Continuous–Variable Quantum Fourier Neural Operator for Solving Partial Differential Equations

Fourier Neural Operators have become a central tool for learning solution operators of partial differential equations, but their spectral layers remain entirely classical and rely on digital Fourier processing. In this work, we introduce the Continuous-Variable Quantum Fourier Neural Operator (CV-QFNO), a Gaussian photonic formulation of the FNO spectral layer. The proposed architecture maps the essential operations of Fourier-domain operator learning, Fourier transformation, mode selection, and channel mixing, onto native continuous-variable optical primitives. In this way, CV-QFNO provides a photonic quantum analogue of the truncated spectral mechanism underlying the classical FNO, while avoiding the compilation overhead and spectral mismatch that arise in qubit-based Quantum FNO constructions. We extend the framework to both one- and two-dimensional operator learning and validate it on standard PDE benchmarks, including Burgers’ equation, heat equation, Navier–Stokes dynamics, and Darcy flow. The results show that the proposed model preserves the predictive accuracy, resolution generalisation, and spectral inductive bias of Fourier neural operators while using a structurally constrained photonic parameterisation. Since all experiments are performed as classical simulations, the contribution should be understood as an architectural and algorithmic blueprint for photonic neural operators, rather than as a demonstration of quantum computational advantage.