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IEEE Access

Lyapunov Control of High-Dimensional Closed Quantum Systems Based on Particle Swarm Optimization

Xiaoke Guan, Sen Kuang, Xiujuan Lu, Jiazhen Yan

Published Jan 1, 2020• 11 citations
Publisher

Abstract

For high-dimensional closed quantum systems, this paper proposes a novel quantum Lyapunov control scheme based on the particle swarm optimization algorithm and achieves a high-probability population transfer of the system to a non-isolated target eigenstate in the LaSalle invariant set under usual smooth Lyapunov control laws. Via a quadratic Lyapunov function with unknown parameters, a control law with the unknown parameters is designed; based on the LaSalle invariance principle and the energy-level connectivity graph, the stability of the system is analyzed; by using the particle swarm optimization algorithm, a set of optimal parameters is obtained to achieve the control goal. In particular, we propose a path planning method based on the energy-level connectivity graph to determine the initial values of the unknown parameters, which is such that the optimization algorithm can efficiently and conveniently find a set of optimal solutions of the unknown parameters. Numerical simulation experiments on a five-level quantum system and a three-qubit system demonstrate that the proposed Lyapunov control scheme based on the particle swarm optimization algorithm has a good control effect.

DOI
10.1109/access.2020.2980132

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For high-dimensional closed quantum systems, this paper proposes a novel quantum Lyapunov control scheme based on the particle swarm optimization algorithm and achieves a high-probability population transfer of the system to a non-isolated target eigenstate in the LaSalle invariant set under usual smooth Lyapunov control laws. Via a quadratic Lyapunov function with unknown parameters, a control law with the unknown parameters is designed; based on the LaSalle invariance principle and the energy-level connectivity graph, the stability of the system is analyzed; by using the particle swarm optimization algorithm, a set of optimal parameters is obtained to achieve the control goal. In particular, we propose a path planning method based on the energy-level connectivity graph to determine the initial values of the unknown parameters, which is such that the optimization algorithm can efficiently and conveniently find a set of optimal solutions of the unknown parameters. Numerical simulation experiments on a five-level quantum system and a three-qubit system demonstrate that the proposed Lyapunov control scheme based on the particle swarm optimization algorithm has a good control effect.
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.
By using a quadratic Lyapunov function with un-known parameters and a particle swarm optimization (PSO) algorithm, this paper proposes a new Lyapunov control scheme for quantum systems. This approach can achieve high-probability population transfer to a given target state. For the case when the target state is an eigenstate of the internal Hamiltonian, we introduce a virtual control law into the system model and design corresponding control law. The stability of the system under the action of the Lyapunov control law is analyzed via the LaSalle invariance principle. For the case when the target state is a superposition state, we design a control law by performing a unitary transformation for the quantum system model under consideration. To achieve desired state transfer, we further introduce a PSO algorithm to search for the unknown parameters contained in the control law. Numerical results on a five-level quantum system and a three-qubit system are presented to demonstrate the effectiveness of the proposed approaches.
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.
This paper introduces the quantum control of Lyapunov functions based on the state distance, the mean of imaginary quantities and state errors. In this paper, the specific control laws under the three forms are given. Stability is analyzed by the LaSalle invariance principle and the numerical simulation is carried out in a 2D test system. The calculation process for the Lyapunov function is based on a combination of the average of virtual mechanical quantities, the particle swarm algorithm and a simulated annealing algorithm. Finally, a unified form of the control laws under the three forms is given.
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.
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.