Random Series and Discrete Path Integral methods: The Levy-Ciesielski implementation

TL;DR

This paper analyzes the relationship between discrete and series path integral methods, introduces a new interpretation of discrete techniques, and establishes convergence rates for Levy-Ciesielski-based approaches in quantum simulations.

Contribution

It provides a novel interpretation linking discrete and series path integral methods and derives convergence rates for Levy-Ciesielski techniques.

Findings

Discrete and series methods are directly connected.

Levy-Ciesielski methods achieve O(1/n^2) convergence.

Sharp estimates for convergence rates of random series methods.

Abstract

We perform a thorough analysis of the relationship between discrete and series representation path integral methods, which are the main numerical techniques used in connection with the Feynman-Kac formula. First, a new interpretation of the so-called standard discrete path integral methods is derived by direct discretization of the Feynman-Kac formula. Second, we consider a particular random series technique based upon the Levy-Ciesielski representation of the Brownian bridge and analyze its main implementations, namely the primitive, the partial averaging, and the reweighted versions. It is shown that the n=2^k-1 subsequence of each of these methods can also be interpreted as a discrete path integral method with appropriate short-time approximations. We therefore establish a direct connection between the discrete and the random series approaches. In the end, we give sharp estimates on the rates of convergence of the partial averaging and the reweighted Levy-Ciesielski random series approach for sufficiently smooth potentials. The asymptotic rates of convergence are found to be O(1/n^2), in agreement with the rates of convergence of the best standard discrete path integral techniques.