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IEEE Conference on Decision and Control (CDC)

Parameter Estimation for Quantum Stochastic Systems

Weichao Liang, Shuixin Xiao

Published Dec 16, 2024• 0 citations
Publisher

Abstract

This paper addresses the problem of parameter estimation for quantum systems undergoing homodyne detection. By generalizing the stability of quantum filters under quantum non-demolition measurements to cases with mismatched parameters, we propose an estimation approach that extends beyond finite parameter sets. Leveraging maximum likelihood estimation, our method offers improved reliability, flexibility, and potential computational savings compared to existing techniques. The asymptotic behavior of the estimator is rigorously analyzed, and simulations on spin- $\frac{1}{2}$ systems demonstrate the effectiveness and robustness of the proposed approach.

DOI
10.1109/cdc56724.2024.10886489

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This paper addresses the problem of parameter estimation for quantum systems undergoing homodyne detection. By generalizing the stability of quantum filters under quantum non-demolition measurements to cases with mismatched parameters, we propose an estimation approach that extends beyond finite parameter sets. Leveraging maximum likelihood estimation, our method offers improved reliability, flexibility, and potential computational savings compared to existing techniques. The asymptotic behavior of the estimator is rigorously analyzed, and simulations on spin- 12\frac{1}{2}21​ systems demonstrate the effectiveness and robustness of the proposed approach.
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This paper introduces a Lyapunov-based control strategy alongside two filtering methods for controlling and estimating the evolution of coherence vector elements from sequential homodyne measurements. The methods include traditional quantum filtering and a novel extended Kalman filter, which explicitly addresses the dynamics of a stochastic master equation with correlated noise, thereby ensuring the quantum properties of the estimated state variable by design. We also explore scenarios where the system’s Hamiltonian is unknown, demonstrating that both filters exhibit reduced performance with increased estimation errors. To address this, we propose a multiple model estimation scheme applicable to either filter. The estimated density operator is then controlled using the proposed switching-based Lyapunov scheme, which guarantees noise-to-state practical stability in probability. We demonstrate the effectiveness of our approach in stabilizing a qubit coupled to a leaky cavity under homodyne detection, even with uncertainty in the resonance frequency.
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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.
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