# Hyperbolic recurrent neural network as the first type of non-Euclidean neural quantum state ansatz

arXiv: https://arxiv.org/abs/2505.22083 · HTML: https://mlsanity.com/paper/2505.22083/

## Key facts

- Cited by 3 papers in the map (data through 14 Oct 2026).
- Builds on 15 of its 58 references that are in the map.
- First posted on arXiv 28 May 2025.

## Builds on (most cited first, 15 of 15)

- [Poincar'e Embeddings for Learning Hierarchical Representations](https://mlsanity.com/paper/1705.08039/) (2017)
- [Hyperbolic Neural Networks](https://mlsanity.com/paper/1805.09112/) (2018)
- [Representation Tradeoffs for Hyperbolic Embeddings](https://mlsanity.com/paper/1804.03329/) (2018)
- [Riemannian Adaptive Optimization Methods](https://mlsanity.com/paper/1810.00760/) (2018)
- [Hyperbolic Entailment Cones for Learning Hierarchical Embeddings](https://mlsanity.com/paper/1804.01882/) (2018)
- [Hyperbolic Deep Neural Networks: A Survey](https://mlsanity.com/paper/2101.04562/) (2021)
- [Hyperbolic Attention Networks](https://mlsanity.com/paper/1805.09786/) (2018)
- [Continuous Hierarchical Representations with Poincar'e Variational Auto-Encoders](https://mlsanity.com/paper/1901.06033/) (2019)
- [Constant Curvature Graph Convolutional Networks](https://mlsanity.com/paper/1911.05076/) (2019)
- [Fully Hyperbolic Neural Networks](https://mlsanity.com/paper/2105.14686/) (2021)
- [Match-SRNN: Modeling the Recursive Matching Structure with Spatial RNN](https://mlsanity.com/paper/1604.04378/) (2016)
- [A Fully Hyperbolic Neural Model for Hierarchical Multi-Class Classification](https://mlsanity.com/paper/2010.02053/) (2020)
- [Recurrent neural network wave functions for Rydberg atom arrays on kagome lattice](https://mlsanity.com/paper/2405.20384/) (2024)
- [Supplementing Recurrent Neural Network Wave Functions with Symmetry and Annealing to Improve Accuracy](https://mlsanity.com/paper/2207.14314/) (2022)
- [Are queries and keys always relevant? A case study on Transformer wave functions](https://mlsanity.com/paper/2405.18874/) (2024)

## Built on by (most cited first, 3 of 3)

- [New non-Euclidean neural quantum states from hyperbolic Lorentz recurrent architectures](https://mlsanity.com/paper/2604.24337/) (2026)
- [Two-dimensional Hyperbolic RNN Neural Quantum State](https://mlsanity.com/paper/2606.25600/) (2026)
- [Hyperbolic Restricted Boltzmann Machine Neural Quantum State](https://mlsanity.com/paper/2609.26032/) (2026)

## Abstract (arXiv)

In this work, we introduce the first type of non-Euclidean neural quantum state (NQS) ansatz, in the form of the hyperbolic GRU (a variant of recurrent neural networks (RNNs)), to be used in the Variational Monte Carlo method of approximating the ground state energy for quantum many-body systems. In particular, we examine the performances of NQS ansatzes constructed from both conventional or Euclidean RNN/GRU and from hyperbolic GRU in the prototypical settings of the one- and two-dimensional transverse field Ising models (TFIM) and the one-dimensional Heisenberg J1J2 and J1J2J3 systems. By virtue of the fact that, for all of the experiments performed in this work, hyperbolic GRU can yield performances comparable to or better than Euclidean RNNs, which have been extensively studied in these settings in the literature, our work is a proof-of-concept for the viability of hyperbolic GRU as the first type of non-Euclidean NQS ansatz for quantum many-body systems. Furthermore, in settings where the Hamiltonian displays a clear hierarchical interaction structure, such as the 1D Heisenberg J1J2 & J1J2J3 systems with the 1st, 2nd and even 3rd nearest neighbor interactions, our results show that hyperbolic GRU definitively outperforms its Euclidean version in almost all instances. The fact that these results are reminiscent of the established ones from natural language processing where hyperbolic GRU almost always outperforms Euclidean RNNs when the training data exhibit a tree-like or hierarchical structure leads us to hypothesize that hyperbolic GRU NQS ansatz would likely outperform Euclidean RNN/GRU NQS ansatz in quantum spin systems that involve different degrees of nearest neighbor interactions. Finally, with this work, we hope to initiate future studies of other types of non-Euclidean NQS beyond hyperbolic GRU.