A Survey of Quantum Lyapunov Control Methods

TL;DR

This survey reviews quantum Lyapunov control methods for closed quantum systems, focusing on convergence conditions for various target states in both non-degenerate and degenerate cases.

Contribution

It systematically summarizes and analyzes quantum Lyapunov control strategies for different target states and system degeneracies, highlighting convergence conditions.

Findings

Convergence conditions vary for different target states.

Methods differ between non-degenerate and degenerate cases.

The survey provides a comprehensive overview of control strategies.

Abstract

The condition of a quantum Lyapunov-based control which can be well used in a closed quantum system is that the method can make the system convergent but not just stable. In the convergence study of the quantum Lyapunov control, two situations are classified: non-degenerate cases and degenerate cases. In this paper, for these two situations, respectively, the target state is divided into four categories: eigenstate, the mixed state which commutes with the internal Hamiltonian, the superposition state, and the mixed state which does not commute with the internal Hamiltonian state. For these four categories, the quantum Lyapunov control methods for the closed quantum systems are summarized and analyzed. Especially, the convergence of the control system to the different target states is reviewed, and how to make the convergence conditions be satisfied is summarized and analyzed.