New non-Euclidean neural quantum states from additional types of hyperbolic recurrent neural networks

Abstract

In this work, we extend the class of previously introduced non-Euclidean neural quantum states (NQS) which consists only of Poincar\'e hyperbolic GRU, to new variants including Poincar\'e RNN as well as Lorentz RNN and Lorentz GRU. In addition to constructing and introducing the new non-Euclidean hyperbolic NQS ansatzes, we generalized the results of our earlier work regarding the definitive outperformances delivered by hyperbolic Poincar\'e GRU NQS ansatzes when benchmarked against their Euclidean counterparts in the Variational Monte Carlo (VMC) experiments involving the quantum many-body settings of the Heisenberg $J_1J_2$ and $J_1J_2J_3$ models, which exhibit hierarchical structures in the forms of the different degrees of nearest-neighbor interactions. Here, in particular, using larger systems consisting of 100 spins, we found that all four hyperbolic RNN/GRU NQS variants always outperformed their respective Euclidean counterparts. Specifically, for all $J_2$ and $(J_2,J_3)$ couplings considered, including $J_2=0.0$, Lorentz RNN NQS and Poincar\'e RNN NQS always outperformd Euclidean RNN NQS, while Lorentz/Poincar\'e GRU NQS always outperformed Euclidean GRU NQS, with a single exception when $J_2=0.0$ for Poincar\'e GRU NQS. Furthermore, among the four hyperbolic NQS ansatzes, depending on the specific $J_2$ or $(J_2, J_3)$ couplings, on four out of eight experiment settings, Lorentz GRU and Poincar\'e GRU took turns to be the top performing variant among all Euclidean and hyperbolic NQS ansatzes considered, while Lorentz RNN, with up to three times fewer parameters, was capable of not only surpassing the Euclidean GRU eight out of eight times but also outperforming both Lorentz GRU and Poincar\'e GRU four out of eight times, to emerge as the best overall hyperbolic NQS ansatz.