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IEEE Conference on Decision and Control (CDC)

Simulation of open quantum dynamics in Markovian environment

Haidong Yuan

Published Dec 1, 2011• 0 citations
Publisher

Abstract

Although conditions for controllability of quantum systems are well studied, most of them consider the infinite time horizon, i.e., whether certain operations are possible given enough time. In this article, I propose a framework studying all the possible operations one can generate at given finite time on a quantum system in Markovian environment. I give a complete characterization of the operations one can simulate at any given time on a qubit in unital Markovian environment and discuss possible extensions.

DOI
10.1109/cdc.2011.6160937

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Although conditions for controllability of quantum systems are well studied, most of them consider the infinite time horizon, i.e., whether certain operations are possible given enough time. In this article, I propose a framework studying all the possible operations one can generate at given finite time on a quantum system in Markovian environment. I give a complete characterization of the operations one can simulate at any given time on a qubit in unital Markovian environment and discuss possible extensions.
Although the conditions for performing arbitrary unitary operations to simulate the dynamics of a closed quantum system are well understood, the same is not true of the more general class of quantum operations (also known as superoperators) corresponding to the dynamics of open quantum systems. We propose a framework for the generation of Markovian quantum dynamics and study the resources needed for universality. For the case of a single qubit, we show that a single nonunitary process is necessary and sufficient to generate all unital Markovian quantum dynamics, whereas a set of processes parametrized by one continuous parameter is needed in general. We also obtain preliminary results for the unital case in higher dimensions.
We study reachable sets of coherently controllable open quantum systems with switchable coupling of two models: we show that finite-dimensional systems coupled to a temperature-zero bath allow for approximately generating every quantum state from every initial state. Also studying d-majorization yields an upper bound for non-zero temperature. Then we consider infinite-dimensional open systems following a unital evolution. For arbitrary drift, bounded control Hamiltonians, and noise generated by a compact normal operator, one can approximately reach any state majorized by the initial one.
In the study of open quantum systems, one typically obtains the decoherence dynamics by solving a master equation. The master equation is derived using knowledge of some basic properties of the system, the environment, and their interaction: One basically needs to know the operators through which the system couples to the environment and the spectral density of the environment. For a large system, it could become prohibitively difficult to even write down the appropriate master equation, let alone solve it on a classical computer. In this paper, we present a quantum algorithm for simulating the dynamics of an open quantum system. On a quantum computer, the environment can be simulated using ancilla qubits with properly chosen single-qubit frequencies and with properly designed coupling to the system qubits. The parameters used in the simulation are easily derived from the parameters of the system +++ environment Hamiltonian. The algorithm is designed to simulate Markovian dynamics, but it can also be used to simulate non-Markovian dynamics provided that this dynamics can be obtained by embedding the system of interest into a larger system that obeys Markovian dynamics. We estimate the resource requirements for the algorithm. In particular, we show that for sufficiently slow decoherence a single ancilla qubit could be sufficient to represent the entire environment, in principle.
Open quantum systems have been aim of several theoretical and experimental studies over the last years.In particular, research related to non-Markovianity and how it influences the dynamics of the system has become relatively important, because this way we can have the return the initial quantum characteristics present in the system.In this thesis two related works were developed about the influence of non-Markovianity in open quantum systems.In the first, it was used a one qubit system in contact with a bosonic reservoir and also under the influence of a classic stochastic pumping, thus generating an effective coupling of the system with the stochastic noise and the thermal reservoir.Thus, it was investigated how the dynamics of the qubit is modified through the production of non-Markovianity in the system due to the effective coupling and also how the mean system energy, with and without pumping, behaves in both Markovian and non-Markovian regime.Additionally, an analysis of non-Markovianity production was performed using exponential, Gaussian and power law stochastic noises.Applying the Andersson, Cresser, Hall and Li (ACHL) measure, it was possible to make a comparison of how much non-Markovianity was produced for each stochastic noise.In the second work, a study was performed on how it is possible to generate the transition between Markovian to non-Markovian regimes through the control of the spin-lattice relaxation time of an ancillary qubit, which acts as an environment.The decoherence of the system was simulated through the formulation of quantum channels.In addition, it was discussed how the relaxation mechanism is modified by the action of an external magnetic field.Finally, an experimental demonstration of the model is presented and the experimental data were compared with the theoretically simulated ones.
We consider the problem of constructing a ``universal set'' of Markovian processes, such that any Markovian open quantum system, described by a one-parameter semigroup of quantum channels, can be simulated through sequential simulations of processes from the universal set. In particular, for quantum systems of dimension ddd, we explicitly construct a universal set of semigroup generators, parametrized by ${d}^{2}\ensuremath{-}3$ continuous parameters, and prove that a necessary and sufficient condition for the dynamical simulation of a ddd-dimensional Markovian quantum system is the ability to implement (a) quantum channels from the semigroups generated by elements of the universal set of generators, and (b) unitary operations on the system. Furthermore, we provide an explicit algorithm for simulating the dynamics of a Markovian open quantum system using this universal set of generators, and show that it is efficient, with respect to this universal set, when the number of distinct Lindblad operators (representing physical dissipation processes) scales polynomially with respect to the number of subsystems.