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SIAM Journal on Control and Optimization

Quantum Measurement-Based Feedback Control: A Nonsmooth Time Delay Control Approach

Shuzhi Sam Ge, Thanh Long Vu, Tong Heng Lee

Published Jan 1, 2012• 25 citations
Publisher

Abstract

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-$1/2$ systems. Simulation results are presented and discussed to show the effectiveness of the proposed control.

DOI
10.1137/100801287

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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.
Feedback control (based on the quantum continuous measurement) of quantum systems inevitably suffers from estimation delays. In this technical note we give a delay-dependent stability criterion for a wide class of nonlinear stochastic systems including quantum spin systems. We utilize a semi-algebraic problem approach to incorporate the structure ofdensitymatrices. To show the effectiveness of the result, we derive a globally stabilizing control law for a quantum spin-1/2 systems in the face of feedback delays.
Feedback control of quantum mechanical systems must take into account the probabilistic nature of quantum measurement. We formulate quantum feedback control as a problem of stochastic nonlinear control by considering separately a quantum filtering problem and a state feedback control problem for the filter. We explore the use of stochastic Lyapunov techniques for the design of feedback controllers for quantum spin systems and demonstrate the possibility of stabilizing one outcome of a quantum measurement with unit probability.
In this paper, feedback control of quantum spin systems is investigated. The quantum control scenario follows the formalism of continuous quantum measurement and quantum state estimation using the so-called stochastic master equation. In quantum feedback control implementations the state estimation is delayed, which is known to degrade the resulting control performance. As yet there are no theoretical means to take such estimation delays into consideration. In this paper we derive a globally stabilizing control law for two-dimensional quantum spin systems taking estimation delays into explicit account. From a control theoretic viewpoint, the obtained result is a delay-dependent stability criterion for a nonlinear stochastic switching system.
Bell states are the maximally entangled states in two-qubit quantum systems, and play a significant role in quantum computation and quantum communication. Feedback control of stochastic quantum systems usually suffers from time-delay problems caused by the computation time for filter states and the control input. This paper addresses the preparation of Bell states in two-qubit systems with a constant delay time. A Lyapunov method is used to design a switching control law and a constant is introduced to compensate for the computation time of the estimated states and the feedback control input, thereby stabilizing the target Bell state globally. Numerical results show the effectiveness of the proposed control law.
Quantum control concerns driving quantum systems to the desired properties or states. This objective is hindered by the non-deterministic nature of quantum systems and the symmetric topology of the quantum state space. This paper introduces a coupling Lyapunov functions approach to counter for the stochastic nature of quantum measurements and break the symmetric topology of resulting filter state space. As an example, we then apply it to synthesize non-smooth control almost surely globally stabilizing the desired Bell state-the maximally entangled two-qubit state with various applications. It turns out that this approach enables the short time convergence of the system state, which is essential in the control of entanglement. More interestingly, its combination with the generalized concept of SWM (simultaneous weak measurement)-induced quantum state reduction allows us to deterministically generate the maximally entangled Bell state with little non-local effects. We use numerical illustration to validate the proposed scheme.