Optimised Trotter decompositions for classical and quantum computing

Abstract

Suzuki–Trotter decompositions of exponential operators like are required in almost every branch of numerical physics. Often the exponent under consideration has to be split into more than two operators , for instance as local gates on quantum computers. We demonstrate how highly optimised schemes originally derived for exactly two operators 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 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 . 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.