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arXiv:2004.05638 · IEEE Conference on Decision and Control (CDC)

On the robustness of stabilizing feedbacks for quantum spin-1/2 systems

Weichao Liang, Nina H. Amini, Paolo Mason

Published Dec 14, 2020• 14 citations
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Abstract

In this paper, we consider stochastic master equations describing the evolution of quantum spin-1/2 systems interacting with electromagnetic fields undergoing continuous-time measurements. We suppose that the initial states and the exact values of the physical parameters are unknown. We prove that the feedback stabilization strategy considered in [1] is robust to these imperfections. This is shown by studying the asymptotic behavior of the coupled stochastic master equations describing the evolutions of the actual state and the estimated one under appropriate assumptions on the feedback controller. We provide sufficient conditions on the feedback controller and a valid domain of estimated parameters which ensure exponential stabilization of the coupled system. Furthermore, our results allow us to answer positively to [2], [Conjecture 4.4] in the case of spin-1/2 systems with unknown initial states, even in presence of imprecisely known physical parameters.

DOI
10.1109/cdc42340.2020.9304009

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In this paper, we consider stochastic master equations describing the evolution of quantum spin-1/2 systems interacting with electromagnetic fields undergoing continuous-time measurements. We suppose that the initial states and the exact values of the physical parameters are unknown. We prove that the feedback stabilization strategy considered in [1] is robust to these imperfections. This is shown by studying the asymptotic behavior of the coupled stochastic master equations describing the evolutions of the actual state and the estimated one under appropriate assumptions on the feedback controller. We provide sufficient conditions on the feedback controller and a valid domain of estimated parameters which ensure exponential stabilization of the coupled system. Furthermore, our results allow us to answer positively to [2], [Conjecture 4.4] in the case of spin-1/2 systems with unknown initial states, even in presence of imprecisely known physical parameters.
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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.
In this article, we consider stochastic master equations describing the evolution of a multiqubit system interacting with electromagnetic fields undergoing continuous-time measurements. By considering multiplezzz-type (Paulizzzmatrix on different qubits) andxxx-type (Paulixxxmatrix on all qubits) measurements and one control Hamiltonian, we provide general conditions on the feedback controller and the control Hamiltonian ensuring almost sure exponential convergence to a predetermined Greenberger–Horne-Zeilinger (GHZ) state, which is assumed to be a common eigenstate of the measurement operators. We provide explicit expressions of feedback controllers satisfying such conditions. We also consider the case of onlyzzz-type measurements and multiple control Hamiltonians. We show that local stability in probability holds true, however due to the absence of random displacements generated byxxx-type measurements, the reachability of a neighborhood of a predetermined GHZ state is not clear. In this case, we provide a heuristic discussion on some conditions which may ensure asymptotic convergence toward the target state. Finally, we demonstrate the effectiveness of our methodology for a three-qubit system through numerical simulations.
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In this paper, we present several refinements of the results in [1] concerning feedback stabilization of invariant subspaces of quantum trajectories. First, we consider a more realistic dynamics that accounts for potential measurement imperfections. Second, the strong controllability assumption can be replaced by a simpler and more natural condition whenever the channel is primitive on its minimal invariant subspaces. Finally, we show that the control strategy can be based on the estimated trajectory (starting from an estimated initial state) while still ensuring exponential stabilization towards the target subspace at the same rate, thereby demonstrating the robustness of the approach.
We consider the question of asymptotic stability of quantum trajectories undergoing quantum non-demolition imperfect measurement, that is to say the convergence of the estimated trajectory towards the true trajectory whose parameters and initial state are not necessarily known. We give conditions on the estimated initial state and regions of validity for the estimated parameters so that this convergence is ensured. We illustrate these results through numerical simulations on the physical example [1] and discuss the asymptotic stability for a more realistic general case where decoherence acts on the system. In this case, the evolution is described by new Kraus operators which do not satisfy the quantum non-demolition property.