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 $H={\sum }_k{A}_k$, 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 $n\le 4$ 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 $n\le 8$. 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.

arXiv comment25 + 5 pages, 6 figures; code and data available, DOI: 10.5281/zenodo.8044499 ; v2: added references; v3: added appendix; v4 (published version): added Yoshida's method

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