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

TL;DR

This paper introduces a continuous-variable quantum Fourier layer (CV--QFL) that leverages the structure of the FFT for optical quantum computing, enabling precise spectral processing and applications to filtering and PDE solving.

Contribution

The paper presents a novel CV--QFL based on Gaussian photonic circuits that directly implements the quantum Fourier transform for spectral tasks.

Findings

CV--QFL achieves machine-precision accuracy on filtering and PDE tasks.

The method enables direct optical processing without classical encoding.

It paves the way for quantum neural operator architectures.

Abstract

Fourier representations play a central role in operator learning methods 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. Beyond these examples, our method naturally extends to optical-input settings in which the signal is already available as a Gaussian optical field. In such scenarios, coherent light coupled into single-mode waveguides can be processed directly by the CV--QFL, bypassing the need for an explicit classical-to-quantum encoding stage. This enables native spectral processing of light and lays the groundwork for new approaches to quantum scientific machine learning, in particular for future neural operator architectures within the CV framework.