Physical-Layer Machine Learning with Multimode Interferometric Photon Counting

TL;DR

This work tackles learning from weak, high-dimensional quantum data by fusing variational quantum processing with multimode interferometric photon counting at the physical layer. It formulates PCA and CCA as variational objectives ϑ_PCA(w) = w^T V w and ϑ_CCA(w) = u^T V w, respectively, and shows that placing a programmable linear-optical circuit before photon counting yields a non-Gaussian, highly informative measurement that surpasses standard homodyne detection in the weak-signal regime. The authors employ gradient-free training, notably particle swarm optimization, to cope with quantum measurement noise, and demonstrate that photon counting achieves higher convergence accuracy than homodyne across varying dimensions M and signal strengths, including robustness to realistic imperfections. A squeezing-enhanced variant further improves performance by coherently amplifying the weak signals, suggesting a scalable framework for robust, quantum-enabled learning of weak, correlated signals in sensing networks with potential applications in fundamental physics searches and precision metrology.

Abstract

The learning of the physical world relies on sensing and data post-processing. When the signals are weak, multidimensional and correlated, the performance of learning is often bottlenecked by the quality of sensors, calling for integrating quantum sensing into the learning of such physical-layer data. An example of such a learning scenario is the stochastic quadrature displacements of electromagnetic fields, modeling optomechanical force sensing, radiofrequency photonic sensing, microwave cavity weak signal sensing, and other applications. We propose a unified protocol that combines machine learning with interferometric photon counting to reduce noise and reveal correlations. By applying variational quantum learning with multimode programmable quantum measurements, we enhance signal extraction. Our results show that multimode interferometric photon counting outperforms conventional homodyne detection proposed in prior works for tasks like principal component analysis (PCA) and cross-correlation analysis (CCA), even below vacuum noise levels. To further enhance the performance, we also integrate entanglement-enhanced modules, in the form of squeezed state distribution and anti-squeezing at detection, into the protocol. Combining multimode interferometric photon counting and multipartite entanglement, the proposed protocol provides a powerful toolbox for learning weak signals.

Figures (8)

• Figure 1: Variational quantum processing of stochastic displacement data. (a) High-dimensional random displacement data --- e.g., from radio-frequency (rf) signals Xia202PRL_RadioQSN or opto-mechanical signals Xia2023OmechDQS --- are collected, measured, and processed to extract certain properties from the data, such as principal components and cross-correlations, using a variational measurement and processing unit. This unit comprises of a programmable linear optical circuit ($\hat{U}_{\rm ML}$), photon counters, and a classical post-processing and machine learning module. Illustrative data (yellow dots) and corresponding measurement configurations are shown for (b) Principal Component Analysis (PCA) and (c) Collective Cross-correlation Analysis (CCA).
• Figure 2: Training history of the accuracy of (a) PCA with photon counting, (b) PCA with homodyne detection, (c) CCA with photon counting, and (d) CCA with homodyne detection. Other parameters are $M=21$ and $\sigma_c=0.02$. The blue solid curve indicates the accuracy of the best-performing setup at each time step $t$, while the orange dashed curve represents the highest accuracy achieved by the global best configuration up to time $t$.
• Figure 3: Accuracy depending on signal strength. Each data point corresponds to a different initial condition for optimization and distinct sampling results. The number of modes is $M=21$. The two cases are (a) PCA (b) CCA.
• Figure 4: Accuracy as a function of the number of modes $M$ for (a) PCA and (b) CCA. Each data point corresponds to a separate training run with distinct initialization and signal sampling. We fix $\sqrt{2M} \sigma_c = 0.2$ to maintain constant total signal strength. Dashed lines indicate the expected random guess limit $1/M$.
• Figure 5: Average training performance of PCA in the presence of dark counts. Curves in different colors correspond to varying levels of effective thermal noise induced by dark counts ($n_{\rm DC}=\Gamma_{\rm DC}\tau$). The number of modes is fixed at $M=21$. Vertical dashed lines indicate the points where the signal-to-noise ratio (SNR) reaches 3% under different thermal noise levels.