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arXiv:2404.09279

Optimal Control of a Markovian Qubit with Unitary Control

Emanuel Malvetti

Published Apr 14, 2024• 1 citation
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Abstract

We study a single Markovian qubit governed by a Lindblad master equation and subject to fast unitary control. Using reduced control systems and optimal control theory we determine (i) controls for cooling and heating such systems in a time-optimal way as well as (ii) the set of stabilizable states in the Bloch ball. No restrictions on the Lindblad equation are assumed, and several known results, for instance for the Bloch equations, are recovered. Furthermore we introduce integral systems, for which the solutions take a particularly nice form. These integral systems include all systems with real Lindblad terms as well as all coolable systems. The method allows for intuitive visualizations and is mostly analytical, making use of only basic numerical methods.

DOI
10.48550/arxiv.2404.09279

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We study a single Markovian qubit governed by a Lindblad master equation and subject to fast unitary control. Using reduced control systems and optimal control theory we determine (i) controls for cooling and heating such systems in a time-optimal way as well as (ii) the set of stabilizable states in the Bloch ball. No restrictions on the Lindblad equation are assumed, and several known results, for instance for the Bloch equations, are recovered. Furthermore we introduce integral systems, for which the solutions take a particularly nice form. These integral systems include all systems with real Lindblad terms as well as all coolable systems. The method allows for intuitive visualizations and is mostly analytical, making use of only basic numerical methods.
We address the problem of cooling a Markovian quantum system to a pure state in the shortest amount of time possible. Here, the system drift takes the form of a Lindblad master equation and we assume fast unitary control. This setting allows for a natural reduction of the control system to the eigenvalues of the state density matrix. We give a simple necessary and sufficient characterization of systems which are (asymptotically) coolable and present a powerful maximum principle, based on majorization, which allows to considerably simplify the search for optimal cooling solutions. With these tools at our disposal, we derive explicit, provably time-optimal cooling protocols for rank one qubit systems, inverted Λ\LambdaΛ-systems on a qutrit, and a certain system consisting of two coupled qubits.
The objective of this work is to study time-minimum and energy-minimum global optimal control for dissipative open quantum systems whose dynamics is governed by the Lindblad equation. The controls appear only in the Hamiltonian. Using recent results regarding the decoupling of such dissipative dynamics into intra- and inter-unitary orbits, we transform the control system into a bi-linear control system on the Bloch ball (the unitary sphere together with its interior). We then design a numerical algorithm to construct an optimal path to achieve a desired point given initial states close to the origin (the singular point) of the Bloch ball. This is done both for the minimum-time and minimum-energy control problems.
We study time-minimum optimal control for a class of quantum two-dimensional dissipative systems whose dynamics are governed by the Lindblad equation and where control inputs acts only in the Hamiltonian. The dynamics of the control system are analyzed as a bi-linear control system on the Bloch ball after a decoupling of such dynamics into intra- and inter-unitary orbits. The (singular) control problem consists of finding a trajectory of the state variables solving a radial equation in the minimum amount of time, starting at the completely mixed state and ending at the state with the maximum achievable purity.The boundary value problem determined by the time-minimum singular optimal control problem is studied numerically. If controls are unbounded, simulations show that multiple local minimal solutions might exist. To find the unique globally minimal solution, we must repeat the algorithm for various initial conditions and find the best solution out of all of the candidates. If controls are bounded, optimal controls are given by bang-bang controls using the Pontryagin minimum principle. Using a switching map we construct optimal solutions consisting of singular arcs. If controls are bounded, the analysis of our model also implies classical analysis done previously for this problem.
Abstract. Open Markovian quantum systems with fast unitary control can be reduced to an equivalent control system on the standard simplex modeling the dynamics of the eigenvalues of the density matrix describing the quantum state. We explore this reduced control system for answering questions on reachability and stabilizability with immediate applications to the cooling of Markovian quantum systems. We show that for certain tasks of interest, the control Hamiltonian can be chosen time-independent.
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.