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arXiv:2403.04648 · IEEE Control Systems Letters

Online Parameter Estimation for Continuously Monitored Quantum Systems

Henrik Glavind Clausen, Pierre Rouchon, Rafał Wiśniewski

Published Jan 1, 2024• 4 citations
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Abstract

In this work, we consider the problem of online (real-time, single-shot) estimation of static or slow-varying parameters along quantum trajectories in quantum dynamical systems. Based on the measurement signal of a continuously monitored quantum system, we propose a recursive algorithm for computing the maximum likelihood (ML) estimate of unknown parameters using an approach based on stochastic gradient ascent on the log-likelihood function. We formulate the algorithm in both discrete-time and continuous-time and illustrate the performance of the algorithm through simulations of a simple two-level system undergoing homodyne measurement from which we are able to track multiple parameters simultaneously.

DOI
10.1109/lcsys.2024.3407608

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In this work, we consider the problem of online (real-time, single-shot) estimation of static or slow-varying parameters along quantum trajectories in quantum dynamical systems. Based on the measurement signal of a continuously monitored quantum system, we propose a recursive algorithm for computing the maximum likelihood (ML) estimate of unknown parameters using an approach based on stochastic gradient ascent on the log-likelihood function. We formulate the algorithm in both discrete-time and continuous-time and illustrate the performance of the algorithm through simulations of a simple two-level system undergoing homodyne measurement from which we are able to track multiple parameters simultaneously.
We investigate the determination of a Hamiltonian parameter in a quantum system undergoing continuous measurement. We demonstrate a computationally rapid method to estimate an unknown and possibly time-dependent parameter, where we maximize the likelihood of the observed stochastic readout. By dealing directly with the raw measurement record rather than the quantum-state trajectories, the estimation can be performed while the data are being acquired, permitting continuous tracking of the parameter during slow drifts in real time. Furthermore, we incorporate realistic nonidealities, such as decoherence processes and measurement inefficiency. As an example, we focus on estimating the value of the Rabi frequency of a continuously measured qubit and compare maximum likelihood estimation to a simpler fast Fourier transform. Using this example, we discuss how the quality of the estimation depends on both the strength and the duration of the measurement; we also discuss the trade-off between the accuracy of the estimate and the sensitivity to drift as the estimation duration is varied.
We review the introduction of likelihood functions and Fisher information in classical estimation theory, and we show how they can be defined in a very similar manner within quantum measurement theory. We show that the stochastic master equations describing the dynamics of a quantum system subject to a definite set of measurements provides likelihood functions for unknown parameters in the system dynamics, and we show that the estimation error, given by the Fisher information, can be identified by stochastic master equation simulations. For large parameter spaces we describe and illustrate the efficient use of Markov chain Monte Carlo sampling of the likelihood function.
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.
While typical online estimation methods can estimate the multiparameter dynamics of many systems, they may not be sufficient for a system with highly noisy measurement and rapid detection rate. In this paper, we create a hybrid estimation method by augmenting the sequential Monte Carlo (SMC) sampler, an online estimation method with an offline technique known as the time-batch estimation technique. By continuously monitoring the system, we may divide signals into batches and average them into an averaged trajectory. The system dynamics is then evolved with batch-averaged Kraus maps, for which we derive a highly efficient approximation. To facilitate the adoption of our algorithm, we present a modular derivation of the SMC methods and showcase our algorithm as an explicit example. We then implement our algorithm on the measurement signals obtained from superconducting-qubit experiments under two types of measurement setting: a fluorescence measurement and a dispersive zzz-measurement. The algorithm's hyperparameter values are chosen from independent numerical simulation, while the accuracy of our estimation is validated by a signal reconstruction method. Our results show that, for the fluorescence case, our algorithm can estimate the system's parameters better than the standard calibration method, and, for the dispersive case, our estimation is capable of finding an unexpected jump in parameter values that the standard calibration method could not find.
We consider the estimation of parameters encoded in the measurement record of a continuously monitored quantum system in the jump unraveling, corresponding to a single-shot scenario, where information is continuously gathered. Here, it is generally difficult to assess the precision of the estimation procedure via the Fisher Information due to intricate temporal correlations and memory effects. In this paper we provide a full set of solutions to this problem. First, for multi-channel renewal processes we relate the Fisher Information to an underlying Markov chain and derive a easily computable expression for it. For non-renewal processes, we introduce a new algorithm that combines two methods: the monitoring operator method for metrology and the Gillespie algorithm which allows for efficient sampling of a stochastic form of the Fisher Information along individual quantum trajectories. We show that this stochastic Fisher Information satisfies useful properties related to estimation on a single run. Finally, we consider the case where some information is lost in data compression/post-selection and provide tools for computing the Fisher Information in this case. All scenarios are illustrated with instructive examples from quantum optics and condensed matter.
We investigate parameter estimation in a Vconfiguration three-level atomic system continuously monitored via photon counting. By unraveling the system's dynamics into individual quantum trajectories, we demonstrate how time-resolved fluorescence records can be processed using maximum likelihood estimation (MLE) to extract key system parameters such as spontaneous decay rates and Rabi frequencies. We analyze the role of waiting-time distributions, likelihood reconstruction, and finite detector efficiencies, showing that trajectory-based MLE achieves the Cramér–Rao bound in the asymptotic limit. Our results highlight the potential of quantum-jump–resolved statistics to enable precision parameter estimation in open quantum systems.