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arXiv:2604.04828

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-phcs.CEcs.LGphysics.comp-phSubmitted Apr 6, 2026• Announced Apr 7, 2026• 0 citations
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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.

Comments
24 pages, 10 figures, 6 tables

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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.
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We introduce a hybrid approach utilising a quantum machine learning surrogate model to approximate the non-linear collision dynamics of the LBM. It effectively offloads the non-unitary operations that challenge pure quantum solvers. The expressivity of the surrogate is built on the ability of parameterised quantum circuits to implement partial Fourier series, with data re-uploading extending the spectrum of representable frequencies. Unlike previous approaches with a fixed relaxation parameter, the surrogate recovers the complete Bhatnagar-Gross-Krook (BGK) collision dynamics across the full physically admissible range of relaxation without retraining. We reassess the relevance of standard variational quantum circuit (VQC) metrics, including expressibility, entanglement, and effective dimension, by relating them directly to task-specific surrogate performance and identifying the key architectural parameters that determine approximation accuracy. The proposed surrogate is validated against the classical BGK collision operator using established benchmark problems, including the Taylor-Green vortex for evaluating energy dissipation and the double shear layer for assessing shear-driven instabilities and nonlinear flow evolution. Our results demonstrate that the hybrid model achieves high accuracy and generalisability while closely replicating classical solutions. These findings suggest that hybrid quantum-classical strategies offer a practical path toward realising the potential of quantum computing in fluid engineering.
Medical image classification is a critical component of modern healthcare; however, accurate diagnosis remains challenging due to limited annotated datasets, class imbalance, and the high dimensionality of medical imaging data. To address these challenges, a hybrid quantum-classical neural network (HQCNN) is proposed, integrating classical deep learning with variational quantum learning for medical image classification. The proposed architecture combines a five-layer convolutional neural network (CNN) for hierarchical feature extraction with a lightweight 4-qubit variational quantum circuit (VQC) incorporating quantum state encoding, superposition and entanglement mechanisms, and a quantum attention-Fourier (QAF) module. This hybrid design aims to improve nonlinear feature representation and quantum parameter efficiency while maintaining a shallow quantum circuit suitable for noisy intermediate-scale quantum (NISQ)-era constraints. Experimental evaluation on six MedMNIST benchmark datasets demonstrated competitive performance across both binary and multi-class classification tasks. HQCNN achieved 98.88% accuracy on the binary subset of PathMNIST (classes 0 vs. 1), 97.61% accuracy on the multi-class OrganAMNIST dataset, and 86.29% accuracy on BreastMNIST. Comparative experiments and statistical analyses demonstrated consistent improvements over the controlled BHQNN baseline, while component-wise ablation studies showed that the QAF module, superposition and entanglement mechanisms, and expressive parameterized rotations contributed to classification performance in a complementary and dataset-dependent manner. Moreover, HQCNN reduced the number of trainable quantum parameters by approximately 55.6% compared with the baseline hybrid quantum neural network (BHQNN). Noise-aware simulations further showed that the model retained relatively stable predictive performance under moderate depolarizing noise, supporting further evaluation under near-term quantum computing conditions. Overall, the results demonstrate that HQCNN provides a parameter-efficient hybrid quantum-classical framework for medical image classification and offers a promising foundation for further investigation of quantum-enhanced medical image analysis.
Abstract Many real-world problems, like modelling environment dynamics, physical processes, time series etc involve solving partial differential equations (PDEs) parameterised by problem-specific conditions. Recently, a deep learning architecture called Fourier neural operator (FNO) proved to be capable of learning solutions of given PDE families for any initial conditions as input. However, it results in a time complexity linear in the number of evaluations of the PDEs while testing. Given the advancements in quantum hardware and the recent results in quantum machine learning methods, we exploit the running efficiency offered by these and propose quantum algorithms inspired by the classical FNO, which result in time complexity logarithmic in the number of evaluations and are expected to be substantially faster than their classical counterpart. At their core, we use the unary encoding paradigm and orthogonal quantum layers and introduce a new quantum Fourier transform in the unary basis. We propose three different quantum circuits to perform a quantum FNO. The proposals differ in their depth and their similarity to the classical FNO. We also benchmark our proposed algorithms on three PDE families, namely Burgers’ equation, Darcy’s flow equation and the Navier–Stokes equation. The results show that our quantum methods are comparable in performance to the classical FNO. We also perform an analysis on small-scale image classification tasks where our proposed algorithms are at par with the performance of classical convolutional neural networks, proving their applicability to other domains as well.
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.