Optimal control of a driven single dissipative qubit is formulated as an inverse problem. We show that direct inversion is possible which allows an analytic construction of optimal control fields. Exact inversion is shown to be possible for dissipative qubits which can be described by a Lindblad equation. It is shown that optimal solutions are not unique. For a qubit with weak coupling to phonons we choose, among the set of exact solutions for the dissipationless qubit, one which minimizes the dissipative contribution in the kinetic equations. Examples are given for state trapping and Z-gate operation. Using analytic expressions for optimal control fields, favorable domains for dynamic stabilization in the Bloch sphere are identified. In the case of approximate inversion, the identified approximate solution may be used as a starting point for further optimization following standard methods.

We present a general state-independent optimal control strategy for weakly dissipative quantum systems which allows the identification of Hamilton operators which approximately perform a specified unitary operation and minimize the adverse effects of the dissipative environment. Compared to traditional approaches, it avoids the need for Lagrange multipliers and the repeated solving of the full kinetic equations of the quantum subsystem. This direct method is outlined for a single qubit raealization, such as the spin of an excess electron in a quantum dot or a Josephson qubit. We show that for this system formulated within the Lindblad equation, optimal solutions for arbitrary unitary single-qubit operations may be found analytically provided there is full control over the system Hamiltonian. Absolute bounds for controllability are derived analytically. Numerical implementations of this approach confirm these results and allow an efficient determination of optimal solutions for various dissipators, starting from various initial guesses, both for limited and full control over the Hamiltonian of the quantum system.

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

A general formulation of optimal control theory for open quantum systems ͑quantum subsystems͒ based on a superoperator method is presented. This approach is applied to a computation of optimized time-dependent fields for solid-state-based realizations of quantum gates. Decoherence is incorporated within the spin-boson model and treated within, essentially, the Bloch-Redfield formalism. Generating a "decoherence-free subspace" for the superoperator dynamically, we identify optimal trajectories in the phase space of the dynamical system for one-and two-qubit subsystems. Numerical analysis shows that, for the ideal case where one has full control over the subsystem's Hamiltonian, any gate operation can be realized with arbitrary small loss of coherence due to dephasing and optimal solutions exist which are independent of both spectral density and the temperature of the environment. At the example of a Josephson charge quantum gate we study the more realistic situation where one has a restricted number of control fields which, while providing a universal gate, cannot completely eliminate dephasing.

A general state-independent optimal control strategy for weakly dissipative quantum systems is presented. It allows the identification of Hamilton operators which approximately perform a preselected unitary operation and minimize the adverse effects from a dissipative environment. This direct method is demonstrated at the example of a single qubit, as realized, for example, by the spin of an excess electron in a quantum dot or a Josephson flux qubit, with Lindblad dissipator. We show that optimal solutions for arbitrary unitary single-qubit operations may be found both analytically and numerically.