Higher Order Decompositions of Ordered Operator Exponentials

TL;DR

This paper introduces a new decomposition scheme for ordered operator exponentials using Lie-Trotter-Suzuki formulae, with rigorous error bounds applicable to non-analytic functions, expanding their theoretical applicability.

Contribution

It provides a novel, rigorous proof for decomposing ordered operator exponentials without superoperators, applicable to non-analytic functions, and discusses limitations in achieving arbitrary order scaling.

Findings

Decomposition scheme based on Lie-Trotter-Suzuki formulae

Rigorous error bounds without using superoperators

Applicability to non-analytic functions

Abstract

We present a decomposition scheme based on Lie-Trotter-Suzuki product formulae to represent an ordered operator exponential as a product of ordinary operator exponentials. We provide a rigorous proof that does not use a time-displacement superoperator, and can be applied to non-analytic functions. Our proof provides explicit bounds on the error and includes cases where the functions are not infinitely differentiable. We show that Lie-Trotter-Suzuki product formulae can still be used for functions that are not infinitely differentiable, but that arbitrary order scaling may not be achieved.