Time-dependent Neural Galerkin Method for Quantum Dynamics

Abstract: We introduce a classical computational method for quantum dynamics that relies on a global-in-time variational principle. Unlike conventional time-stepping approaches, our scheme computes the entire state trajectory over a finite time window by minimizing a loss function that enforces the Schr\"odinger's equation. The variational state is parametrized with a Galerkin-inspired ansatz based on a time-dependent linear combination of time-independent Neural Quantum States. This structure is particularly well-suited for exploring long-time dynamics and enables bounding the error with the exact evolution via the global loss function. We showcase the method by simulating global quantum quenches in the paradigmatic Transverse-Field Ising model in both 1D and 2D, uncovering signatures of ergodicity breaking and absence of thermalization in two dimensions. Overall, our method is competitive compared to state-of-the-art time-dependent variational approaches, while unlocking previously inaccessible dynamical regimes of strongly interacting quantum systems.