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arXiv:2409.03302 · Communications Physics

Fourier neural operators for learning dynamics in quantum spin systems

Freya Shah, Taylor L. Patti, Julius Berner, Bahareh Tolooshams, Jean Kossaifi, Animashree Anandkumar

Published Apr 29, 2026• 3 citations
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Abstract

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.

DOI
10.1038/s42005-026-02644-1

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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.
Time-periodic quantum systems exhibit a rich variety of far-from-equilibrium phenomena and serve as ideal platforms for quantum engineering and control. However, simulating their dynamics with conventional numerical methods remains challenging due to the exponential growth of Hilbert space dimension and rapid spreading of entanglement. In this work, we introduce Fourier neural operators (FNOs) as an efficient, accurate, and scalable framework for nonequilibrium quantum dynamics. Parameterized in Fourier space, FNO naturally captures temporal correlations and remains minimally dependent on discretization of time. We demonstrate the versatility of FNO through three complementary learning paradigms: reconstructing effective Floquet Hamiltonians, predicting expectation values of local observables, and learning quantum information spreading. For each learning task, FNO achieves remarkable accuracy, while attaining a significant speedup, compared to exact numerical methods. Moreover, FNO possesses capabilities beyond that of conventional methods, such as predicting all local observables from a subset of measurements without information about the Hamiltonian, as well as extrapolating beyond the time window provided by training data, enabling access to observables and operator-spreading dynamics that might be beyond the coherence time. By employing a spatially local basis, we argue that the computational cost of FNOs scales only polynomially with the system size. Our results establish FNO as a versatile and scalable computation framework that integrates numerical simulations and experimental data seamlessly, with direct implications for extracting meaningful physics from measurements by near-term quantum computers.
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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.
Capturing the dynamics of quantum many-body systems under time-dependent driving protocols is a central challenge for numerical simulations. Existing methods such as tensor networks and time-dependent neural quantum states, however, must be re-run for every protocol. In this work, we introduce the Neural Operator Quantum State (NOQS) as a foundation model for quantum dynamics. Rather than solving the Schrödinger equation for individual trajectories, our approach aims to \emph{learn the solution operator} that maps entire driving protocols to time-evolved quantum states. Once trained, the NOQS predicts time evolution under unseen protocols in a single forward pass, requiring no additional optimization. We validate NOQS on the two-dimensional Ising model with time-dependent longitudinal and transverse fields, demonstrating accurate prediction not only for unseen in-distribution protocols, but also for qualitatively different, out-of-distribution functional forms of driving. Further, a single NOQS model can be transferred between different temporal resolutions, and can be efficiently fine-tuned with sparse experimental measurements to improve predictions across all observables at negligible cost. Our work introduces a new paradigm for quantum dynamics simulation and provides a practical computational-experimental interface for driven quantum systems.
Quantum optimal control is a key tool for steering quantum dynamics, but its computational cost grows rapidly with the Hilbert space dimension. Here, we introduce a Fourier Neural Operator (FNO)-based framework for learning high dimensional molecular quantum dynamics and accelerating the inverse design of control protocols. Given an initial molecular population distribution, laser frequency, and polarization, the FNO predicts molecular-motional population dynamics up to 10710^7107 times faster than GPU-accelerated numerical propagation with CUDA-Q Dynamics. Using this fast and differentiable surrogate, we develop the FNO stochastic pulse-measurement planner (FNO-SPMP), which constructs pulse sequences to purify an initially mixed Boltzmann distribution. We demonstrate the protocol in an 888-dimensional subspace of the hydronium molecule at 20 K, achieving a target-state population of 0.98 with a sequence success rate of up to 86.2%. In a shared discrete control space, FNO-SPMP achieves nearly twice the success rate of a reinforcement-learning baseline while using roughly half as many quantum control pulses and reducing pulse-sequence generation time from approximately 10 hours to 10-20 minutes. These results show that operator-learning surrogates can enable inverse design in quantum systems whose Hilbert spaces are too large for conventional direct optimization.
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.