Optimised Trotter Decompositions for Classical and Quantum Computing

Suzuki-Trotter decompositions of exponential operators like $\exp(Ht)$ are required in almost every branch of numerical physics. Often the exponent under consideration has to be split into more than two operators $H=\sum_k A_k$, for instance as local gates on quantum computers. We demonstrate how highly optimised schemes originally derived for exactly two operators $A_{1,2}$ can be applied to such generic Suzuki-Trotter decompositions, providing a formal proof of correctness as well as numerical evidence of efficiency. A comprehensive review of existing symmetric decomposition schemes up to order $n\le4$ is presented and complemented by a number of novel schemes, including both real and complex coefficients. We derive the theoretically most efficient unitary and non-unitary 4th order decompositions. The list is augmented by several exceptionally efficient schemes of higher order $n\le8$. Furthermore we show how Taylor expansions can be used on classical devices to reach machine precision at a computational effort at which state of the art Trotterization schemes do not surpass a relative precision of $10^{-4}$. Finally, a short and easily understandable summary explains how to choose the optimal decomposition in any given scenario.