We present an inference method utilizing artificial neural networks for
parameter estimation of a quantum probe monitored through a single continuous
measurement. Unlike existing approaches focusing on the diffusive signals
generated by continuous weak measurements, our method harnesses quantum
correlations in discrete photon-counting data characterized by quantum jumps.
We benchmark the precision of this method against Bayesian inference, which is
optimal in the sense of information retrieval. By using numerical experiments
on a two-level quantum system, we demonstrate that our approach can achieve a
similar optimal performance as Bayesian inference, while drastically reducing
computational costs. Additionally, the method exhibits robustness against the
presence of imperfections in both measurement and training data. This approach
offers a promising and computationally efficient tool for quantum parameter
estimation with photon-counting data, relevant for applications such as quantum
sensing or quantum imaging, as well as robust calibration tasks in
laboratory-based settings.Comment: 15 pages, 8 figures, code is available at
http://github.com/CarlosSMWolff/ParamEst-N