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Australian Control Conference

Approximate bang-bang Lyapunov control for closed quantum systems

Sen Kuang, Daoyi Dong, Ian R. Petersen

Published Nov 1, 2014• 6 citations
Publisher

Abstract

This paper proposes a new approximate bang-bang Lyapunov control that can achieve rapid state control for quantum systems. A construction method is presented to design the degrees of freedom in the Lyapunov function so that the control law can guarantee the convergence of the system to a target eigenstate being isolated in the invariant set. Simulation experiments on a three-level system demonstrate that the proposed method can achieve good performance for rapid state control of quantum systems.

DOI
10.1109/aucc.2014.7358685

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This paper proposes a new approximate bang-bang Lyapunov control that can achieve rapid state control for quantum systems. A construction method is presented to design the degrees of freedom in the Lyapunov function so that the control law can guarantee the convergence of the system to a target eigenstate being isolated in the invariant set. Simulation experiments on a three-level system demonstrate that the proposed method can achieve good performance for rapid state control of quantum systems.
This paper addresses the critical objective of achieving rapid state transfer in quantum systems. Increasing control gain to achieve fast convergence can result in poor convergence. To circumvent this, a novel continuous, nonsmooth control is devised, exploiting Lyapunov conditions for quantum systems described by von Neumann equations. This approach ensures swift convergence to the target eigenstate while minimizing control effort and enhancing fidelity compared to prevalent techniques such as bang-bang control, approximate bang-bang control, and standard Lyapunov control. Several illustrative examples are provided to corroborate and validate the efficacy of the proposed methodology.
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.
A Lyapunov-based method is presented for stabilizing and controlling of closed quantum systems. The proposed method is constructed upon a novel quantum Lyapunov function of the system state trajectory tracking error. A positive-definite operator in the Lyapunov function provides additional degrees of freedom for the designer. The stabilization process is analyzed regarding two distinct cases for this operator in terms of its vanishing or non-vanishing commutation with the Hamiltonian operator of the undriven quantum system. To cope with the global phase invariance of quantum states as a result of the quantum projective measurement postulate, equivalence classes of quantum states are defined and used in the proposed Lyapunov-based analysis and design. Results show significant improvement in both the set of stabilizable quantum systems and their invariant sets of state trajectories generated by designed control signals. The proposed method can potentially be applied for high-fidelity quantum control purposes in quantum computing frameworks.
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.
A control strategy of quantum states of open quantum system based on Lyapunov method is proposed. On the basis of choosing the average of observable operators as a Lyapunov function, the control laws are designed in interaction picture. We analyze the largest invariant set of the controlled system via Barbalat's Lemma. Based on some assumptions of Hamiltonians, we propose a sufficient condition of the observable operator to guarantee the convergence of the open quantum system, and proof that the proposed strategy can make any pure target state in decoherence-free subspace be globally asymptotic stable under the sufficient condition. Also, we provide a method to construct the observable operator via Schmidt orthogonalization. Numerical experiments on a three-level system validate the effectiveness of the proposed control strategy.
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.