Optimised Trotter Decompositions for Classical and Quantum Computing

Abstract

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 = k A k , for instance as local gates on quantum computers. We demonstrate how highly optimised schemes originally derived for exactly two operators A1,2 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 ≤ 4 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 ≤ 8. 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.1 Symplectic integrators rely on the property that a large number of high order commutators vanishes [1]. This means that using the coefficients of a symplectic integrator for a general splitting method leads to a reduced efficiency starting from order n ≥ 4 and might not even give the expected order at all for n ≥ 6.