# Hybrid Fourier Neural Operator for Surrogate Modeling of Laser Processing with a Quantum-Circuit Mixer

arXiv: https://arxiv.org/abs/2604.04828 · HTML: https://mlsanity.com/paper/2604.04828/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 6 of its 12 references that are in the map.
- First posted on arXiv 6 Apr 2026.

## Builds on (most cited first, 6 of 6)

- [Gaussian Error Linear Units (GELUs)](https://mlsanity.com/paper/1606.08415/) (2016)
- [TQml Simulator: optimized simulation of quantum machine learning](https://mlsanity.com/paper/2506.04891/) (2025)
- [Information plane and compression-gnostic feedback in quantum machine learning](https://mlsanity.com/paper/2411.02313/) (2024)
- [Predictive control of blast furnace temperature in steelmaking with hybrid depth-infused quantum neural networks](https://mlsanity.com/paper/2504.12389/) (2025)
- [Superposed parameterised quantum circuits](https://mlsanity.com/paper/2506.08749/) (2025)
- [A Fast and Generalizable Fourier Neural Operator-Based Surrogate for Melt-Pool Prediction in Laser Processing](https://mlsanity.com/paper/2602.06241/) (2026)

## Abstract (arXiv)

Data-driven surrogates can replace expensive multiphysics solvers for parametric PDEs, yet building compact, accurate neural operators for three-dimensional problems remains challenging: in Fourier Neural Operators, dense mode-wise spectral channel mixing scales linearly with the number of retained Fourier modes, inflating parameter counts and limiting real-time deployability. We introduce HQ-LP-FNO, a hybrid quantum-classical FNO that replaces a configurable fraction of these dense spectral blocks with a compact, mode-shared variational quantum circuit mixer whose parameter count is independent of the Fourier mode budget. A parameter-matched classical bottleneck control is co-designed to provide a rigorous evaluation framework. Evaluated on three-dimensional surrogate modeling of high-energy laser processing, coupling heat transfer, melt-pool convection, free-surface deformation, and phase change, HQ-LP-FNO reduces trainable parameters by 15.6% relative to a classical baseline while lowering phase-fraction mean absolute error by 26% and relative temperature MAE from 2.89% to 2.56%. A sweep over the quantum-channel budget reveals that a moderate VQC allocation yields the best temperature metrics across all tested configurations, including the fully classical baseline, pointing toward an optimal classical-quantum partitioning. The ablation confirms that mode-shared mixing, naturally implemented by the VQC through its compact circuit structure, is the dominant contributor to these improvements. A noisy-simulator study under backend-calibrated noise from ibm-torino confirms numerical stability of the quantum mixer across the tested shot range. These results demonstrate that VQC-based parameter-efficient spectral mixing can improve neural operator surrogates for complex multiphysics problems and establish a controlled evaluation protocol for hybrid quantum operator learning in practice.