the infamous sign problem (e.g., complex Langevin [13,14,5,6,15,16], contour deformation techniques including Lefschetz thimble methods [2,[17][18][19]3,20,7,21,22,8,9,[23][24][25][26][27][28][29][30]11], and tensor renormalization group [31][32][33][34][35][36][37][38][39]10]), which can enables us to investigate real-time qua...…”