Finding Exponential Product Formulas of Higher Orders

TL;DR

This paper reviews methods for constructing higher-order exponential product formulas, which are useful for approximating system dynamics while preserving symmetries, focusing on fractal decomposition and quantum analysis techniques.

Contribution

It introduces two algorithms for generating higher-order exponential product formulas: fractal decomposition and quantum analysis-based correction methods.

Findings

Fractal decomposition enables recursive construction of higher-order formulas.

Quantum analysis allows direct computation of correction terms.

The methods improve approximation accuracy while conserving symmetries.

Abstract

In the present article, we review a continual effort on generalization of the Trotter formula to higher-order exponential product formulas. The exponential product formula is a good and useful approximant, particularly because it conserves important symmetries of the system dynamics. We focuse on two algorithms of constructing higher-order exponential product formulas. The first is the fractal decomposition, where we construct higher-order formulas recursively. The second is to make use of the quantum analysis, where we compute higher-order correction terms directly. As interludes, we also have described the decomposition of symplectic integrators, the approximation of time-ordered exponentials, and the perturbational composition.