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Optimal quantum control based on Lyapunov stability theorem

Shuang Cong

Published Jan 1, 2008• 1 citation
OpenAlex

Abstract

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.

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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.
By means of the achievement of bilinear systems theory research in the macroscopic fields, associated with the characteristic that the mathematical model of quantum mechanical system is bilinear, an iterative optimal controller for a quantum mechanical system is designed through the derivation of the optimal controller with two points boundary values problem. Also by the simulation experiment of states transfer control of the designed quantum system, the effects to control results with different control parameters are detailly analyzed.
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.
The unitary matrix was constructed by using eigenstates of the controlled quantum system so as to perform a coordinate rotation on the controlled system.In such a way the local phases of all states of the system were eliminated.Then the error between the controlled state and the target state after transformation was selected as the Lyapunov function,and the system control law was designed based on Lyapunov stability theorem with the pre-condition that the system is kept stable.The detailed design procedure of the controller needed for preparation of the system states and the comparison analysis of the results were given.Finally,the system simulation experiments on a spin-1/2 particle system were implemented to prepare both the eigen and superposition target states with the proposed control method,and the relation between the system state evolution time and control value was analyzed with different parameters.
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.
Abstract Based on Bures distance, a Lyapunov function that represents the distance between a desired state and the actual state of a quantum system is selected. Considering the cases that an initial state is and is not orthogonal to the desired state respectively, we propose a class of control strategies with state feedback that ensures the stability of the closed-loop control system. Especially, the asymptotic stability of the control system is analyzed, deduced and proved in detail. Finally, a simulation experiment on a spin-1/2 particle system is done and the relation between the system state evolution time and control value is analyzed with different parameters. Research results have general theoretical meaning for control of quantum systems.