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Applied Physics Letters

State-independent optimal control of dissipative qubits

Markus Wenin, W. Pötz

Published Mar 10, 2008• 7 citations
Publisher

Abstract

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.

DOI
10.1063/1.2894200

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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.
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.
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 ZZZ-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.
Optimal quantum control in dissipative environments is formulated combining the optimal control theory from the Wilson group with the projection operator technique. In the weak response limit, a set of formally exact equations defines the optimal weak field for arbitrary target states and arbitrary system-bath interactions. Perturbative approximations are then made to derive general equations for weakly dissipative non-Markovian systems, which are amenable to analytical and/or numerical solutions. Potential applications of these results are discussed.
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.
Solid-state implementations of a qubit based on semiconductor quantum dots or Josephson junctions are studied within a driven spin-boson model with Ohmic spectral function. It is shown that direct or indirect optimization allows selection of simple control fields which drive the dissipative system from a given initial state to a selected target state provided that the system’s intrinsic minimal response times are respected. The control field can be used to establish interference between the system-bath and system-control interactions, leading to control over effective transition rates and the dissipative action of the environment.
Superconducting nanoelectrical circuits are promising candidates for the physical implementation of the basic building block of a quantum computer, the qubit. We investigate how optimal control theory can be applied to optimize the dynamics of Josephson qubits. For the example of the charge qubit, several numerical methods are employed to search for external control fields which, by current technology, are realistic and induce the desired unitary time evolution within the system (i.e. the desired gate operation) as faithfully as possible in presence of dissipation, decoherence and leakage. Associated calculations which model the environment microscopically are time-intensive so that parallel computing methods are beneficial in the sampling over control fields. In particular, we discuss the performance of differential-evolution-algorithm based optimizations on a cluster. Using a simpler Lindblad model for environmental effects, we compare the performance of a conjugate-gradient approach to that of a genetic algorithm.