Error bounds for the truncated Baker--Campbell--Hausdorff and Zassenhaus formulas in unitary problems

The Baker--Campbell--Hausdorff (BCH) formula plays a critical role in many branches of mathematics and physics.

It expresses the logarithm of the product of exponentials of non-commuting operators as an infinite series of nested commutators of the operators involved.

The Zassenhaus formula is the dual of the BCH formula: the exponential of a sum of operators is written as an infinite product of exponentials involving the operators and their commutators.

In practical computations, however, one typically has to truncate the expansions, and so understanding the error committed by the resulting approximations and eventually providing suitable bounds for this error is of paramount interest.

In this work we present a general strategy to derive rigorous error bounds and explicit error constants for the BCH and Zassenhaus formulas when the operators involved are skew-adjoint, as is the case for quantum evolution problems.