This paper gives clear, usable error bounds for two classical formulas that rewrite combinations of non-commuting operators. The Baker–Campbell–Hausdorff (BCH) formula writes the logarithm of a product of exponentials as a series of nested commutators. The Zassenhaus formula writes the exponential of a sum as a product of exponentials. When these series are cut off in practical calculations, we need to know how big the error is. The authors provide a general way to get explicit error bounds for the unitary case that appears in quantum problems.

Concretely, the authors work mainly with the case where the operators are self-adjoint (so their exponentials are unitary). They derive explicit inequalities for the error when the series are truncated. For example, truncating the BCH series after the first two terms (A + B + 1/2[A,B]) produces an error bounded by 1/4 times the norm of [A,[A,B]] plus 1/12 times the norm of [B,[A,B]]. Here [X,Y] means the commutator X Y − Y X, and [A,[A,B]] is a “nested” commutator. For the Zassenhaus formula, they give an optimal estimate for the analogous two-term truncation: the error is at most 1/6‖[A,[A,B]]‖ + 1/3‖[B,[A,B]]‖. They also explain how to obtain bounds when more terms are kept and for symmetric versions of the BCH expansion.

At a high level their method uses two simple ideas from the theory of linear operators. First, they compare two time-dependent unitary evolutions and bound their difference by the time integral of the difference of the generators. Second, they expand conjugation by an exponential into a finite Taylor series plus a remainder, which lets them control the size of the neglected nested commutators. Combining these steps gives explicit constants in the final bounds, not just order estimates.

These bounds matter because truncated BCH and Zassenhaus formulas are used in many numerical and physical calculations. A concrete example is quantum simulation by product formulas, where errors from truncation affect how many quantum gates and how much circuit depth are needed. Bounds that depend on nested commutators are helpful when the operators nearly commute, because then those nested commutators can be small and the bounds show the approximation is good.