Quantum correlation measurements have emerged as a powerful tool for localizing multiple emitters with subwavelength separation. Multi-emitter localization consists of two distinct tasks: identifying the emitter count $N$ and estimating their positions $\left\{x^{i}\right\}$. Existing fit-and-select frameworks fit the measured intensity and autocorrelation data using several models corresponding to different values of $N$ and then select the best-supported model. This procedure lets errors from the multi-parameter optimization of emitter positions propagate into the choice of model, degrading performance under photon-limited and high-background conditions. We propose a method that directly infers the emitter count and positions from the multimodal data via learned inference. Learned inference improves both the emitter-counting accuracy and localization precision compared to the fit-and-select method. It achieves more than tenfold improvement in precision and requires up to five times fewer photons to achieve practically meaningful localization with the largest gains observed under low photon counts, strong background, narrowly spaced emitter configurations, and spatially truncated data. This photon-efficient quantum-assisted localization framework offers a promising route toward real-time biological imaging and single-particle tracking in densely labeled samples.
Quantum correlation measurements have emerged as a powerful tool for localizing multiple emitters with subwavelength separation. Multi-emitter localization consists of two distinct tasks: identifying the emitter count N and estimating their positions {xi}. Existing fit-and-select frameworks fit the measured intensity and autocorrelation data using several models corresponding to different values of N and then select the best-supported model. This procedure lets errors from the multi-parameter optimization of emitter positions propagate into the choice of model, degrading performance under photon-limited and high-background conditions. We propose a method that directly infers the emitter count and positions from the multimodal data via learned inference. Learned inference improves both the emitter-counting accuracy and localization precision compared to the fit-and-select method. It achieves more than tenfold improvement in precision and requires up to five times fewer photons to achieve practically meaningful localization with the largest gains observed under low photon counts, strong background, narrowly spaced emitter configurations, and spatially truncated data. This photon-efficient quantum-assisted localization framework offers a promising route toward real-time biological imaging and single-particle tracking in densely labeled samples.
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.
Identifying single photon emitters from large, inhomogeneous candidate populations is key to realizing many quantum applications. This requires measuring the emitters second order autocorrelation function, whose statistical reliability is fundamentally limited by acquisition time. Machine learning classifiers can accelerate identification from sparse data, but their performance relative to physics-based inference has not been systematically examined. Here, we introduce sequential Bayesian inference for single photon emitter classification and benchmark it against Levenberg-Marquardt fitting and a feedforward neural network. We use synthetic training and test data calibrated against real Hanbury Brown-Twiss measurements from hexagonal boron nitride emitters, enabling evaluation against an exactly known ground-truth emitter number under realistic noise and background conditions. All three approaches achieve high, near-perfect accuracy with sufficient integration time, but differ greatly in convergence rate and robustness under sparse photon statistics. The neural network is most robust at short integration times. The Bayesian classifier reaches near-perfect accuracy fastest, while retaining full physical interpretability. Levenberg-Marquardt fitting remains a valuable, fully interpretable method, achieving the highest recall despite being the slowest to converge. These results lead to several key conclusions. No single method dominates across all performance metrics. Relying on any one metric alone can give a misleading picture of classifier performance, particularly under sparse photon statistics. Physics-based and data-driven methods are complementary rather than competing approaches. Together, these findings provide practical guidance for selecting and combining classification strategies for scalable single-photon-source screening and other quantum-emitter characterization tasks.
Characterization of the photon statistics of an optical field is central to identifying light sources in various quantum-enabled technologies, from super-resolved quantum imaging to precise quantum sensing and spectroscopy. In practice, however, estimating photon-correlation functions with sufficient accuracy requires long acquisition times and computationally intensive statistical estimators, severely limiting real-time operation and the study of rapidly varying or weak sources. Here, we show that learned inference based on artificial neural networks enables rapid estimation of second-order coherence directly from photon-number-resolved measurements, outperforming the precision of conventional statistical methods with processing times three orders of magnitude shorter. Our approach is source-agnostic and remains accurate in few-photon illumination regimes where the statistics of coherent, partially coherent, thermal, and super-thermal light become nearly indistinguishable. We experimentally validate the method across more than 4000 distinct classical light sources, demonstrating equal or improved precision with drastically reduced data and computational requirements. These results establish machine-learning-enabled photon-statistics inference as a new operational route to rapid optical-coherence diagnostics, enabling real-time monitoring of classical light sources in regimes previously inaccessible to conventional estimators.
Abstract 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.
The advancement and scaling of quantum technology has made the learning and identification of quantum systems and devices in highly multidimensional parameter spaces a pressing task for a variety of applications. In many cases, the integration of real-time feedback control and adaptive choice of measurement settings places strict demands on the speed of this task. Here we present a joint model selection and parameter estimation algorithm that is fast and operable on a large number of model parameters. The algorithm is based on variational Bayesian inference, which approximates the target posterior distribution by optimizing a tractable family of distributions, making it more scalable than exact inference methods relying on sampling and that generally suffer from high variance and computational cost in high-dimensional spaces. We show how a regularizing prior can be used to select between competing models, each comprising a different number of parameters, identifying the simplest model that explains the experimental data. The regularization can further separate the degrees of freedom, e.g., quantum systems in the environment or processes, which contribute to major features in the observed dynamics, with respect to others featuring small coupling, which only contribute to a background. As an application of the introduced framework, we consider the problem of the identification of multiple individual nuclear spins with a single electron spin quantum sensor, relevant for nanoscale nuclear magnetic resonance and for the implementation of multiqubit quantum networking nodes. With the number of environmental spins unknown a priori, our Bayesian approach is able to correctly identify the model, i.e., the number of spins and their couplings. We benchmark the algorithm on both simulated and experimental data, using standard figures of merit, and demonstrate that we can estimate dozens of parameters within minutes. Our methodology is compatible with the implementation of real-time adaptive choice of experimental settings in multiparameter estimation, which has the potential to greatly decrease the estimation time in a variety of metrology, tomography, and characterization tasks by prioritizing the measurements with the highest content of information.
Identification of a suitable source of single photons via second-order autocorrelation function measurement within an array of thousand possible candidates is a routine, key step of any practical realization in quantum optics. Within this work, we have shown that machine learning algorithms enable high precision classification between “single” and “not single” quantum emitters based on sparse autocorrelation data measurement and require < 1 s acquisition time, while conventional methods demand > 1 min. Machine learning assisted classification, done on a sparse 1-s dataset, provides ~85% accuracy of “single”/“not single” emitter identification versus only 57% of the conventional Levenberg-Marquardt approach.