Population control of quantum states based on invariant subsets under a diagonal Lyapunov function

This paper deduces and analyzes the invariant set in quantum Lyapunov control, explores the principles for constructing and adjusting diagonal elements of a diagonal Lyapunov function, and achieves the convergence to any goal state in some invariant subset of closed loop systems by using dynamical system theory and energy-level connectivity graph. Research results show that if a goal state is an eigenstate of the inner Hamiltonian, then it is very easy to achieve convergence to the goal state with a high probability; and if a goal state is a superposition state in some invariant subset, then it is possible to achieve satisfactory control when the diagonal elements are properly constructed.