Two-dimensional Hyperbolic RNN Neural Quantum State

Research Question and Approach

This paper investigates whether hyperbolic geometry—which naturally accommodates hierarchical, tree-like structures—can improve the performance of Neural Quantum States (NQS) when modeling many-body quantum systems. The author introduces a new two-dimensional hyperbolic NQS architecture, the Lorentz 2DRNN, and benchmarks it against standard Euclidean 2DRNNs using the 2D Transverse Field Ising Model (2DTFIM). Additionally, the study evaluates one-dimensional hyperbolic NQS (Poincaré and Lorentz RNN/GRU) by mapping the 2D lattice into a 1D sequence, testing these models across various magnetic field strengths and lattice sizes.

The study demonstrates that hyperbolic NQS consistently outperform their Euclidean counterparts in the 2DTFIM setting. Specifically, the Lorentz 2DRNN shows superior performance at the system's phase transition point. The author suggests this is due to the AdS/CFT correspondence, where the hyperbolic geometry of the neural network aligns with the hyperbolic spatial geometry dual to the conformal field theory describing the system at criticality. In the 1D mapping experiments, the hierarchical structure induced by the 2D-to-1D conversion, combined with the CFT physics at the critical point, further validates the effectiveness of hyperbolic ansatzes.

Standard NQS architectures often struggle to capture the complex correlations in quantum systems that exhibit hierarchical structures or critical behavior. By utilizing hyperbolic space, which allows for low-distortion embeddings of these structures, this work provides a promising alternative to traditional Euclidean neural networks. These findings offer a proof-of-concept that non-Euclidean geometries can provide more accurate approximations for quantum many-body wavefunctions, potentially opening new avenues for simulating quantum phase transitions.