# Partitioned Hybrid Quantum Fourier Neural Operators for Scientific Quantum Machine Learning

arXiv: https://arxiv.org/abs/2507.08746 · HTML: https://mlsanity.com/paper/2507.08746/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- Builds on 10 of its 37 references that are in the map.
- First posted on arXiv 11 Jul 2025.

## Builds on (most cited first, 10 of 10)

- [Fourier Neural Operator for Parametric Partial Differential Equations](https://mlsanity.com/paper/2010.08895/) (2020)
- [Variational Quantum Algorithms](https://mlsanity.com/paper/2012.09265/) (2020)
- [The power of quantum neural networks](https://mlsanity.com/paper/2011.00027/) (2020)
- [Continuous-variable quantum neural networks](https://mlsanity.com/paper/1806.06871/) (2018)
- [Orthogonal Deep Neural Networks](https://mlsanity.com/paper/1905.05929/) (2019)
- [Quantum Vision Transformers](https://mlsanity.com/paper/2209.08167/) (2022)
- [Hybrid quantum programming with PennyLane Lightning on HPC platforms](https://mlsanity.com/paper/2403.02512/) (2024)
- [What can we learn from quantum convolutional neural networks?](https://mlsanity.com/paper/2308.16664/) (2023)
- [On Physics-Informed Neural Networks for Quantum Computers](https://mlsanity.com/paper/2209.14754/) (2022)
- [What is Quantum Parallelism, Anyhow?](https://mlsanity.com/paper/2405.07222/) (2024)

## Built on by (most cited first, 2 of 2)

- [A Fast and Generalizable Fourier Neural Operator-Based Surrogate for Melt-Pool Prediction in Laser Processing](https://mlsanity.com/paper/2602.06241/) (2026)
- [WTHaar-Net: a Hybrid Quantum-Classical Approach](https://mlsanity.com/paper/2603.02497/) (2026)

## Abstract (arXiv)

We introduce the Partitioned Hybrid Quantum Fourier Neural Operator (PHQFNO), a generalization of the Quantum Fourier Neural Operator (QFNO) for scientific machine learning. PHQFNO 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 PHQFNO using PennyLane with PyTorch integration and evaluate it on Burgers' equation, incompressible and compressible Navier-Stokes equations. We show that PHQFNO recovers classical FNO accuracy. On incompressible Navier-Stokes, PHQFNO achieves higher accuracy than its classical counterparts. Finally, we perform a sensitivity analysis under input noise, confirming improved stability of PHQFNO over classical baselines.