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Entropy

Quantum Control Design by Lyapunov Trajectory Tracking and Optimal Control

Hongli Yang, Guohui Yu, Ivan Ganchev Ivanov

Published Nov 15, 2024• 0 citations
Publisher

Abstract

In this paper, we investigate a Lyapunov trajectory tracking design method that incorporates a Schrödinger equation with a dipole subterm and polarizability. Our findings suggest that the proposed control law can overcome the limitations of certain existing control laws that do not converge. By integrating a quadratic performance index, we introduce an optimal control law, which we subsequently analyze for stability and optimality. We also simulate the spin-1/2 particle system to illustrate our results. These findings are further validated through numerical illustrations involving a 3D, 5D system, and a spin-1/2 particle system.

DOI
10.3390/e26110978

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In this paper, we investigate a Lyapunov trajectory tracking design method that incorporates a Schrödinger equation with a dipole subterm and polarizability. Our findings suggest that the proposed control law can overcome the limitations of certain existing control laws that do not converge. By integrating a quadratic performance index, we introduce an optimal control law, which we subsequently analyze for stability and optimality. We also simulate the spin-1/2 particle system to illustrate our results. These findings are further validated through numerical illustrations involving a 3D, 5D system, and a spin-1/2 particle system.
By means of the design method for bilinear systems in the macroscopic field in association with the characteristic that the mathematical model of pure state quantum system is bilinear,and of selecting the performance index of the controlled system,the optimal control strategy in quantum systems was proposed based on the Lyapunov stability theorem.The detailed design process of optimal control law was derived.Some simulation experiments were performed on a spin-1/2 particle system.Experimental results demonstrate the effectiveness of the method proposed.At the same time,the influence of different parameters on the system control effects is analyzed.
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.
For quantum state trajectory tracking of density matrix in Liouville equation of quantum systems, with the help of concept in quantum system control, one can apply unitary transformation both to controlled system and free-evolutionary target system such as to change the time-variant and non-stationary target system into a stationary state. Therefore, the quantum state trajectory tracking problem becomes a steering one. State steering control law of the system transformed is designed by means of the Lyapunov stability theorem. Finally, numerical simulation experiments are given for a five-level energy quantum system. The comparison analysis of original system’s trajectory tracking with other method illustrates the advantage in control time of the method proposed.
We investigate in depth the Lyapunov-based quantum control methods including the relation between the Lyapunov-based control and the optimal control,the Lyapunov function and the performance index,the range of application and the existing problems.Combining our results with intrinsic properties of quantum system,we develop several control schemes with improvements for different objectives such as the regulation of eigenstates,superposition state or mixed state.The design ideas,the physical significance and the range of suitable applications are analyzed.The designs method employing Lyapunov stability theorem in controlling quantum system is systemized.
A Lyapunov-based method is presented for stabilizing and controlling of closed quantum systems. The proposed method is constructed upon a novel quantum Lyapunov function of the system state trajectory tracking error. A positive-definite operator in the Lyapunov function provides additional degrees of freedom for the designer. The stabilization process is analyzed regarding two distinct cases for this operator in terms of its vanishing or non-vanishing commutation with the Hamiltonian operator of the undriven quantum system. To cope with the global phase invariance of quantum states as a result of the quantum projective measurement postulate, equivalence classes of quantum states are defined and used in the proposed Lyapunov-based analysis and design. Results show significant improvement in both the set of stabilizable quantum systems and their invariant sets of state trajectories generated by designed control signals. The proposed method can potentially be applied for high-fidelity quantum control purposes in quantum computing frameworks.