  Markovian master equations are a ubiquitous tool in the study of open quantum
systems, but deriving them from first principles involves a series of
compromises. On the one hand, the Redfield equation is valid for fast
environments (whose correlation function decays much faster than the system
relaxation time) regardless of the relative strength of the coupling to the
system Hamiltonian, but is notoriously non-completely-positive. On the other
hand, the Davies equation preserves complete positivity but is valid only in
the ultra-weak coupling limit and for systems with a finite level spacing,
which makes it incompatible with arbitrarily fast time-dependent driving. Here
we show that a recently derived Markovian coarse-grained master equation
(CGME), already known to be completely positive, has a much expanded range of
applicability compared to the Davies equation, and moreover, is locally
generated and can be generalized to accommodate arbitrarily fast driving. This
generalization, which we refer to as the time-dependent CGME, is thus suitable
for the analysis of fast operations in gate-model quantum computing, such as
quantum error correction and dynamical decoupling. Our derivation proceeds
directly from the Redfield equation and allows us to place rigorous error
bounds on all three equations: Redfield, Davies, and coarse-grained. Our main
result is thus a completely positive Markovian master equation that is a
controlled approximation to the true evolution for any time-dependence of the
system Hamiltonian, and works for systems with arbitrarily small level spacing.
We illustrate this with an analysis showing that dynamical decoupling can
extend coherence times even in a strictly Markovian setting.
