Online Parameter Estimation for Continuously Monitored Quantum Systems

TL;DR

This work tackles online maximum-likelihood parameter estimation for static or slowly varying parameters in continuously monitored quantum systems described by stochastic master equations. It develops a recursive online gradient-ascent framework based on the quantum filter and its sensitivity equations, applicable to both discrete-time Kraus-map updates and continuous-time diffusive SMEs. Through a two-level system under homodyne monitoring, the method demonstrates simultaneous tracking of multiple parameters and real-time adaptability, with learning-rate choices balancing tracking speed and robustness to quantum noise. The approach leverages the tractable finite-dimensional quantum filtering solution to enable online ML without resorting to full offline batch processing. This has potential implications for real-time calibration and high-precision quantum sensing, where parameters evolve slowly yet must be inferred from noisy measurement records.

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 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.

Figures (1)

• Figure 1: Two examples of sample trajectories for the parameter estimates for the two-level quantum system described by the diffusive master equation \ref{['eq:quantum_sys']} with Hamiltonian and Lindblad operator given by \ref{['eq:sim_example']}. The learning rate in both examples is constant and given by $\gamma=\frac{\kappa_0}{1000}=10^{-4}$. Simulations are done using the discrete-time model \ref{['eq:full_online_GA_DT']} and Kraus map \ref{['eq:sde_kraus']} with a time-discretization of $\Delta t=10^{-2}$. The solid lines correspond to the estimated parameters, whereas the dashed lines correspond to the true parameter values.