# Optimized Lie-Trotter-Suzuki decompositions for two and three non-commuting terms

## TL;DR

This paper develops optimized Lie-Trotter-Suzuki decompositions up to sixth order for efficiently approximating operator exponentials in quantum and classical systems, improving accuracy over previous methods.

## Contribution

It introduces new optimized decompositions up to order six for multiple non-commuting operators, with minimized error coefficients, enhancing simulation efficiency.

## Key findings

- Optimized decompositions significantly reduce approximation errors.
- Results are close to previous optimal decompositions for two terms.
- Decompositions are applicable to lattice models with finite-range interactions.

## Abstract

Lie-Trotter-Suzuki decompositions are an efficient way to approximate operator exponentials $\exp(t H)$ when $H$ is a sum of $n$ (non-commuting) terms which, individually, can be exponentiated easily. They are employed in time-evolution algorithms for tensor network states, digital quantum simulation protocols, path integral methods like quantum Monte Carlo, and splitting methods for symplectic integrators in classical Hamiltonian systems. We provide optimized decompositions up to order $t^6$. The leading error term is expanded in nested commutators (Hall bases) and we minimize the 1-norm of the coefficients. For $n=2$ terms, several of the optima we find are close to those in McLachlan, SlAM J. Sci. Comput. 16, 151 (1995). Generally, our results substantially improve over unoptimized decompositions by Forest, Ruth, Yoshida, and Suzuki. We explain why these decompositions are sufficient to efficiently simulate any one- or two-dimensional lattice model with finite-range interactions. This follows by solving a partitioning problem for the interaction graph.