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Journal of the Franklin Institute

Stabilizing a class of mixed states for stochastic quantum systems via switching control

Jie Wen, Yuanhao Shi, Xiaonong Lu

Published Feb 7, 2018• 21 citations
Publisher
DOI
10.1016/j.jfranklin.2018.01.031

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The global stabilization of mixed states for finite dimensional stochastic quantum systems with non-regular measurement operator and non-diagonal free Hamiltonian is investigated in this paper. A two-part switching control strategy is proposed, in which the constant control is used to steer the system state to enter the convergence domain, while the control law which is designed based on Lyapunov stability theorem is used to attract the system state in convergence domain to the target state. The convergence of the switching control is strictly proved. Moreover, the numerical experiments on a three dimensional stochastic quantum system are implemented to demonstrate the effectiveness of the control proposed.
Rapid stabilization of general stochastic quantum systems is investigated based on the rapid stability of stochastic differential equations. We introduce a Lyapunov–LaSalle-like theorem for a class of nonlinear stochastic systems first, based on which a unified framework of rapidly stabilizing stochastic quantum systems is proposed. According to the proposed unified framework, we design the switching state feedback controls to achieve the rapid stabilization of single-qubit systems, two-qubit systems, and N-qubit systems. From the unified framework, the state space is divided into two state subspaces, and the target state is located in one state subspace, while the other system equilibria are located in the other state subspace. Under the designed state feedback controls, the system state can only transit through the boundary between the two state subspaces no more than two times, and the target state is globally asymptotically stable in probability. In particular, the system state can converge exponentially in (all or part of) the state subspace where the target state is located. Moreover, the effectiveness and rapidity of the designed state feedback controls are shown in numerical simulations by stabilizing GHZ states for a three-qubit system.