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IEEE Conference on Decision and Control (CDC)

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

Sen Kuang, Shuang Cong, Yuesheng Lou

Published Dec 1, 2009• 3 citations
Publisher

Abstract

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.

DOI
10.1109/cdc.2009.5399699

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This paper studies the population control problem associated with the equilibrium states of mixed-state quantum systems by using a Lyapunov function with degrees of freedom. The control laws are designed by ensuring the monotonicity of the Lyapunov function; main results on the largest invariant set in the sense of LaSalle are given; and the strict expression of any state in the largest invariant set is normally deduced in the framework of Bloch vectors. By analyzing the obtained largest invariant set and the Lyapunov function itself, this paper also discusses the determination problem of the degrees of freedom. Numerical simulation experiments on a three-level system show the validity of research results.
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This paper proposes a new multilayer Lyapunov control scheme based on energy-level connectivity graphs of quantum systems, which enables a high-probability population transfer to any eigenstate in high-dimensional quantum systems. We provide a graph connectivity result for stability analysis and explore the selectivity characteristic of the Lyapunov control fields. A systematic layering method for the energy-level connectivity graph is presented to achieve a layer-by-layer population transfer to the target state. Several applications including a five-level quantum system, a three-qubit superconducting system, and a spin chain of five qubits are presented to demonstrate the effectiveness of the multilayer Lyapunov control method.
Based on the analysis of the invariant set in quantum Lyapunov control under a performance index with degrees of freedom, the population control of eigenstates is studied. For isolated target eigenstates, this paper shows the existence of the degrees of freedom that guarantee exact convergence and obtains rigorous construction relation of those degrees of freedom. For non-isolated target eigenstates, we propose a path-planning-based method. The method considers transition paths of controlled systems, so can achieve intuitive and high-probability transition to desired target eigenstate accompanied with appropriate design of those degrees of freedom.
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A Lyapunov-based control scheme is presented to drive closed quantum systems into any target eigenstate with as high population as possible by the quantum-behaved particle swarm optimization (PSO) algorithm. Based on a Lyapunov function with a Hermitian operator to be constructed, a control law with the unknown parameters contained in the Hermitian operator is designed. To achieve high-population state transfer to the target state, we first initialize those unknown parameters by choosing a path to the target state in its energy-level connectivity graph and setting their values along the path. Then, a set of optimal parameters is found by the quantum-behaved PSO algorithm. Finally, numerical simulation experiments are performed on a five-level quantum system and a four-qubit system to demonstrate the effectiveness of the control scheme in this paper.