Optimal Control for Open Quantum Systems: Qubits and Quantum Gates

This article provides a review of recent developments in the formulation and execution of optimal control strategies for the dynamics of quantum systems. A brief introduction to the concept of optimal control, the dynamics of of open quantum systems, and quantum information processing is followed by a presentation of recent developments regarding the two main tasks in this context: state-specific and state-independent optimal control. For the former, we present an extension of conventional theory (Pontryagin's principle) to quantum systems which undergo a non-Markovian time-evolution. Owing to its importance for the realization of quantum information processing, the main body of the review, however, is devoted to state-independent optimal control. Here, we address three different approaches: an approach which treats dissipative effects from the environment in lowest-order perturbation theory, a general method based on the time--evolution superoperator concept, as well as one based on the Kraus representation of the time-evolution superoperator. Applications which illustrate these new methods focus on single and double qubits (quantum gates) whereby the environment is modeled either within the Lindblad equation or a bath of bosons (spin-boson model). While these approaches are widely applicable, we shall focus our attention to solid-state based physical realizations, such as semiconductor- and superconductor-based systems. While an attempt is made to reference relevant and representative work throughout the community, the exposition will focus mainly on work which has emerged from our own group.Comment: 27 pages, 18 figure

Time-Optimal Frictionless Atom Cooling in Harmonic Traps

Frictionless atom cooling in harmonic traps is formulated as a time-optimal control problem and a synthesis of optimal controlled trajectories is obtained. This work has already been used to determine the minimum time for transition between two thermal states and to show the emergence of the third law of classical thermodynamics from quantum thermodynamics. It can also find application in the fast adiabatic-like expansion of Bose-Einstein condensates, with possible applications in atom interferometry. This paper is based on our recently published article in SIAM J. Control Optim.Comment: Submitted to 51st IEEE Conference on Decision and Control as a SIAM regular paper, it is a shorter version of our recently published article in SIAM J. Control Optim., vol. 49, pp. 2440-2462, 2011. It contains an elegant proof of the main technical point using the symmetries of the system, and a discussion of the implications of the results on finite time thermodynamic processe

Noncommutative optimal control and quantum networks

Optimal control is formulated based on a noncommutative calculus of operator derivatives. The use of optimal control methods in the design of quantum systems relies on the differentiation of an operator-valued function with respect to the relevant operator. Noncommutativity between the operator and its derivative leads to a generalization of the conventional method of control for classical systems. This formulation is applied to quantum networks of both spin and bosonic particles for the purpose of quantum state control via quantum random walks