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

Abstract

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.

Figures (10)

• Figure 1: Process-space coverage of high-fidelity simulation data generated with FLOW-3D WELD®benoit2026lpfno. The dataset spans laser power $P$ and scan speed $V_{\mathrm{scan}} \in [0.1\,\mathrm{m/s}, 1.0\,\mathrm{m/s}]$, and is constructed on a grid uniform in normalized enthalpy $H^*$ and power. Markers denote the training, validation, and super-resolution test sets used for neural operator learning, (Figure adapted from benoit2026lpfno).
• Figure 2: HQ-LP-FNO for 3D laser--material heat transfer. (a) Input features are lifted by a pointwise projection $P$, processed by $L$ hybrid Fourier layers, and projected back by $Q$ to predict $T$ and $\alpha$. (b) One hybrid Fourier layer: a spectral path (PHQFNOSpectralConv3d) runs in parallel with a pointwise linear map $W_\ell$, outputs are summed and passed through GELU (except in the last Fourier layer, where no nonlinearity is applied). (c) Partitioned hybrid spectral convolution: after a 3D real FFT, output channels are split into a classical branch (channels $[C_q:C_{\mathrm{out}}]$) with learnable complex spectral weights, and a quantum branch (channels $[0:C_q]$) that applies a mode-shared variational quantum circuit to the first $C_q$ input channels. The two branches are concatenated and mapped back via an inverse 3D real FFT.
• Figure 3: Qualitative comparison of classical and hybrid predictions at $C_q=5$ for representative conduction and keyhole cases. The first two rows show temperature fields from each model using identical slice positions, colormaps, and color limits. The third row shows their difference on a diverging colormap centered at zero, where blue indicates lower temperature for the hybrid model and red indicates higher.
• Figure 4: Temperature error maps across process space at $C_q=5$. Rows show the classical model, the hybrid model, and their difference. Columns correspond to the $(P,V_{\mathrm{scan}})$ and $(P,H^*)$ parameter planes. In the difference maps, blue indicates lower error for the hybrid model and red indicates higher error.
• Figure 5: (a) Average FIM showing diagonal dominance. (b) FIM rank vs. number of trainable layers. (c) Eigenvalue distribution across depths. (d--e) Fourier coefficient spectrum, real and imaginary parts.