Abstract. This paper considers a simplified model of open quantum systems undergoing imperfect measurements obtained via a projection filter approach. We use this approximate filter in the feedback stabilization problem, specifically in the case of quantum nondemolition (QND) measurements. The feedback design relies on the structure of the exponential family utilized for the projection process. We demonstrate that the introduced feedback guarantees exponential convergence of the original filter equation toward a predefined target state, corresponding to an eigenstate of the measurement operator.
Abstract. This paper considers a simplified model of open quantum systems undergoing imperfect measurements obtained via a projection filter approach. We use this approximate filter in the feedback stabilization problem, specifically in the case of quantum nondemolition (QND) measurements. The feedback design relies on the structure of the exponential family utilized for the projection process. We demonstrate that the introduced feedback guarantees exponential convergence of the original filter equation toward a predefined target state, corresponding to an eigenstate of the measurement operator.
In this paper, we examine an open quantum system that is subject to indirect and imperfect measurements. Specifically, for quantum non-demolition (QND) measurements, we show that the evolution of the system is confined to a properly selected manifold. This allows us to express the exact solution of the quantum filter equation in terms of a lower-dimensional stochastic differential equation, significantly reducing the computational complexity. Motivated by a quantum state reduction result for the predicted dynamics under QND measurement, we propose, in the case when the Hamiltonian and the coupling operator do not commute, a projection filter approach for the quantum state estimation. The efficiency of the proposed quantum projection filter is verified by numerical simulations. The latter suggest that, even when paired with a stabilizing feedback control mechanism that depends on the projected state, the filter maintains its robustness. More generally, our numerical results suggest the effectiveness of the adopted quantum projection filter approach as a method for state estimation and feedback stabilization of open quantum systems.
In this letter, we consider an open quantum system undergoing imperfect and indirect measurement. For quantum non-demolition (QND) measurement, we show that the system evolves on an appropriately chosen manifold and we express the exact solution of the quantum filter equation in terms of the solution of a lower dimensional stochastic differential equation. In order to further reduce the dimension of the system under study, we consider the projection on the lower dimensional manifold originally introduced in Gao et al. (2019) for the case of perfect measurements. An error analysis is performed to evaluate the precision of this approximate quantum filter, focusing on the case of QND measurement. Simulations suggest the efficiency of the proposed quantum projection filter, even in presence of a stabilizing feedback control which depends on the projection filter.
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.
This review develops a control-theoretic perspective on measurement-based feedback for continuously monitored open quantum systems, with the main analysis focused on finite-dimensional systems governed by diffusive stochastic master equations. We introduce the relevant state-space, invariant-subspace, and quantum non-demolition structures, interpret quantum filtering as nonlinear observer dynamics, and review open-loop asymptotics, filter stability, state-feedback stabilization, robustness, and reduced-order observer-based control. Particular emphasis is placed on a recurrence--contraction framework, in which Hamiltonian feedback removes non-target invariant obstructions while measurement-induced dynamics provide local exponential contraction. Although the detailed analysis is developed for finite-dimensional diffusive models, the underlying measurement--estimation--feedback architecture is relevant across a broad range of quantum platforms. We further discuss implementation challenges and open problems involving scalable estimation, sampling and delay, adaptation, hybrid and non-Markovian dynamics, practical stability, optimal control, and learning-based design. By organizing these developments around invariance, estimation, recurrence, and contraction, the review provides a tutorial bridge between measurement-based quantum feedback and nonlinear stochastic control.
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.
The exponential quantum projection filtering algorithm is applied in measurement-based quantum feedback control and a specific switching control law is designed to realize the rapid preparation of a target Bell state. First, we use an exponential projection filter to obtain an approximate estimate of the system state, which can reduce the calculation time spent in the measurement process. Then, the quantum state space is partitioned into two subsets to separate the target state from other equilibrium states, two Lyapunov functions are constructed and the corresponding control laws are designed in the two subsets. It is shown that the switching control scheme in this paper ensures that the system trajectory crosses the boundary of the two subsets no more than twice from any initial state, which effectively reduces the time that the system converges to the target Bell state. Numerical simulation on a two-qubit system verifies the effectiveness of the proposed control scheme.