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

Mateusz Papierz; Asel Sagingalieva; Alix Benoit; Toni Ivas; Elia Iseli; Alexey Melnikov

quant-phApr 6, 2026

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

Authors: Mateusz Papierz, Asel Sagingalieva, Alix Benoit, Toni Ivas, Elia Iseli, Alexey Melnikov

Organizations: Terra Quantum AG, Kornhausstrasse 25, 9000 St. Gallen, Switzerland · Laboratory for Advanced Materials Processing, Feuerwerkerstrasse, Thun, 3603, BE, Switzerland

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 & tables

Figure 1: Process-space coverage of high-fidelity simulation data generated with FLOW-3D WELD ® [ 7 ] . 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 [ 7 ] ).

Symbol	Value	Description
$\eta$	$0.35\text{\,}$	Absorptivity
$\rho$	$4420\text{\,}\mathrm{kg}\text{\,}{\mathrm{m}}^{-1}$	Density
$C_{p}$	$750\text{\,}\mathrm{J}\text{\,}{\mathrm{kg}}^{-1}\text{\,}{\mathrm{K}}^{-1}$	Specific heat capacity at Tsolidus=1873K
$\Delta T_{m}$	$1573\text{\,}\mathrm{K}$	Temperature rise to melting
$D$	$8.1\text{\times}{10}^{-6}\text{\,}\mathrm{m}$	Thermal diffusivity
$\sigma$	$50\text{\,}\mathrm{\SIUnitSymbolMicro m}$	Beam radius (1/e 2 )

Table 1: Parameters used for normalized enthalpy evaluation (same as the classical study).

Quantity	Symbol	Reference value
Length scale	$L$	$1\text{\times}{10}^{-4}\text{\,}\mathrm{m}$
Temperature scale	$T$	$3000\text{\,}\mathrm{K}$
Velocity scale	$V$	$1\text{\times}{10}^{-1}\text{\,}\mathrm{m}\text{\,}{\mathrm{s}}^{-1}$
Power scale	$P$	$10\text{\,}\mathrm{W}$
Normalized enthalpy	$H$	$7.5$

Table 2: Reference scales used for normalization (SI units), same as the classical study.

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.

Component	Classical FNO	Hybrid FNO
Mode-wise spectral weights	$8\,C^{2}\,N_{\mathrm{m}}$	$8\,C(C-C_{q})\,N_{\mathrm{m}}$
Mode-shared quantum branch	–	$3\,d(n_{\mathrm{q}}-1)+4C_{q}n_{\mathrm{q}}+n_{\mathrm{q}}+2C_{q}$

Table 3: Trainable spectral-branch parameters per Fourier layer.

Temperature field $\tilde{T}$
	LP-FNO	HQ-LP-FNO	CM-LP-FNO
	(classical)	(VQC, $C_{q}\!=\!5$ )	(MLP, $C_{q}\!=\!5$ )
Abs. mean [K]	18.2 [14.1, 21.4]	15.2 [14.0, 15.8]	15.4 [13.2, 17.5]
RMSE [K]	37.3 [31.4, 45.3]	39.7 [38.4, 40.7]	32.1 [28.6, 33.8]
Rel. mean [%]	2.9 [2.2, 3.4]	2.6 [2.4, 2.7]	2.3 [2.0, 2.6]
Rel. RMSE [%]	5.9 [5.0, 7.2]	6.7 [6.5, 6.9]	4.7 [4.2, 5.0]

Table 4: Error metrics on the held-out test set (mean

[\min,\max]

across 8 folds,

C_{q}=5

). CM-LP-FNO replaces the VQC with a parameter-matched mode-shared bottleneck MLP. Absolute temperature errors are reported in kelvin (rescaled with

T_{\mathrm{ref}}=

3000\text{,}\mathrm{K}$$ ), other fields are dimensionless. Best mean value per metric is shown in bold (lower is better for error metrics, higher is better for IoU).

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.

	$C_{q}=0$	$C_{q}=3$	$C_{q}=5$	$C_{q}=8$
	(classical)
Params (M)	184.327	167.047	155.527	138.248
Rel. mean $\tilde{T}$ [%]	$2.53\pm 0.48$	$2.40\pm 0.50$	$2.58\pm 0.61$	$2.73\pm 0.52$
Rel. RMSE $\tilde{T}$ [%]	$5.45\pm 1.02$	$5.31\pm 1.08$	$5.56\pm 1.13$	$5.61\pm 1.07$
IoU( $f_{l}$ )	$0.9047\pm 0.0085$	$0.9016\pm 0.0079$	$0.9003\pm 0.0114$	$0.90165\pm 0.00868$

Table 5: Quantum-channel-budget sweep (mean

\pm

std across 3 folds). IoU(

\alpha

) is omitted as it exceeds

0.999

for all settings.

Appendix figures & tables7 assets

Supplementary material from the paper’s appendix.

Appendix

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.

Figure 6: Mean squared error between the noiseless and noisy outputs of the quantum spectral mixer as a function of the number of measurement shots under the ibm_torino -derived noise model. Points show the mean over 10 repeated runs, the shaded band indicates

\pm

one standard deviation.

Category	Parameter	Value
FNO backbone
	Spatial dimension	3
	Number of FNO layers ( $L$ )	3
	Latent channel width ( $C$ )	32
	Retained Fourier modes $(m_{x},m_{y},m_{z})$	$[25,\,20,\,15]$
	Padding	9 (constant)

Appendix

Table 6: FNO architecture and training hyperparameters. We follow [ 7 ] ; for hybrid models we use a cosine-annealing schedule with a higher base learning rate.

Figure 7: $\alpha$ error maps across process space at $C_{q}=5$ , analogous to Fig. 4 for temperature.

Figure 8: $f_{l}$ error maps across process space at $C_{q}=5$ , analogous to Fig. 4 for temperature

Figure 9: $\alpha$ IoU error ( $1-\text{IoU}$ ) maps across process space at $C_{q}=5$ .

Figure 10: $f_{l}$ IoU error ( $1-\text{IoU}$ ) maps across process space at $C_{q}=5$ .

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Partial differential equations (PDEs) are central to modeling physical and engineering systems, but repeatedly solving parametric PDEs remains computationally expensive. Operator learning enables fast surrogate inference, yet typically requires large input-output paired datasets generated by costly high-fidelity PDE solvers. Unsupervised operator learning frameworks alleviate data dependency but remain hindered by computational bottlenecks. To address this, we propose Neural Variational Quantum Linear Solver (NVQLS), the first hybrid quantum-classical operator learning framework leveraging the Legendre--Galerkin weak formulation. We critically resolve the sign ambiguity in VQLS energy minimization, preventing erroneous solution representations. Additionally, we introduce a neural embedding, a novel encoding scheme to map varying forcings and PDE coefficients into parameterized quantum circuit representations. These structural innovations provide theoretical computational complexity advantages under efficient state preparation schemes, while achieving superior accuracy compared to a representative classical baseline. Validations on 1D and 2D parametric PDEs under diverse boundary conditions demonstrate NVQLS's capability to simultaneously process varying inputs, offering a scalable unsupervised approach to quantum-enhanced operator learning.

Abstract

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