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arXiv:2602.19792

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

Mateusz Molenda, Lewis A. Clark, Marcin Płodzień, Jan Kolodynski

quant-phSubmitted Feb 23, 2026• Announced Feb 24, 2026• 0 citations• Revised Sep 25, 2026 (v2)
v1 · Feb 23, 2026v2 · Sep 25, 2026
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Abstract

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.

Comments
20 pages (+ appendices), 12 figures, local estimation performance analysis added, comments are always welcome

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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.
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Quantum-enhanced sensing typically relies on preparing specific entangled states. However, engineering these states in scalable many-body systems remains challenging due to the difficulty of designing robust preparation protocols and accounting for realistic noise. We show that evolving a simple product state under native time-independent Hamiltonian dynamics in Rydberg atom arrays generates complex many-body correlations that can potentially enable quantum-enhanced parameter estimation reaching the Heisenberg limit (HL). The key challenge is then shifted to extracting the metrological information encoded in spatial correlations of measurement patterns. We introduce a neural correlation learning framework that combines Bayesian inference with neural networks trained on calibration data. Operating in a two-stage calibration-and-sensing protocol, we numerically demonstrate that this framework effectively extracts many-body correlations to saturate the Cramér--Rao bound dictated by the classical Fisher information. Consequently, it achieves quantum-enhanced sensitivity surpassing the standard quantum limit (SQL). Furthermore, the learned estimator is robust to realistic noise, and remains accurate in the absence of rare measurement patterns. Our results establish a hardware-efficient and scalable paradigm for quantum sensing in interacting many-body systems without requiring engineered entangled states.
Abstract We show that ensembles of deep neural networks, called deep ensembles, can be used to perform quantum parameter estimation while also providing a means for quantifying uncertainty in parameter estimates, which is a key advantage of using Bayesian inference for parameter estimation that is lost when using existing machine learning methods. We show that optimizing for both accurate parameter estimates and well calibrated uncertainty estimates does not lead to degradation in the former as opposed to only optimizing for accuracy. We also show that the drift detection capabilities of these ensemble models can be used to detect drift in the experimental data used during inference. This approach is also shown to provide much faster inference time than both likelihood-based and likelihood-free Bayesian inference. These results suggest that such models could enable accurate, real-time parameter estimation with quantified uncertainty, making them promising candidates for deployment in experimental settings.