Optimized Lie-Trotter-Suzuki decompositions for two and three non-commuting operators
Lie-Trotter-Suzuki decompositions are an efficient way to approximate operator exponentials exp(tH) 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 sixth order. The leading error term is expanded in nested commutators (Hall bases) and we minimize the 1-norm of the coefficients. For two terms, several of the optima we find are close to those in [McLachlan, SlAM J. Sci. Comput. 16, 151 (1995)]. For three terms, our results substantially improve over unoptimized decompositions by Forest, Ruth, Yoshida, and Suzuki. We explain why these decompositions are sufficient to efficiently simulate one- and two-dimensional systems with finite-range interactions. This follows by solving a partitioning problem for the interaction graph.
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