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.
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 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.
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(logK) for an encoded block of size K 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.
In the realm of computational science and engineering, constructing models that reflect real-world phenomena requires solving partial differential equations (PDEs) with different conditions. Recent advancements in neural operators, such as deep operator network (DeepONet), which learn mappings between infinite-dimensional function spaces, promise efficient computation of PDE solutions for a new condition in a single forward pass. However, classical DeepONet entails quadratic complexity concerning input dimensions during evaluation. Given the progress in quantum algorithms and hardware, here we propose to utilize quantum computing to accelerate DeepONet evaluations, yielding complexity that is linear in input dimensions. Our proposed quantum DeepONet integrates unary encoding and orthogonal quantum layers. We benchmark our quantum DeepONet using a variety of PDEs, including the antiderivative operator, advection equation, and Burgers' equation. We demonstrate the method's efficacy in both ideal and noisy conditions. Furthermore, we show that our quantum DeepONet can also be informed by physics, minimizing its reliance on extensive data collection. Quantum DeepONet will be particularly advantageous in applications in outer loop problems which require exploring parameter space and solving the corresponding PDEs, such as uncertainty quantification and optimal experimental design.
Abstract Fourier Neural Operators (FNOs) excel on functional data tasks, such as those originating from partial differential equations. This renders them effective at simulating the time evolution of quantum wavefunctions, a computationally challenging, yet coveted task. In this manuscript, we use FNOs to model the evolution of random quantum spin systems, chosen due to their representative dynamics. We find that traditional neural networks, such as U-Net, exhibit limited extrapolation beyond the training time interval, whereas FNOs capture the underlying time-evolution operator, generalizing to unseen times. Additionally, we apply FNOs to a compact set of Hamiltonian observables (~ poly( n )) instead of 2 n -component wavefunctions, greatly reducing the size of our inputs, outputs, and FNO models. This Hamiltonian observable-based method demonstrates that FNOs can distill information from high to low-dimensional spaces. We perform numerical experiments on a 20-qubit system, extrapolating Hamiltonian observables to twice the training time with a relative error of 5.8%. Notably, relative to numerical time-evolution methods, FNO achieves an inference speedup of approximately 10 4 × for 20-qubit systems, underscoring its computational efficiency. This extrapolation of observables past training times stands to fundamentally increase the simulatability of quantum systems beyond the limitations of quantum device coherence and tensor network circuit-depth.
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.
Recently, physics-informed neural networks (PINNs) have aroused an upsurge in the field of scientific computing including solving partial differential equations (PDEs), which convert the task of solving PDEs into an optimization challenge by adopting governing equations and definite conditions or observation data as loss functions. Essentially, the underlying logic of PINNs is based on the universal approximation and differentiability properties of classical neural networks (NNs). Recent research has revealed that quantum neural networks (QNNs), known as parameterized quantum circuits, also exhibit universal approximation and differentiability properties. This observation naturally suggests the application of PINNs to QNNs. In this work, we introduce a physics-informed quantum neural network (PI-QNN) by employing the QNN as the function approximator for solving forward and inverse problems of PDEs. The performance of the proposed PI-QNN is evaluated by various forward and inverse PDE problems. Numerical results indicate that PI-QNN demonstrates superior convergence over PINN when solving PDEs with exact solutions that are strongly correlated with trigonometric functions. Moreover, its accuracy surpasses that of PINN by two to three orders of magnitude, while requiring fewer trainable parameters. However, the computational time of PI-QNN exceeds that of PINN due to its operation on classical computers. This limitation may improve with the advent of commercial quantum computers in the future. Furthermore, we briefly investigate the impact of network architecture on PI-QNN performance by examining two different QNN architectures. The results suggest that increasing the number of trainable network layers can enhance the expressiveness of PI-QNN. However, an excessive number of data encoding layers significantly increases computational time, rendering the marginal gains in performance insufficient to compensate for the shortcomings in computational efficiency.