Rapid Lyapunov control of finite-dimensional quantum systems

TL;DR

This paper introduces two innovative Lyapunov control strategies for quantum systems that enable rapid convergence to target states, improving control speed and robustness against environmental decoherence.

Contribution

It proposes switching and approximate bang-bang Lyapunov control methods with stability guarantees and a construction approach for rapid quantum state control.

Findings

Enhanced control speed demonstrated in numerical examples

Guarantees of convergence to target eigenstates

New control functions with bang-bang properties

Abstract

Rapid state control of quantum systems is significant in reducing the influence of relaxation or decoherence caused by the environment and enhancing the capability in dealing with uncertainties in the model and control process. Bang-bang Lyapunov control can speed up the control process, but cannot guarantee convergence to a target state. This paper proposes two classes of new Lyapunov control methods that can achieve rapidly convergent control for quantum states. One class is switching Lyapunov control where the control law is designed by switching between bang-bang Lyapunov control and standard Lyapunov control. The other class is approximate bang-bang Lyapunov control where we propose two special control functions which are continuously differentiable and yet have a bang-bang type property. Related stability results are given and a construction method for the degrees of freedom in the Lyapunov function is presented to guarantee rapid convergence to a target eigenstate being isolated in the invariant set. Several numerical examples demonstrate that the proposed methods can achieve improved performance for rapid state control of quantum systems.