Simple Ways to improve Discrete Time Evolution

TL;DR

This paper improves Suzuki-Trotter decompositions for exponential operators, offering optimized schemes for multiple operators and suggesting alternative methods like Taylor or Chebyshev expansions for better efficiency.

Contribution

It adapts optimized two-operator schemes to multi-operator decompositions and introduces a stable polynomial expansion method using complex zeros.

Findings

Optimized schemes can be applied to multi-operator Suzuki-Trotter decompositions.

Alternative methods like Taylor and Chebyshev expansions can outperform traditional decompositions.

A numerically stable polynomial expansion method using complex zeros is derived.

Abstract

Suzuki-Trotter decompositions of exponential operators like $\exp(Ht)$ are required in almost every branch of numerical physics. Often the exponent under consideration has to be split into more than two operators, for instance as local gates on quantum computers. In this work, we demonstrate how highly optimised schemes originally derived for exactly two operators can be applied to such generic Suzuki-Trotter decompositions. After this first trick, we explain what makes an efficient decomposition and how to choose from the large variety available. Furthermore we demonstrate that many problems for which a Suzuki-Trotter decomposition might appear to be the canonical ansatz, are better approached with different methods like Taylor or Chebyshev expansions. In particular, we derive an efficient and numerically stable method to implement truncated polynomial expansions based on a linear factorisation using their complex zeros.