Researchers have developed a general strategy to derive rigorous error bounds and explicit error constants for the Baker-Campbell-Hausdorff (BCH) and Zassenhaus formulas. These formulas are fundamental mathematical tools in various branches of physics and mathematics, especially in problems involving non-commuting operators, such as quantum evolution. The ability to quantify the error in their truncated approximations is crucial for the reliability of computations.

The BCH formula expresses the logarithm of the product of exponentials of non-commuting operators as an infinite series of nested commutators. Conversely, the Zassenhaus formula, its dual, writes the exponential of a sum of operators as an infinite product of exponentials involving the operators and their commutators. In practice, these series must be truncated for computations, which introduces an error. Understanding and bounding this error is of paramount importance to ensure the accuracy of approximations.

This new work focuses on cases where the involved operators are skew-adjoint, a condition met in numerous quantum evolution problems. By providing a methodology to derive explicit error bounds, this research enhances confidence in the computational applications of these formulas in fields like quantum mechanics, where precision in describing the time evolution of systems is fundamental. This will allow for a more robust evaluation of results obtained through truncated approximations.