Backpropagating Hybrid Monte Carlo algorithm for fast Lefschetz thimble calculations

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