Partitioned Hybrid Quantum Fourier Neural Operators for Scientific Quantum Machine Learning

We introduce the Partitioned Hybrid Quantum Fourier Neural Operator (PH-QFNO), a generalization of the Quantum Fourier Neural Operator (QFNO) for scientific machine learning. PH-QFNO partitions the Fourier operator computation across classical and quantum resources, enabling tunable quantum-classical hybridization and distributed execution across quantum and classical devices. The method extends QFNOs to higher dimensions and incorporates a message-passing framework to distribute data across different partitions. Input data are encoded into quantum states using unary encoding, and quantum circuit parameters are optimized using a variational scheme. We implement PH-QFNO using PennyLane with PyTorch integration and evaluate it on Burgers' equation, incompressible and compressible Navier-Stokes equations. We show that PH-QFNO recovers classical FNO accuracy. On incompressible Navier-Stokes, PH-QFNO achieves higher accuracy than its classical counterparts. Finally, we perform a sensitivity analysis under input noise, confirming improved stability of PH-QFNO over classical baselines.