Continuous-Variable Quantum Fourier Layer: Applications to Filtering and PDE Solving

Fourier representations play a central role in operator learning for partial differential equations and are increasingly being explored in quantum machine learning architectures. The classical fast Fourier transform (FFT), particularly in its Cooley–Tukey decomposition, exhibits a structure that naturally matches continuous-variable quantum circuits. This correspondence establishes a direct structural isomorphism between the Cooley–Tukey butterfly network and Gaussian photonic gates, enabling the FFT to be realized as a native optical computation in continuous-variable quantum computing. Building on this observation, we introduce a continuous-variable Quantum Fourier Layer (CV–QFL) based on a bipartite Gaussian encoding and a Cooley–Tukey quantum Fourier transform, enabling exact two-dimensional spectral processing within a Gaussian photonic circuit. We test the CV–QFL on two representative tasks: spectral low-pass filtering and Fourier-domain integration of the heat equation. In both cases, the results match the classical reference to machine precision. More broadly, this work lays the foundation for continuous-variable approaches to quantum scientific computing and for the development of native spectral architectures in quantum machine learning.