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

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.