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Markovian master equations are a ubiquitous tool in the study of open quantumsystems, but deriving them from first principles involves a series ofcompromises. On the one hand, the Redfield equation is valid for fastenvironments (whose correlation function decays much faster than the systemrelaxation time) regardless of the relative strength of the coupling to thesystem Hamiltonian, but is notoriously non-completely-positive. On the otherhand, the Davies equation preserves complete positivity but is valid only inthe ultra-weak coupling limit and for systems with a finite level spacing,which makes it incompatible with arbitrarily fast time-dependent driving. Herewe show that a recently derived Markovian coarse-grained master equation(CGME), already known to be completely positive, has a much expanded range ofapplicability compared to the Davies equation, and moreover, is locallygenerated and can be generalized to accommodate arbitrarily fast driving. Thisgeneralization, which we refer to as the time-dependent CGME, is thus suitablefor the analysis of fast operations in gate-model quantum computing, such asquantum error correction and dynamical decoupling. Our derivation proceedsdirectly from the Redfield equation and allows us to place rigorous errorbounds on all three equations: Redfield, Davies, and coarse-grained. Our mainresult is thus a completely positive Markovian master equation that is acontrolled approximation to the true evolution for any time-dependence of thesystem Hamiltonian, and works for systems with arbitrarily small level spacing.We illustrate this with an analysis showing that dynamical decoupling canextend coherence times even in a strictly Markovian setting.

Completely positive master equation for arbitrary driving and small level spacing