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

The Baker-Campbell-Hausdorff formula, a crucial tool in mathematics and physics, has been reexamined to establish error bounds for its truncated versions, as well as those of its dual, the Zassenhaus formula. These formulas are essential in handling non-commuting operators, which is a common challenge in quantum mechanics. By understanding the limitations of these truncated formulas, researchers can better navigate the complexities of quantum computing and its applications. The study focuses on unitary problems, where the operators involved are unitary, meaning they preserve the norm of the vectors they act upon. Establishing precise error bounds for these formulas is critical, as small errors can propagate and significantly impact the accuracy of quantum computations¹. This research has significant implications for the development of quantum computing, particularly in the context of cryptography, where the security of quantum protocols relies on the precise manipulation of quantum operators.