Abstract
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.
Highlights
Modeling experiments requires taking into account that physical systems are open, i.e., not ideally isolated from their environments
Note that we had to make these assumptions for any y of bounded norm, not just density operators
H allows us to get a tighter bound, but the difference is most drastic for large τB/τSB when the bound is weaker than the trivial ρBM,I (t) − ρC,I (t) ≤ 1 + cBM
Summary
Modeling experiments requires taking into account that physical systems are open, i.e., not ideally isolated from their environments. The time-dependent CGME is compatible with the assumption of arbitrarily fast gates, often made in the circuit model of quantum computing, e.g., in the analysis of fault-tolerant quantum error correction [42] This assumption was shown to be incompatible with the derivation of the Davies master equation [43]. We compare the range of applicability with other master equations, and note the spatial locality of the Lindblad generators of the CGME At this point we are ready to address the case of time-dependent Hamiltonians. This equation is well suited for the simulation of open system adiabatic quantum computation, and for dynamical decoupling, which involves the opposite limit of very fast system dynamics.
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