Efficient Prediction of SO(3)-Equivariant Hamiltonian Matrices via SO(2) Local Frames
Haiyang Yu, Yuchao Lin, Xuan Zhang, Xiaofeng Qian, Shuiwang Ji
TL;DR
This paper introduces QHNetV2, an efficient SO(3)-equivariant neural network that predicts Hamiltonian matrices by leveraging SO(2) local frames, avoiding costly tensor products and demonstrating superior performance on molecular datasets.
Contribution
The paper proposes a novel SO(2)-equivariant network architecture that achieves SO(3) equivariance efficiently without tensor products, improving electronic structure predictions.
Findings
Achieves superior accuracy on QH9 and MD17 datasets.
Demonstrates strong generalization across diverse molecular structures.
Eliminates the need for costly SO(3) tensor products in equivariant learning.
Abstract
We consider the task of predicting Hamiltonian matrices to accelerate electronic structure calculations, which plays an important role in physics, chemistry, and materials science. Motivated by the inherent relationship between the off-diagonal blocks of the Hamiltonian matrix and the SO(2) local frame, we propose a novel and efficient network, called QHNetV2, that achieves global SO(3) equivariance without the costly SO(3) Clebsch-Gordan tensor products. This is achieved by introducing a set of new efficient and powerful SO(2)-equivariant operations and performing all off-diagonal feature updates and message passing within SO(2) local frames, thereby eliminating the need of SO(3) tensor products. Moreover, a continuous SO(2) tensor product is performed within the SO(2) local frame at each node to fuse node features, mimicking the symmetric contraction operation. Extensive experiments…
Peer Reviews
Decision·Submitted to ICLR 2026
1. The theory of connecting the SO(2) to minimal frame averaging is interesting. Appendix C shows how frame averaging collapses to the canonical rotation when the local model is equivariant to the stabilizer. 2. Empirical improvements are clear and practically relevant. Laudable MAE reductions on QH9 (including OOD) and solid speedups vs a strong SO(3)-TP baseline demonstrate usefulness (Table 1 & 3, pp. 9, 14).
1. The scalability of the model remains unclear. The authors should evaluate and compare performance with SPHNet and QHNet across systems of increasing Hamiltonian size to assess how well the proposed approach scales. 2. The discussion of related work is underdeveloped. Given that the contribution lies within the design of equivariant architectures, the paper should discuss prior SO(3) and SO(2)-equivariant models and spherical scalarization models (e.g., e3nn, SE(3)-Transformers, TFN, Allegro,
1. Comprehensive use of SO(2) operations. The model systematically generalizes SO(2)-equivariant operations beyond the message passing step, extending them to the FFN, gate, and linear layers, and allowing an entirely SO(2)-based neural architecture. This contributes to higher efficiency and stability. 2. Strong and broad empirical performance. The model achieves state-of-the-art accuracy on QH9 and competitive results on MD17, showing that the proposed SO(2) local frame formulation generalizes
1. Limited novelty of the core operator. The SO(2) linear/tensor-product mechanism has already been introduced and optimized in eSCN [1] and subsequently adopted in EquiformerV2 [2] and eSEN [3]; moreover, it has been used in downstream Hamiltonian modeling such as DeepH-2 (which employs EquiformerV2’s SO(2) convolution) [4]. As a result, the present paper’s contribution appears primarily as extending these existing SO(2) operations to additional layers (FFN, gate) and applying them to Hamiltoni
1. Use of local SO(2) frames to achieve global SO(3) equivariance to reduce costs 2. Improved efficiency gains with comparable or better accuracy. 3. Promising direction for scalable learning of quantum Hamiltonians.
1. While the idea of achieving global SO(3) equivariance through SO(2) operations is elegant, it is not entirely novel. Similar concepts have been explored in previous works, such as eSCN, which the authors themselves acknowledge. The main contribution here lies in applying this framework to Hamiltonian prediction. As a result, the methodological innovation is somewhat limited, and the paper would benefit from a clearer articulation of what new theoretical or algorithmic insights distinguish it
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Taxonomy
TopicsMachine Learning in Materials Science · Advanced Graph Neural Networks · Quantum many-body systems
MethodsLib · Sparse Evolutionary Training