Unlocking photodetection for quantum sensing with Bayesian likelihood-free methods and deep learning

To operate quantum sensors at their quantum limit in real time, it is crucial to identify efficient data inference tools for rapid parameter estimation. In photodetection, the key challenge is the fast interpretation of click-patterns that exhibit non-classical statistics-the very features responsible for the quantum enhancement of precision. We achieve this goal by comparing Bayesian likelihood-free methods with ones based on deep learning (DL). While the former are more conceptually intuitive, the latter, once trained, provide significantly faster estimates with comparable precision and yield similar predictions of the associated errors, challenging a common misconception that DL lacks such capabilities. We first verify both approaches for an analytically tractable, yet multiparameter, scenario of a two-level system emitting uncorrelated photons. Our main result, however, is the application to a driven non-linear optomechanical device emitting non-classical light with complex multiclick correlations. Although both methods facilitate inference, only DL provides the execution speed required for real-time performance while naturally learning complex multi-photon click-patterns. Our results pave the way for real-time sensing and dynamical control of quantum devices that leverage non-classical effects in photodetection.