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arXiv:2504.18205 · Physical Review Applied

Estimation of the second-order coherence function using quantum reservoir and ensemble methods

Dogyun Ko, Stanisław Świerczewski, Andrzej Opala, Michał Matuszewski, Amir Rahmani

Published Nov 19, 2025• 1 citation
arXivHTMLPDF

Abstract

We propose a machine-learning-based approach enhanced by quantum reservoir computing (QRC) to estimate the zero-time second-order correlation function ${g}^{(2)}(0)$. Typically, measuring ${g}^{(2)}(0)$ requires single-photon detectors and time-correlated measurements. Machine learning may offer practical solutions by training a model to estimate ${g}^{(2)}(0)$ solely from average intensity measurements. In our method, emission from a given quantum source is first processed in QRC. During the inference phase, only intensity measurements are used, and these are then passed to a software-based decision-tree ensemble model. We evaluate this hybrid quantum-classical approach across a variety of quantum optical systems and demonstrate that it provides accurate estimates of ${g}^{(2)}(0)$. We further extend our analysis to assess the ability of a trained model to generalize beyond its training distribution, both to the same system under different physical parameters and to fundamentally different quantum sources. While the model may yield reliable estimates within specific regimes, its performance across distinct systems is generally limited.

DOI
10.1103/ty2d-kgkr

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We propose a machine-learning-based approach enhanced by quantum reservoir computing (QRC) to estimate the zero-time second-order correlation function g(2)(0){g}^{(2)}(0)g(2)(0). Typically, measuring g(2)(0){g}^{(2)}(0)g(2)(0) requires single-photon detectors and time-correlated measurements. Machine learning may offer practical solutions by training a model to estimate g(2)(0){g}^{(2)}(0)g(2)(0) solely from average intensity measurements. In our method, emission from a given quantum source is first processed in QRC. During the inference phase, only intensity measurements are used, and these are then passed to a software-based decision-tree ensemble model. We evaluate this hybrid quantum-classical approach across a variety of quantum optical systems and demonstrate that it provides accurate estimates of g(2)(0){g}^{(2)}(0)g(2)(0). We further extend our analysis to assess the ability of a trained model to generalize beyond its training distribution, both to the same system under different physical parameters and to fundamentally different quantum sources. While the model may yield reliable estimates within specific regimes, its performance across distinct systems is generally limited.
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.
Forecasting dynamical systems is of importance to numerous real-world applications. Typically, these are obtained using first-principles-based models such as through the use of differential equations. When these equations are unknown, non-intrusive techniques must be utilized to build predictive models from data alone. Machine learning (ML) methods have recently been used for such tasks with the added advantage of significant reductions in time-to-solution for predictions during deployment. However, many state-of-the-art ML-based methods for forecasting rely on neural networks, which are expensive to train and require large amounts of memory. In this work, we propose a quantum mechanics inspired ML strategy for forecasting nonlinear dynamical systems that results in reduced training time and memory requirements. This approach, denoted the quantum reservoir computing technique (QRC), is a hybrid quantum-classical framework employing an ensemble of interconnected small quantum systems via classical linear feedback connections. By mapping the dynamical state to a suitable quantum representation amenable to unitary operations, QRC is able to predict complex nonlinear dynamical systems in a stable and accurate manner. We demonstrate the efficacy of this framework through benchmark forecasts of the NOAA Optimal Interpolation Sea Surface Temperature dataset and compare the performance of QRC to other ML methods.
A distinctive feature of Quantum Reservoir Computing (QRC) is the ability to directly embed quantum input states into the reservoir dynamics. However, the resulting output is fundamentally linear for a single input state, preventing QRC from naturally computing nonlinear functionals such as purity or entropy. We overcome this limitation with a quantum-classical hybrid architecture combining a qubit quantum reservoir with a classical echo state network (ESN), allowing both nonlinear functional approximation and effective temporal processing. We systematically study performance under two information regimes: full-tomography and partial information (single-axis measurements), with the latter demonstrating that the hybrid system outperforms its standalone components in both linear and nonlinear tasks due to the enhanced information retrieval provided by the quantum reservoir. Building on these results, we apply an online monitoring protocol that explicitly accounts for measurement back-action and finite measurement ensembles, enabling a realistic assessment of performance under experimental conditions. These results establish hybrid quantum-classical reservoir computing (HRC) architectures as a practical and scalable route for enhanced quantum machine learning on near-term qubit hardware.
Reservoir computing (RC) is an effective method for predicting chaotic systems by using a high-dimensional dynamic reservoir with fixed internal weights, while keeping the learning phase linear, which simplifies training and reduces computational complexity compared to fully trained recurrent neural networks (RNNs). Quantum reservoir computing (QRC) uses the exponential growth of Hilbert spaces in quantum systems, allowing for greater information processing, memory capacity, and computational power. However, the original QRC proposal requires coherent injection of inputs multiple times, complicating practical implementation. We present a hybrid quantum-classical approach that implements memory through classical post-processing of quantum measurements. This approach avoids the need for multiple coherent input injections and is evaluated on benchmark tasks, including the chaotic Mackey-Glass time series prediction. We tested our model on two physical platforms: a fully connected Ising model and a Rydberg atom array. The optimized model demonstrates promising predictive capabilities, achieving a higher number of steps compared to previously reported approaches.
Quantum reservoir computing has emerged as a promising paradigm within the field of quantum machine learning, harnessing the inherent properties of quantum systems to optimise and enhance information processing capabilities. Here, we explore the potential of quantum-inspired machine learning methodologies by leveraging the complex dynamics of quantum reservoirs to address computationally challenging tasks with enhanced efficiency and accuracy. To this end, we employ an open quantum system model comprising two-level atomic ensembles coupled to Lorentzian photonic cavities to construct a quantum physical reservoir computer layer for a recurrent neural network. We evaluate the effectiveness of this approach by applying it to a standard machine learning image-recognition problem and benchmarking its performance against a conventional neural network of similar architecture, but lacking the quantum physical reservoir computer layer. Remarkably, as the dataset size increases, the quantum physical reservoir computer outperforms the conventional neural network, requiring fewer training epochs and a smaller dataset to achieve comparable accuracy. Furthermore, we employ the quantum physical reservoir computing approach to model the dynamics of open quantum systems, focusing on atomic system ensembles interacting with a structured photonic reservoir associated with a photonic band-gap material. Our results reveal that the quantum reservoir computer provides equally powerful representations for quantum dynamical problems, maintaining effectiveness even under constraints of limited training data.
[eng] Quantum photonic technologies have achieved considerable success across a range of disciplines, including quantum computing, quantum communications, and quantum metrology. Conversely, in the current era of information, the ability to discern patterns within voluminous data sets is highly sought after, with Machine Learning (ML) having emerged as a powerful solution to address this challenge. The processing speed of photonic hardware, in conjunction with the potential for computation enhancement resulting from quantum properties, positions quantum photonics as a promising platform for the implementation of ML algorithms. In this thesis, we focus on reservoir computing (RC) as an ML framework that is suitable for time series processing. It provides fast and energy-efficient training in comparison to more conventional methods and is also appropriate for physical implementations. In this thesis, we explore the possibilities brought by quantum photonic setups in the field of quantum RC. Specifically, we propose and study different platform designs that are able to perform different temporal benchmark tasks by providing tunable memory and nonlinearity. Furthermore, we leverage the capabilities of state-of-the-art quantum photonic hardware and identify novel strategies to address key challenges in RC in quantum substrates, as well as expand the potential of quantum machine learning. Firstly, an approach is provided to overcome some of the limitations of quantum projective measurements, thereby enabling information processing in real time. This is achieved by generating a physical ensemble of quantum reservoirs that recirculate through an optical fiber inside a feedback loop. We investigate the detrimental effects of sampling noise due to finite ensembles and provide a strategy to mitigate them. Secondly, the consequences of increasing quantum squeezing in the dynamics of such setups were studied, and it was found that it monotonically improves the noise robustness of the platforms in realistic scenarios. Finally, we propose a novel design in which photonic cluster states, a specific kind of highly entangled quantum network, are used as resources to implement quantum RC. In this configuration, external input information is encoded into the quantum network via quantum teleportation by engineering the measurement basis of the protocol. Furthermore, the platform has the capacity to implement global quantum gates through local operations (single-mode measurements), thus establishing the foundation for distributed quantum ML.