Completely positive master equation for arbitrary driving and small level spacing

Evgeny Mozgunov1 and Daniel Lidar1,2,3,4

1Center for Quantum Information Science & Technology, University of Southern California, Los Angeles, California 90089, USA
2Department of Electrical and Computer Engineering, University of Southern California, Los Angeles, California 90089, USA
3Department of Chemistry, University of Southern California, Los Angeles, California 90089, USA
4Department of Physics and Astronomy, University of Southern California, Los Angeles, California 90089, USA

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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.

Real quantum devices available in the lab today are never perfectly isolated from their environment. ​If the isolation is good enough, the effect of the environment can be either neglected altogether, or approximated by a simple mathematical model. We investigate this intuition in great detail, trying to answer exactly how good the isolation needs to be for a simple mathematical model to be within, say, 5% of what an experimentalist would actually see in the lab. Usually, this question is answered on a case-by-case basis by attempting to fit the experimental results with the model and observing the error of the fit. Here we approach this question in some generality, as a rigorous mathematical theorem that we prove under very weak assumptions.

Our result states that as long as the dynamics of the environment is much faster than its effect on the system, so the environment has time to equilibrate regardless of what the system is doing, then there is a mathematical model that can be trusted. The generality of our result is such that it applies to any number of qubits, as well as other quantum systems such as atoms and quantum dots. The control applied to the quantum system by the experimentalist is included in our model and can have a nontrivial interplay with the open system effects. Our model captures this interplay in many relevant cases, both for future theoretical results in quantum information, and for the simulation of quantum experiments.

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