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

Area of Science:

• Quantum Computing
• Artificial Intelligence
• Computational Physics

Background:

• Fourier Neural Operators (FNOs) are key for solving partial differential equations (PDEs) but rely on classical digital Fourier processing.
• Existing quantum FNOs face compilation overhead and spectral mismatch issues.

Purpose of the Study:

• Introduce the Continuous-Variable Quantum Fourier Neural Operator (CV-QFNO) as a photonic formulation of the FNO spectral layer.
• Develop a quantum analogue of FNOs using continuous-variable optical primitives.

Main Methods:

• Mapped FNO spectral layer operations (Fourier transform, mode selection, channel mixing) onto Gaussian photonic primitives.
• Extended the CV-QFNO framework to one- and two-dimensional operator learning.
• Validated the model on standard PDE benchmarks: Burgers' equation, heat equation, Navier-Stokes, and Darcy flow.

Main Results:

• CV-QFNO preserves predictive accuracy, resolution generalization, and spectral inductive bias of classical FNOs.
• The photonic parameterization is structurally constrained.
• All experiments were conducted as classical simulations, serving as an architectural blueprint.

Conclusions:

• The CV-QFNO presents a viable blueprint for photonic neural operators.
• This approach avoids compilation overhead and spectral mismatch inherent in qubit-based quantum FNOs.
• The study highlights a path towards quantum-enhanced PDE solving without immediate quantum advantage demonstration.