16 pags., 5 figs., 1 tab., 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., E R was supported by Nippon Telegraph and Telephone Corporation (NTT) Research during the early stages of this work. S A and
M K acknowledge support from the Knut and Alice Wallenberg Foundation through the Wallenberg Centre
for Quantum Technology (WACQT). M G L acknowledges financial support from the Spanish Ministerio de
Ciencia e Innovación, through project PID2020-115864RB-I00 and grant PRE2021-098697. F N is supported
in part by Nippon Telegraph and Telephone Corporation (NTT) Research, the Japan Science and Technology
Agency (JST) [via the Quantum Leap Flagship Program (Q-LEAP) and the Moonshot R&D Grant No.
JPMJMS2061], the Asian Office of Aerospace Research and Development (AOARD) (via Grant No.
FA2386-20-1-4069), and the Office of Naval Research Global (ONR) (via GrantNo. N62909-23-1-2074). C S
M acknowledges that the project that gave rise to these results received the support of a fellowship from ‘la
Caixa’ Foundation (ID 100010434) and from the European Union’s Horizon 2020 research and innovation
programme under the Marie Skłodowska-Curie Grant Agreement No.847648, with fellowship code
LCF/BQ/PI20/11760026, and financial support from the MCINN projects PID2021-126964OB-I00
(QENIGMA) and the Proyecto Sinérgico CAM 2020 Y2020/TCS- 6545 (NanoQuCo-CM). We acknowledge
the usage of the HOKUSAI BigWaterfall (HBW) cluster from the Information System Division of RIKEN.