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We derive a universal nonperturbative bound on the distance between unitaryevolutions generated by time-dependent Hamiltonians in terms of the differenceof their integral actions. We apply our result to provide explicit error boundsfor the rotating-wave approximation and generalize it beyond the qubit case. Wediscuss the error of the rotating-wave approximation over long time and in thepresence of time-dependent amplitude modulation. We also show how our universalbound can be used to derive and to generalize other known theorems such as thestrong-coupling limit, the adiabatic theorem, and product formulas, which arerelevant to quantum-control strategies including the Zeno control and thedynamical decoupling. Finally, we prove generalized versions of the Trotterproduct formula, extending its validity beyond the standard scaling assumption.

One bound to rule them all: from Adiabatic to Zeno