# Learning local equivariant representations for quantum operators

arXiv: https://arxiv.org/abs/2407.06053 · HTML: https://mlsanity.com/paper/2407.06053/

## Key facts

- Cited by 8 papers in the map (data through 14 Oct 2026).
- Builds on 23 of its 46 references that are in the map.
- First posted on arXiv 8 Jul 2024.

## Builds on (most cited first, 23 of 23)

- [Neural Message Passing for Quantum Chemistry](https://mlsanity.com/paper/1704.01212/) (2017)
- [E(n) Equivariant Graph Neural Networks](https://mlsanity.com/paper/2102.09844/) (2021)
- [Tensor field networks: Rotation- and translation-equivariant neural networks for 3D point clouds](https://mlsanity.com/paper/1802.08219/) (2018)
- [E(3)-Equivariant Graph Neural Networks for Data-Efficient and Accurate Interatomic Potentials](https://mlsanity.com/paper/2101.03164/) (2021)
- [MACE: Higher Order Equivariant Message Passing Neural Networks for Fast and Accurate Force Fields](https://mlsanity.com/paper/2206.07697/) (2022)
- [3D Steerable CNNs: Learning Rotationally Equivariant Features in Volumetric Data](https://mlsanity.com/paper/1807.02547/) (2018)
- [Equiformer: Equivariant Graph Attention Transformer for 3D Atomistic Graphs](https://mlsanity.com/paper/2206.11990/) (2022)
- [Learning Local Equivariant Representations for Large-Scale Atomistic Dynamics](https://mlsanity.com/paper/2204.05249/) (2022)
- [Deep Potential Molecular Dynamics: a scalable model with the accuracy of quantum mechanics](https://mlsanity.com/paper/1707.09571/) (2017)
- [EquiformerV2: Improved Equivariant Transformer for Scaling to Higher-Degree Representations](https://mlsanity.com/paper/2306.12059/) (2023)
- [e3nn: Euclidean Neural Networks](https://mlsanity.com/paper/2207.09453/) (2022)
- [Clebsch-Gordan Nets: a Fully Fourier Space Spherical Convolutional Neural Network](https://mlsanity.com/paper/1806.09231/) (2018)
- [Reducing SO(3) Convolutions to SO(2) for Efficient Equivariant GNNs](https://mlsanity.com/paper/2302.03655/) (2023)
- [On the Expressive Power of Geometric Graph Neural Networks](https://mlsanity.com/paper/2301.09308/) (2023)
- [DeePMD-kit: A deep learning package for many-body potential energy representation and molecular dynamics](https://mlsanity.com/paper/1712.03641/) (2017)
- [SE(3)-equivariant prediction of molecular wavefunctions and electronic densities](https://mlsanity.com/paper/2106.02347/) (2021)
- [TensorNet: Cartesian Tensor Representations for Efficient Learning of Molecular Potentials](https://mlsanity.com/paper/2306.06482/) (2023)
- [Spherical Channels for Modeling Atomic Interactions](https://mlsanity.com/paper/2206.14331/) (2022)
- [N-body Networks: a Covariant Hierarchical Neural Network Architecture for Learning Atomic Potentials](https://mlsanity.com/paper/1803.01588/) (2018)
- [Efficient and Equivariant Graph Networks for Predicting Quantum Hamiltonian](https://mlsanity.com/paper/2306.04922/) (2023)
- [A Survey of Geometric Graph Neural Networks: Data Structures, Models and Applications](https://mlsanity.com/paper/2403.00485/) (2024)
- [Equivariant representations for molecular Hamiltonians and N-center atomic-scale properties](https://mlsanity.com/paper/2109.12083/) (2021)
- [TraceGrad: a Framework Learning Expressive SO(3)-equivariant Non-linear Representations for Electronic-Structure Hamiltonian Prediction](https://mlsanity.com/paper/2405.05722/) (2024)

## Built on by (most cited first, 8 of 8)

- [Bohrium + SciMaster: Building the Infrastructure and Ecosystem for Agentic Science at Scale](https://mlsanity.com/paper/2512.20469/) (2025)
- [Efficient Prediction of SO(3)-Equivariant Hamiltonian Matrices via SO(2) Local Frames](https://mlsanity.com/paper/2506.09398/) (2025)
- [Deep Learning Accelerated Quantum Transport Simulations in Nanoelectronics: From Break Junctions to Field-Effect Transistors](https://mlsanity.com/paper/2411.08800/) (2024)
- [Deep Neural Network for Phonon-Assisted Optical Spectra in Semiconductors](https://mlsanity.com/paper/2502.00798/) (2025)
- [Physics-Informed Long-Range Coulomb Correction for Machine-learning Hamiltonians](https://mlsanity.com/paper/2603.20007/) (2026)
- [A Density-Matrix Framework for Electronic-Structure Analysis of Electrolytes for Lithium Batteries](https://mlsanity.com/paper/2607.25597/) (2026)
- [Equivariant Flow Matching for Electron Density Prediction](https://mlsanity.com/paper/2610.02651/) (2026)
- [Physics-Aligned Electronic Ground-State Learning Improves Generalization](https://mlsanity.com/paper/2610.10298/) (2026)

## Abstract (arXiv)

Predicting quantum operator matrices such as Hamiltonian, overlap, and density matrices in the density functional theory (DFT) framework is crucial for material science. Current methods often focus on individual operators and struggle with efficiency and scalability for large systems. Here we introduce a novel deep learning model, SLEM (strictly localized equivariant message-passing) for predicting multiple quantum operators, that achieves state-of-the-art accuracy while dramatically improving computational efficiency. SLEM's key innovation is its strict locality-based design for equivariant representations of quantum tensors while preserving physical symmetries. This enables complex many-body dependency without expanding the effective receptive field, leading to superior data efficiency and transferability. Using an innovative SO(2) convolution and invariant overlap parameterization, SLEM reduces the computational complexity of high-order tensor products and is therefore capable of handling systems requiring the f and g orbitals in their basis sets. We demonstrate SLEM's capabilities across diverse 2D and 3D materials, achieving high accuracy even with limited training data. SLEM's design facilitates efficient parallelization, potentially extending DFT simulations to systems with device-level sizes, opening new possibilities for large-scale quantum simulations and high-throughput materials discovery.