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Continuous-Variable Quantum Fourier Neural Operator for Solving Partial Differential Equations

Paolo Marcandelli, Stefano Mariani, Martina Siena, Stefano Markidis

Published Jul 1, 2026• 0 citations
Publisher

Abstract

Fourier Neural Operators have become a central tool for learning solution operators of partial differential equations, but their spectral layers remain entirely classical and rely on digital Fourier processing. In this work, we introduce the Continuous-Variable Quantum Fourier Neural Operator (CV-QFNO), a Gaussian photonic formulation of the FNO spectral layer. The proposed architecture maps the essential operations of Fourier-domain operator learning, Fourier transformation, mode selection, and channel mixing, onto native continuous-variable optical primitives. In this way, the CV-QFNO provides a photonic quantum analogue of the truncated spectral mechanism underlying the classical FNO, while avoiding the compilation overhead and spectral mismatch that arise in qubit-based Quantum FNO constructions. We extended the framework to both one- and two-dimensional operator learning and validated it on standard PDE benchmarks, including Burgers' equation, heat equation, Navier-Stokes dynamics, and Darcy flow. The results show that the proposed model preserves the predictive accuracy, resolution generalisation, and spectral inductive bias of Fourier neural operators while using structurally constrained photonic parameterisation. Since all the experiments were performed as classical simulations, the contribution should be understood as an architectural and algorithmic blueprint for photonic neural operators rather than as a demonstration of quantum computational advantage.

DOI
10.3390/e28070737

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Fourier Neural Operators have become a central tool for learning solution operators of partial differential equations, but their spectral layers remain entirely classical and rely on digital Fourier processing. In this work, we introduce the Continuous-Variable Quantum Fourier Neural Operator (CV-QFNO), a Gaussian photonic formulation of the FNO spectral layer. The proposed architecture maps the essential operations of Fourier-domain operator learning, Fourier transformation, mode selection, and channel mixing, onto native continuous-variable optical primitives. In this way, the CV-QFNO provides a photonic quantum analogue of the truncated spectral mechanism underlying the classical FNO, while avoiding the compilation overhead and spectral mismatch that arise in qubit-based Quantum FNO constructions. We extended the framework to both one- and two-dimensional operator learning and validated it on standard PDE benchmarks, including Burgers' equation, heat equation, Navier-Stokes dynamics, and Darcy flow. The results show that the proposed model preserves the predictive accuracy, resolution generalisation, and spectral inductive bias of Fourier neural operators while using structurally constrained photonic parameterisation. Since all the experiments were performed as classical simulations, the contribution should be understood as an architectural and algorithmic blueprint for photonic neural operators rather than as a demonstration of quantum computational advantage.
Fourier representations play a central role in operator learning methods for partial differential equations and are increasingly being explored in quantum machine learning architectures. The classical fast Fourier transform (FFT), particularly in its Cooley--Tukey decomposition, exhibits a structure that naturally matches continuous-variable quantum circuits. This correspondence establishes a direct structural isomorphism between the Cooley-Tukey butterfly network and Gaussian photonic gates, enabling the FFT to be realized as a native optical computation in continuous-variable quantum computing. Building on this observation, we introduce a continuous-variable Quantum Fourier Layer (CV--QFL) based on a bipartite Gaussian encoding and a Cooley-Tukey quantum Fourier transform, enabling exact two-dimensional spectral processing within a Gaussian photonic circuit. We test the CV--QFL on two representative tasks: spectral low-pass filtering and Fourier-domain integration of the heat equation. In both cases, the results match the classical reference to machine precision. Beyond these examples, our method naturally extends to optical-input settings in which the signal is already available as a Gaussian optical field. In such scenarios, coherent light coupled into single-mode waveguides can be processed directly by the CV--QFL, bypassing the need for an explicit classical-to-quantum encoding stage. This enables native spectral processing of light and lays the groundwork for new approaches to quantum scientific machine learning, in particular for future neural operator architectures within the CV framework.
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.
Fourier representations play a central role in operator learning for partial differential equations and are increasingly being explored in quantum machine learning architectures. The classical fast Fourier transform (FFT), particularly in its Cooley–Tukey decomposition, exhibits a structure that naturally matches continuous-variable quantum circuits. This correspondence establishes a direct structural isomorphism between the Cooley–Tukey butterfly network and Gaussian photonic gates, enabling the FFT to be realized as a native optical computation in continuous-variable quantum computing. Building on this observation, we introduce a continuous-variable Quantum Fourier Layer (CV–QFL) based on a bipartite Gaussian encoding and a Cooley–Tukey quantum Fourier transform, enabling exact two-dimensional spectral processing within a Gaussian photonic circuit. We test the CV–QFL on two representative tasks: spectral low-pass filtering and Fourier-domain integration of the heat equation. In both cases, the results match the classical reference to machine precision. More broadly, this work lays the foundation for continuous-variable approaches to quantum scientific computing and for the development of native spectral architectures in quantum machine learning.
Neural Physics recasts local discretisations of partial differential equations (PDEs) as fixed convolutional operators, providing a physics-preserving alternative to data-driven surrogate modelling in scientific machine learning. However, existing realizations remain largely confined to classical AI hardware and do not directly connect to quantum structured operator design. To bridge this gap, we introduce a \emph{Quantum Neural Physics} framework and develop a Hybrid Quantum-Classical CNN Multigrid Solver (HQC-CNNMG). The proposed method maps analytically prescribed stencil operators to local quantum convolutional primitives and embeds them within a classical multilevel W-cycle architecture, combining the operator-centric view of scientific ML with the numerical rigor of multigrid solvers. Using amplitude encoding together with the Linear Combination of Unitaries (LCU) and the Quantum Fourier Transform (QFT), the resulting local quantum operators admit logarithmic-depth implementation, with circuit depth scaling as O(log⁡K)\mathcal{O}(\log K)O(logK) for an encoded block of size KKK under the idealized parallel circuit model considered here. Numerical experiments on Poisson, transient diffusion, convection--diffusion, and incompressible Navier--Stokes problems demonstrate numerical consistency, stable multilevel behaviour, and workflow-level feasibility on noiseless simulators. Comparisons with representative quantum linear solver paradigms further show that the main strength of HQC-CNNMG lies in its balanced trade-off among local circuit depth, numerical robustness, and compatibility with PDE structure, rather than in fully quantum global inversion.
Continuous-variables (CV) quantum optics is a natural formalism for neural networks (NNs) due to its ability to reproduce the information processing of such trainable interconnected systems. In quantum optics, Gaussian operators induce affine mappings on the quadratures of optical modes while non-Gaussian resources (the challenging piece for physical implementation) originate the nonlinear effects, unlocking quantum analogs of an artificial neuron. This work presents a novel experimentally-feasible framework for continuous-variable quantum optical neural networks (QONNs) developed with available photonic components: coherent states as input encoding, a general Gaussian transformation followed by multi-mode photon subtractions as the processing layer, and homodyne detection as outputs readout. The closed-form expressions of such architecture are derived demonstrating the family of adaptive activations and the quantum-optical neurons that emerge from the amount of photon-subtracted modes, proving that the proposed design satisfies the Universal Approximation Theorem within a single layer. To classically simulate the QONN training, the high-performance QuaNNTO library has been developed based on Wick-Isserlis expansion and Bogoliubov transformations, allowing multi-layer exact expectation values of non-Gaussian states without truncating the infinite-dimensional Hilbert space. Experiments on supervised learning and state-preparation tasks show balanced-resource efficiency with strong expressivity and generalization capabilities, illustrating the potential of the architecture for scalable photonic quantum machine learning and for quantum applications such as complex non-Gaussian gate synthesis.
We introduce the Partitioned Hybrid Quantum Fourier Neural Operator (PH-QFNO), a generalization of the Quantum Fourier Neural Operator (QFNO) for scientific machine learning. PH-QFNO 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 PH-QFNO using PennyLane with PyTorch integration and evaluate it on Burgers' equation, incompressible and compressible Navier-Stokes equations. We show that PH-QFNO recovers classical FNO accuracy. On incompressible Navier-Stokes, PH-QFNO achieves higher accuracy than its classical counterparts. Finally, we perform a sensitivity analysis under input noise, confirming improved stability of PH-QFNO over classical baselines.