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International Journal of Control

Lyapunov-based prescribed-time stabilisation control of quantum systems

Eram Taslima, Shyam Kamal, Ram Khelawan Saket, Thach Ngoc Dinh

Published Sep 2, 2024• 6 citations
Publisher

Abstract

This paper addresses the prescribed-time stabilisation control problem for quantum systems characterised by a single control input. A control law based on the relative state distance is formulated. Through the application of prescribed-time Lyapunov control techniques, the stabilisation of the quantum system to one of the eigenstates of the internal Hamiltonian within a prescribed-time is established. Notably, in contrast to existing quantum system control methods, our proposed approach ensures stabilisation within a specified time, allowing for the flexible selection of the settling-time upper bound according to preference, independent of the initial conditions.

DOI
10.1080/00207179.2024.2395328

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In closed quantum systems, if controlled systems are not strongly regular or there exists at least one eigenstate that is directly uncoupled to the target state, then such systems control will become degenerate cases. This paper proposes a quantum control method designed by an implicit Lyapunov function to transfer states of the quantum systems in these two degenerate cases, especially the method proposed is suitable for multi-control Hamiltonian systems. The state distance is selected as the Lyapunov function in this paper. Numerical simulation experiments on a 4-level system are done, and the experiment results indicate the effectiveness of the implicit Lyapunov control method proposed for degenerate cases and multi-control Hamiltonians.
By using a quadratic Lyapunov function with un-known parameters and a particle swarm optimization (PSO) algorithm, this paper proposes a new Lyapunov control scheme for quantum systems. This approach can achieve high-probability population transfer to a given target state. For the case when the target state is an eigenstate of the internal Hamiltonian, we introduce a virtual control law into the system model and design corresponding control law. The stability of the system under the action of the Lyapunov control law is analyzed via the LaSalle invariance principle. For the case when the target state is a superposition state, we design a control law by performing a unitary transformation for the quantum system model under consideration. To achieve desired state transfer, we further introduce a PSO algorithm to search for the unknown parameters contained in the control law. Numerical results on a five-level quantum system and a three-qubit system are presented to demonstrate the effectiveness of the proposed approaches.
A Lyapunov-based approach of trajectory tracking in Liouville equation for quantum systems is proposed. By selecting density matrix of controlled system as the system state, and defining the error between controlled state and target one as an error state, the paper regards this error as a new controlled state of system. In such a way the quantum state trajectory tracking problem is changed into the state regulation problem by driving the error state to zero. The control law is designed according to the Lyapunov stability theorem, which can be applied to track different kinds of quantum states including pure state, superposition state and mixed-state. Finally, numerical simulation experiments and results analysis are given.