A sharper Magnus expansion bound woven in binary branches

Abstract

The Magnus expansion provides an exponential representation of one-parameter operator families, expressed as a series expansion in its generators. This is useful for example in quantum mechanics for expressing a unitary evolution determined by a time-dependent Hamiltonian generator of the dynamics. The solution is constructed as a series expansion in terms of increasingly complex nested commutators that rapidly become challenging to compute directly. This work establishes a universal upper bound, agnostic to the generator, on the error incurred when the Magnus expansion is truncated at an arbitrary given order. The main technical ingredient of the proof is the binary tree representation introduced by Iserles and Norsett from which we derive a recursion formula to delimit the magnitude of any term in the expansion. We complement our analytic results for the truncation error with explicit calculation of the first 24 terms in the Magnus series, illustrating that they follow the scaling behaviour we have derived. With these findings we aim to contribute to the understanding of the accuracy and limitations of the Magnus expansion technique, and to provide a sharper bound for approximating quantum dynamics without requiring assumptions on the structure of their generators.