State Estimation and Control for Stochastic Quantum Dynamics with Homodyne Measurement: Stabilizing Qubits under Uncertainty (2403.07021v3)

Abstract: This paper introduces a Lyapunov-based control approach with homodyne measurement. We study two filtering approaches: (i) the traditional quantum filtering and (ii) a modified version of the extended Kalman filtering. We examine both methods in order to directly estimate the evolution of the coherence vector elements, using sequential homodyne current measurements. The latter case explicitly addresses the dynamics of a stochastic master equation with correlated noise, transformed into a state-space representation, ensuring by construction the quantum properties of the estimated state variable. In addition, we consider the case where the quantum-mechanical Hamiltonian is unknown, and the system experiences uncertainties. In this case, we show as expected that both filters lose performance, exhibiting large expected estimation errors. To address this problem, we propose a simple multiple model estimation scheme that can be directly applied to any of the studied filters. We then reconstruct the estimated density operator ( \hat{\rho} ), describing the full state of the system, and subject it to a control scheme. The proposed switching-based Lyapunov control scheme, which is fed with ( \hat{\rho} ), guarantees noise-to-state practically stable in probability of the desired stationary target set with respect to the estimation error variance. We demonstrate our approach's efficacy in stabilizing a qubit coupled to a leaky cavity under homodyne detection in the presence of uncertainty in resonance frequency.