Time-Sliced Approximation to Path Integral and Lie–Trotter–Kato Product Formula

Takashi Ichinose

Daisuke Fujiwara, Naoto Kumano‐go

Akira Inomata, Georg Junker

Cristian Predescu

A fast algorithm for path sampling in path-integral Monte Carlo simulations is proposed. The algorithm utilizes the Lévy-Ciesielski implementation of Lie-Trotter products to achieve a mathematically proven computational cost of n log2(n) with the number of time slices n, despite the fact that each path variable is updated separately, for reasons of optimality. In this respect, we demonstrate that updating a group of random variables simultaneously results in loss of efficiency.

Kevin E. Schmidt, Michael A. Lee

Path integrals are a powerful method for calculating real time, finite temperature, and ground state properties of quantum systems. By exploiting some remarkable properties of the symmetric Trotter formula and the discrete Fourier transform, we arrive at a high-accuracy method for removing ``time slice'' errors in Trotter-approximated propagators. We provide an explicit demonstration of the method applied to the two-body density matrix of $^{4}\mathrm{He}$. Our method is simultaneously fast, high precision, and computationally simple and can be applied to a wide variety of quantum propagators.

H.J. Groenewold