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Proceedings of the ... American Control Conference/Proceedings of the American Control Conference

An Adaptive Observer Design for State and Parameter Estimation of Quantum Systems via Averaging Theory

Eram Taslima, Shyam Kamal, Ram Khelawan Saket, Thach Ngoc Dinh

Published Jul 8, 2025• 0 citations
Publisher

Abstract

This paper presents an adaptive asymptotic observer-based approach for estimating the states and the atom-laser coupling constant (parameter) of a two-level, closed quantum system driven by a constant control input. Using the interaction frame and averaging theory, the proposed observer ensures asymptotic estimation of both the system states and the unknown parameter. In addition to the asymptotic observer, we also propose an adaptive finite-time observer for the estimation of states in finite time, considering the parameter is unknown. The proposed finite-time observer is designed to provide the minimum output horizon required for accurate state and parameter estimation. To ensure the estimation of the unknown parameter, the extended Lyapunov method is employed. Through simulations on a two-level quantum dynamical system, the proposed observers are demonstrated to exhibit robust performance in the presence of noise in the control and output measurement.

DOI
10.23919/acc63710.2025.11107918

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This paper presents an adaptive observer-based framework for the simultaneous state and parameter estimation of a closed two-level quantum system governed by the Liouville–von Neumann equation. The system, driven by an external laser field with unknown atom-field coupling strength, is reformulated in the Bloch sphere representation, wherein the nonlinear dynamics of the spin variables are described. To estimate the unmeasured internal states and the unknown coupling parameter, we propose a cascaded strategy combining a nonlinear adaptive observer with the Dynamic Regressor Extension and Mixing (DREM) technique. The observer reconstructs the full Bloch vector from partial measurement (population difference), and the resulting regressors enable a scalar DREM formulation for the unknown parameter. Unlike traditional gradient-based identification, the DREM-based update ensures improved noise robustness and accelerated convergence even for scalar parameters. Numerical simulations confirm the accuracy and robustness of the proposed estimation strategy, demonstrating effective reconstruction of both the quantum state and the Hamiltonian parameter under realistic noisy measurements.
This paper introduces an observer-based parameter estimation technique for two-level quantum systems, incorporating an observer parameter to accelerate convergence. The proposed approach outperforms existing methods by achieving faster convergence, even when initial parameter estimates are significantly inaccurate. The convergence of the observer is rigorously established using the averaging technique. The method is particularly advantageous when estimation must be performed with the limited time output data. Additionally, the algorithm demonstrates strong robustness against noise in both control inputs and measurement outputs. Furthermore, the simulation results show that the proposed framework can be extended to multi-level quantum systems as well, making it a versatile tool for quantum parameter estimation.
Accurate estimation of unknown quantum states using limited number of samples is very important but difficult task using fixed measurement basis states since the best local unbiased estimator (LUE) for estimating the parameter depends, in general, on the unknown parameter itself. In order to solve this problem, Nagaoka advocated an adaptive quantum state estimation (AQSE) procedure [1], where the measurement basis states are adaptively selected for each measurement result. Fujiwara proved the strong consistency and asymptotic efficiency for AQSE [2]. Recently, AQSE has been demonstrated by us for the linear polarizations of photons [3]. However, AQSE was designed for a static quantum state, and was not applicable to dynamic states where quantum states change in time. Here, we propose a new protocol, sequential adaptive quantum state estimation (SAQSE) for dynamic quantum states. Our numerical simulation (shown below) clearly show that the estimated value given by SAQSE can follow the true value changing in time whereas AQSE fails. We believe this new protocol will open up the use of quantum state estimation for various applications including biology, e.g. monitoring the motion of fluorescent molecules.
Continuous measurement of quantum systems gives rise to stochastic dynamics of the conditional quantum state, described by diffusive stochastic master equations. In this paper, we study parameter estimation for such equations when the Hamiltonian and measurement operators depend on unknown parameters. Based on multiple independent observed trajectories with a known initial state, we construct a contrast function using the deterministic averaged state and define a maximum contrast estimator for the unknown parameter. We prove strong consistency and asymptotic normality of the parameter estimator in a fixed-time, many-trajectory asymptotic regime. A key point is that the covariance matrix appearing in the asymptotic normality is given in a form that naturally leads to a consistent covariance estimator. This covariance estimator is computable from the observed data together with the deterministic averaged dynamics, so the asymptotic normality result can be used to construct standard errors and assess uncertainty for the parameter estimator.
We develop a model-free framework for stabilizing quantum states using only empirical finite-difference evaluations of a measurement-derived Lyapunov observable. The controller requires no knowledge of the Hamiltonian, dissipative structure, or generator of the dynamics, and relies solely on discrete measurement data. The approach combines three key elements: sign-based Lyapunov descent, adaptive gain amplification, and a finite-difference analogue of LaSalle's invariance principle. We provide rigorous conditions under which these mechanisms guarantee asymptotic stabilization along the sampling instants in the drift-free case and practical input-to-state stability (ISS) in the presence of unknown drift and noise. The resulting feedback law is simple, derivative-free, and experimentally feasible. A qubit example illustrates the complete closed-loop scheme and the predicted ISS-type behavior. Although demonstrated on a single qubit, the theory applies to arbitrary finite-dimensional quantum systems and offers a foundation for further developments in stochastic, subspace, and multi-qudit model-free quantum control.
Research in the field of quantum control usually assumes complete knowledge of the state and parameters. In this paper, a driven and damped two-level quantum system under continuous observation is considered. Its time evolution is governed by an operator-valued stochastic master equation and conditioned on the recorded measurement data. The system is transformed to state space representation and its observability is proven analytically. An extended Kalman filter is employed in order to both reconstruct the density operator describing the full state of the system, and to dynamically identify key parameters. Convincing results are obtained in numerical simulations. Overall, a practical engineering approach to quantum control is presented.