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arXiv:0802.0606 · Journal of Physics A Mathematical and Theoretical

Complete positivity for time-dependent qubit master equations

Michael J. W. Hall

Published Jun 4, 2008• 1 citation
arXivHTMLPDF

Abstract

All instances of the vector cross product in the published version should be replaced by the dot product. These instances include, in particular, the definition of the Lindblad operators following equation (1) in section 1; equation (5) in section 2; the second (unnumbered) equation in section 5.1; and the derivation of equation (29) in section 5.3.

DOI
10.1088/1751-8121/41/26/269801

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All instances of the vector cross product in the published version should be replaced by the dot product. These instances include, in particular, the definition of the Lindblad operators following equation (1) in section 1; equation (5) in section 2; the second (unnumbered) equation in section 5.1; and the derivation of equation (29) in section 5.3.
We present a quantum algorithm for simulating a family of Markovian master equations that can be realized through a probabilistic application of unitary channels and state preparation. Our approach employs a second-order product formula for the Lindblad master equation, achieved by decomposing the dynamics into dissipative and Hamiltonian components and replacing the dissipative segments with randomly compiled, easily implementable elements. The sampling approach eliminates the need for ancillary qubits to simulate the dissipation process and reduces the gate complexity in terms of the number of jump operators. We provide a rigorous performance analysis of the algorithm. We also extend the algorithm to time-dependent Lindblad equations, generalize the family of Markovian master equations it can be applied to, and explore applications beyond the Markovian noise model. A new error bound, in terms of the diamond norm, for second-order product formulas for time-dependent Liouvillians is provided that might be of independent interest.
The Lindblad master equation can always be transformed into a first-order linear ordinary differential equation (1ODE) for the coherence vector. We pose the inverse problem: given a finite-dimensional, nonhomogeneous 1ODE, does a corresponding Lindblad equation exist? If so, what are the corresponding Hamiltonian and Lindblad operators? We provide a general solution to this problem, including a complete positivity test in terms of the parameters of the 1ODE. We also derive a host of properties relating the two representations (master equation and 1ODE) of independent interest.
Abstract Realistic models of quantum systems must include dissipative interactions with a thermal environment. For weakly-damped systems, while the Lindblad-form Markovian master equation is invaluable for this task, it applies only when the frequencies of any subset of the system’s transitions are degenerate, or their differences are much greater than the transitions’ linewidths. Outside of these regimes the only available efficient description has been the Bloch–Redfield master equation, the efficacy of which has long been controversial due to its failure to guarantee the positivity of the density matrix. The ability to efficiently simulate weakly-damped systems across all regimes is becoming increasingly important, especially in quantum technologies. Here we solve this long-standing problem by deriving a Lindblad-form master equation for weakly-damped systems that is accurate for all regimes. We further show that when this master equation breaks down, so do all time-independent Markovian equations, including the B-R equation. We thus obtain a replacement for the B-R equation for thermal damping that is no less accurate, simpler in structure, completely positive, allows simulation by efficient quantum trajectory methods, and unifies the previous Lindblad master equations. We also show via exact simulations that the new master equation can describe systems in which slowly-varying transition frequencies cross each other during the evolution. System identification tools, developed in systems engineering, play an important role in our analysis. We expect these tools to prove useful in other areas of physics involving complex systems.