Fourier Neural Operators for Learning Dynamics in Quantum Spin Systems

Fourier Neural Operators (FNOs) excel on tasks using functional data, such as those originating from partial differential equations. Such characteristics render them an effective approach for simulating the time evolution of quantum wavefunctions, which is a computationally challenging, yet coveted task for studying quantum systems. In this manuscript, we use FNOs to model the evolution of quantum spin systems, so chosen due to their representative quantum dynamics. We explore two distinct FNO architectures, examining their performance for learning and predicting time evolution on both random and low-energy input states. We find that standard neural networks in fixed dimensions, such as U-Net, exhibit limited ability to extrapolate beyond the training time interval, whereas FNOs reliably capture the underlying time-evolution operator, generalizing effectively to unseen times. Additionally, we apply FNOs to a compact set of Hamiltonian observables ($\sim\text{poly}(n)$) instead of the entire $2^n$ quantum wavefunction, which greatly reduces the size of our FNO inputs, outputs and model dimensions. Moreover, this Hamiltonian observable-based method demonstrates that FNOs can effectively distill information from high-dimensional spaces into lower-dimensional spaces. Using this approach, we perform numerical experiments on a 20-qubit system and extrapolate Hamiltonian observables to twice the training time with a relative error of $5.8\%$. Relative to numerical time-evolution methods, FNO achieves an inference speedup of approximately $10^{4}\times$ for 20-qubit systems. The extrapolation of Hamiltonian observables to times later than those used in training is of particular interest, as this stands to fundamentally increase the simulatability of quantum systems past both the coherence times of contemporary quantum architectures and the circuit-depths of tractable tensor networks.