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arXiv:2007.04211 · SIAM Journal on Control and Optimization

Robust Feedback Stabilization of N -Level Quantum Spin Systems

Weichao Liang, Nina H. Amini, Paolo Mason

Published Jan 1, 2021• 34 citations
arXivHTMLPDF

Abstract

In this paper, we consider $N$-level quantum angular momentum systems interacting with electromagnetic fields undergoing quantum nondemolition measurements in continuous-time. We suppose unawareness of the initial state and physical parameters, entailing the introduction of an additional state representing the estimated quantum state. The evolution of the quantum state and its estimation is described by a coupled stochastic master equation. Here, we study the asymptotic behavior of such a system in the presence of a feedback controller. We provide sufficient conditions on the feedback controller and on the estimated parameters that guarantee exponential stabilization of the coupled stochastic system toward an eigenstate of the measurement operator. Furthermore, we estimate the corresponding rate of convergence. We also provide parametrized feedback laws satisfying such conditions. Our results show the robustness of the feedback stabilization strategy considered in [W. Liang, N. H. Amini, and P. Mason, SIAM J. Control Optim., 57 (2019), pp. 3939--3960] in the case of imprecise initialization of the estimated state and with respect to the unknown physical parameters.

DOI
10.1137/20m1348972

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In this paper, we consider NNN-level quantum angular momentum systems interacting with electromagnetic fields undergoing quantum nondemolition measurements in continuous-time. We suppose unawareness of the initial state and physical parameters, entailing the introduction of an additional state representing the estimated quantum state. The evolution of the quantum state and its estimation is described by a coupled stochastic master equation. Here, we study the asymptotic behavior of such a system in the presence of a feedback controller. We provide sufficient conditions on the feedback controller and on the estimated parameters that guarantee exponential stabilization of the coupled stochastic system toward an eigenstate of the measurement operator. Furthermore, we estimate the corresponding rate of convergence. We also provide parametrized feedback laws satisfying such conditions. Our results show the robustness of the feedback stabilization strategy considered in [W. Liang, N. H. Amini, and P. Mason, SIAM J. Control Optim., 57 (2019), pp. 3939--3960] in the case of imprecise initialization of the estimated state and with respect to the unknown physical parameters.
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On Exponential Stabilization of NNN-Level Quantum Angular Momentum Systems
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In this paper, we consider stochastic master equations describing the evolutions of quantum systems interacting with electromagnetic fields undergoing continuous-time measurements. In particular, we study feedback control of quantum spin1/2 systems in the case of unawareness of initial states and in presence of measurement imperfections. We prove that the fidelity between the true quantum filter and its estimation (with arbitrary initial state) converges to one under appropriate assumption on the feedback controller. This shows the asymptotic convergence of such filters. In the more general case of spin-J systems, we discuss heuristically the asymptotic behavior of the true quantum filter and the associated estimation, and the possibility of exponentially stabilizing such systems towards an eigenvector of the measurement operator by an appropriate feedback.
Stabilizing control synthesis is one of the central subjects in control theory and engineering, and it always has to deal with unavoidable uncertainties in practice. In this study, we propose an adaptive parameter tuning algorithm for robust stabilizing quantum feedback control of N-level quantum angular momentum systems with a robust stabilizing controller proposed by [Liang, Amini, and Mason, SIAM J. Control Optim., 59 (2021), pp. 669-692]. The proposed method ensures local convergence to the target state. Besides, numerical experiments indicate its global convergence if the learning parameters are adequately determined.
The exponential stabilization of the eigenstates for two-level quantum systems based on continuous noise-assisted feedback control is investigated in this paper. Due to the symmetric structure of state space, quantum systems may converge to an undesired eigenstate under continuous measurement. In order to overcome the convergence obstacle, the continuous noise-assisted feedback strategy, in which the control signal is driven by an exogenous Brownian noise, is proposed to make the quantum systems converge to the target eigenstates exponentially, and the noise gain is designed as a continuously differentiable function of quantum state, which is negatively related to the distance between system state and the target eigenstate. In particular, the exponential convergence is proved based on the stability theory of stochastic differential equations. Moreover, the numerical simulations on a two-level quantum system are performed to verify the effectiveness of noise-assisted feedback control and study the influences of physical parameters and control parameters on convergence rate.
In practical applications, quantum systems are inevitably subject to significant uncertainties, including unknown initial states, imprecise physical parameters, and unmodeled environmental noise, all of which pose major challenges to robust quantum feedback control. This paper proposes a feedback stabilization strategy based on a reduced quantum filter that achieves robustness against time-varying Hamiltonian perturbations and additional dissipative effects, without requiring prior knowledge of the initial state or exact system parameters. The proposed filter estimates only O(N) real variables corresponding to the diagonal elements of the system density matrix in a quantum non-demolition basis in contrast to the O(N^2) variables required by a full stochastic master equation, where N is the Hilbert space dimension. This dimensionality reduction substantially simplifies real-time computation and feedback implementation while preserving both convergence and robustness guarantees. Rigorous analysis further establishes global exponential stability of the target subspace. The results provide a scalable framework for robust and efficient measurement-based feedback control applicable to high-dimensional perturbed open quantum systems.