Discrete real-time learning of quantum-state subspace evolution of many-body systems in the presence of time-dependent control fields

A method is presented for discrete real-time learning (DRTL) of the evolution of a many-body quantum state in the presence of time-dependent control fields. The DRTL method is based on the recent adoption of the artificial neural network (ANN) ansatz within the framework of the time-dependent variational Monte Carlo method in learning large-scale quantum many-body dynamics. The method leverages accurate short-time quantum-state propagation with the Schr & ouml;dinger equation and the efficient stochastic quantum natural gradient descent algorithm in machine learning. Importantly, the DRTL method is devoid of high numerical costs resulting from the exponentially large many-body Hilbert space by replacing the full state propagation with short-time propagations of a sequence of partial stochastic samples of the ANN state and may be considered as a synergism between the Schr & ouml;dinger equation and the ANN that streamlines the computation of the many-body controlled quantum dynamics. Specifically, this method minimizes the normed distances between the time-dependent variational ANN ansatz and a set of partial quantum states over a sequence of stochastically sampled subspaces that are connected to each other in small discrete time steps. We demonstrate the high accuracy and stability of the DRTL method for the spin excitation dynamics of 4 x 4, 6 x 6, 8 x 8, and 10 x 10 two-dimensional (2D) Heisenberg spin-1/2 lattices driven by time-dependent control fields applied to a corner spin along with strong exchange coupling between the nearest-neighbored spins. It is found that the number of ANN parameters for properly following the dynamically occupied subspaces (i.e., starting from the pure tensor-product spin configuration with all spins down) approximately increases quadratically with the number of spins in the system. Importantly, we show that the DRTL method can accurately and efficiently follow the small, yet importantly, occupied parts of the Hilbert space of the 2D spin square lattices during the interaction with the strong time-dependent control fields over a physically significant time period. This finding provides a basis for future work to perform quantum optimal control simulations of 2D spin lattices and other multiparticle systems.