Interpretable Quantum Photonic Generative Models

We propose a physical generative model for learning correlated multivariate data based on homodyne detection of photonic continuous-variable (CV) circuits. We use an $N$-mode photonic circuit in which two-mode squeezing provides correlations, local phase shifts tune the strength and sign of these correlations, and an adjustable interferometer redistributes them across modes. We train the circuit by matching its Pearson correlation matrix to the one obtained from data. We apply affine post-processing to adjust the marginals independently while keeping the relations intact. We test the model on the multivariate distribution of standardized log returns of financial assets, showing that the photonic model can represent rich correlation structures. We reproduce empirical correlations with residuals below $10^{-4}$ for five variables and recover the correct correlation signs for $94\% \pm 1\%$ of 190 correlation pairs at twenty variables, with $86\%$ of the residuals below $0.1$. As a separate technical contribution, we develop an approach for simulating Kerr gates with small nonlinearity via a Hubbard-Stratonovich expansion, making them suitable for CV-based simulators. Our results establish CV photonic circuits as a platform for interpretable generative learning, suitable for representing and fast sampling of distributions across different application domains.