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The Picard-Lefschetz theory has been attracting much attention as a tool toevaluate a multi-variable integral with a complex weight, which appears invarious important problems in theoretical physics. The idea is to deform theintegration contour based on Cauchy's theorem using the so-called gradient flowequation. In this paper, we propose a fast Hybrid Monte Carlo algorithm forevaluating the integral, where we "backpropagate" the force of the fictitiousHamilton dynamics on the deformed contour to that on the original contour,thereby reducing the required computational cost by a factor of the systemsize. Our algorithm can be readily extended to the case in which one integratesover the flow time in order to solve not only the sign problem but also theergodicity problem that occurs when there are more than one thimblescontributing to the integral. This enables, in particular, efficientidentification of all the dominant saddle points and the associated thimbles.We test our algorithm by calculating the real-time evolution of the wavefunction using the path integral formalism.

Backpropagating Hybrid Monte Carlo algorithm for fast Lefschetz thimble calculations