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IEEE Conference on Decision and Control (CDC)

Non-smooth Lyapunov function-based global stabilization for 2-dimensional quantum filters

Shuzhi Sam Ge, Thanh-Long Vu, Wei He, Chang Chieh Hang

Published Dec 1, 2010• 3 citations
Publisher

Abstract

This paper addresses the global stabilization problem for 2-dimensional quantum filters via non-smooth Lyapunov functions. Due to the symmetric topology of filter state space, the smooth controls synthesized via smooth Lyapunov stochastic stability theory fail to obtain the global stabilizability. As such, for the first time, we introduce a non- smooth Lyapunov-like theory for generic stochastic nonlinear systems, which includes a continuous Lyapunov-like theorem and a discontinuous Lyapunov-like theorem for stability in probability. Applying the non-smooth Lyapunov-like theory, switching control and saturation-form control are constructed for 2-dimensional quantum filters with the consideration of sliding motion. The non-smooth property enables these controls to deal with the symmetric topology of filter state space and to globally asymptotically render the filter state to the final desired state almost surely. The effectiveness of the proposed controls is illustrated through the control design for the Spin-1/2 system. Simulation results are presented and discussed.

DOI
10.1109/cdc.2010.5717373

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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.
This paper addresses the problem of measurement-based feedback control for quantum systems, in which the time to compute the filter-based control input is taken into account by considering the input delay. It starts with a new Lyapunov–LaSalle-like theorem for delay-dependent stochastic stability of a class of time delay stochastic nonlinear systems. Nonsmooth time delay control is then constructed to compensate for the control-computation time, which is known but arbitrarily long, while globally stabilizing the quantum filters almost surely. The nonsmooth property enables the control to deal with the symmetric topology of filter state space. The effectiveness of the proposed control is illustrated through the global stabilization of the spin-1/21/21/2 systems. Simulation results are presented and discussed to show the effectiveness of the proposed control.
The global stabilization of an eigenstate for finite dimensional stochastic quantum systems with non-diagonal free Hamiltonian and non-regular measurement operator is studied in this paper. A two-part switching control strategy is proposed, in which a constant control is used to steer the system state to enter the convergence domain; another 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 target eigenstate from an arbitrary initial state is strictly proved. Moreover, the numerical experiments on a three dimensional stochastic quantum system are implemented to demonstrate the effectiveness of the control proposed.
In this paper, a stabilization problem for a quantum 1/2 spin system is discussed. The system is assumed to be observed and controlled with a sampled data system. We introduce a discrete time stochastic model of the dynamics and propose a switching feedback controller. We show that the controller attains global stabilization at the target states under a condition on a sampling time by using a stochastic Lyapunov function for the discrete time systems. Finally, we demonstrate numerical simulations to show the effectiveness of our proposing controller.
This paper presents a detailed Lyapunov-based theory to control and stabilize continuously-measured quantum systems, which are driven by Stochastic Schrodinger Equation (SSE). Initially, equivalent classes of states of a quantum system are defined and their properties are presented. With the help of equivalence classes of states, we are able to consider global phase invariance of quantum states in our mathematical analysis. As the second mathematical modelling tool, the conventional Ito formula is further extended to non-differentiable complex functions. Based on this extended Ito formula, a detailed stochastic stability theory is developed to stabilize the SSE. Main results of this proposed theory are sufficient conditions for stochastic stability and asymptotic stochastic stability of the SSE. Based on the main results, a solid mathematical framework is provided for controlling and analyzing quantum system under continuous measurement, which is the first step towards implementing weak continuous feedback control for quantum computing purposes.