One bound to rule them all: from Adiabatic to Zeno

TL;DR

This paper introduces a universal bound on the difference between quantum evolutions driven by different Hamiltonians, applicable to various quantum control techniques and approximations, with broad implications for quantum dynamics analysis.

Contribution

It provides a nonperturbative, universal bound on unitary evolution differences, extending and unifying several key theorems in quantum control and dynamics.

Findings

Derived a universal nonperturbative bound for quantum evolutions

Extended the validity of the Trotter product formula beyond standard assumptions

Provided explicit error bounds for the rotating-wave approximation

Abstract

We derive a universal nonperturbative bound on the distance between unitary evolutions generated by time-dependent Hamiltonians in terms of the difference of their integral actions. We apply our result to provide explicit error bounds for the rotating-wave approximation and generalize it beyond the qubit case. We discuss the error of the rotating-wave approximation over long time and in the presence of time-dependent amplitude modulation. We also show how our universal bound can be used to derive and to generalize other known theorems such as the strong-coupling limit, the adiabatic theorem, and product formulas, which are relevant to quantum-control strategies including the Zeno control and the dynamical decoupling. Finally, we prove generalized versions of the Trotter product formula, extending its validity beyond the standard scaling assumption.