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Chinese Control Conference (CCC)

Quantum-behaved PSO-based Lyapunov control of closed quantum systems

Song Liu, Sen Kuang

Published Jul 26, 2021• 0 citations
Publisher

Abstract

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.

DOI
10.23919/ccc52363.2021.9549738

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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.
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.
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 unified control method for closed quantum systems is proposed, which can transfer the state of closed quantum system from any initial state to any desired target state. In particular, the proposed control method can work under the conditions that free Hamiltonian of the non-ideal quantum system is not strongly regular, and (or) the Hamiltonian is not fully connected. The simulation experiments of the state transition control between different states of three, four and five qubit systems are carried out to verify that the proposed control method is suitable for all kinds of non-ideal quantum systems and the advantages of Lyapunov-based unified control method. The design process of the proposed method, the calculation process of each parameter in each experiment and the analysis of each experimental result are given in detail.
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.
The problem of quantum state control in open quantum systems based on the Lyapunov method is studied. The control laws are designed to drive the system to a pure target state in decoherence-free subspace (DFS) in the interaction picture, where the average of an observable operator P is chosen as a Lyapunov function. By analyzing the positive limit set via Barbalat’s lemma, we propose a sufficient condition on the observable operator such that the positive limit set only contains the target state. Under this condition, it is proved that the control laws can drive the open system to any pure target state in DFS in the interaction picture. Also, we provide a method to construct the observable operator via Schmidt orthogonalization. Numerical simulations on a three-level system validate the effectiveness of the proposed control strategy.