# HyQuRP: Hybrid quantum-classical neural network with rotational and permutational equivariance

arXiv: https://arxiv.org/abs/2602.06381 · HTML: https://mlsanity.com/paper/2602.06381/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 39 of its 64 references that are in the map.
- First posted on arXiv 6 Feb 2026.

## Builds on (most cited first, 30 of 39)

- [PointNet: Deep Learning on Point Sets for 3D Classification and Segmentation](https://mlsanity.com/paper/1612.00593/) (2016)
- [PointNet++: Deep Hierarchical Feature Learning on Point Sets in a Metric Space](https://mlsanity.com/paper/1706.02413/) (2017)
- [ShapeNet: An Information-Rich 3D Model Repository](https://mlsanity.com/paper/1512.03012/) (2015)
- [Dynamic Graph CNN for Learning on Point Clouds](https://mlsanity.com/paper/1801.07829/) (2018)
- [Point Transformer](https://mlsanity.com/paper/2011.00931/) (2020)
- [Group Equivariant Convolutional Networks](https://mlsanity.com/paper/1602.07576/) (2016)
- [PCT: Point cloud transformer](https://mlsanity.com/paper/2012.09688/) (2020)
- [E(3)-Equivariant Graph Neural Networks for Data-Efficient and Accurate Interatomic Potentials](https://mlsanity.com/paper/2101.03164/) (2021)
- [Supervised learning with quantum enhanced feature spaces](https://mlsanity.com/paper/1804.11326/) (2018)
- [Point-BERT: Pre-training 3D Point Cloud Transformers with Masked Point Modeling](https://mlsanity.com/paper/2111.14819/) (2021)
- [Multi-view Convolutional Neural Networks for 3D Shape Recognition](https://mlsanity.com/paper/1505.00880/) (2015)
- [Deep Learning for 3D Point Clouds: A Survey](https://mlsanity.com/paper/1912.12033/) (2019)
- [Masked Autoencoders for Point Cloud Self-supervised Learning](https://mlsanity.com/paper/2203.06604/) (2022)
- [General E(2)-Equivariant Steerable CNNs](https://mlsanity.com/paper/1911.08251/) (2019)
- [PointNeXt: Revisiting PointNet++ with Improved Training and Scaling Strategies](https://mlsanity.com/paper/2206.04670/) (2022)
- [The power of quantum neural networks](https://mlsanity.com/paper/2011.00027/) (2020)
- [Relation-Shape Convolutional Neural Network for Point Cloud Analysis](https://mlsanity.com/paper/1904.07601/) (2019)
- [Vector Neurons: A General Framework for SO(3)-Equivariant Networks](https://mlsanity.com/paper/2104.12229/) (2021)
- [Learning SO(3) Equivariant Representations with Spherical CNNs](https://mlsanity.com/paper/1711.06721/) (2017)
- [Clebsch-Gordan Nets: a Fully Fourier Space Spherical Convolutional Neural Network](https://mlsanity.com/paper/1806.09231/) (2018)
- [PointMamba: A Simple State Space Model for Point Cloud Analysis](https://mlsanity.com/paper/2402.10739/) (2024)
- [Revisiting Point Cloud Shape Classification with a Simple and Effective Baseline](https://mlsanity.com/paper/2106.05304/) (2021)
- [Lorentz Group Equivariant Neural Network for Particle Physics](https://mlsanity.com/paper/2006.04780/) (2020)
- [Better than classical? The subtle art of benchmarking quantum machine learning models](https://mlsanity.com/paper/2403.07059/) (2024)
- [Surface Representation for Point Clouds](https://mlsanity.com/paper/2205.05740/) (2022)
- [SE(3) Equivariant Graph Neural Networks with Complete Local Frames](https://mlsanity.com/paper/2110.14811/) (2021)
- [Exploiting symmetry in variational quantum machine learning](https://mlsanity.com/paper/2205.06217/) (2022)
- [Point Cloud Mamba: Point Cloud Learning via State Space Model](https://mlsanity.com/paper/2403.00762/) (2024)
- [Theory for Equivariant Quantum Neural Networks](https://mlsanity.com/paper/2210.08566/) (2022)
- [Group-Invariant Quantum Machine Learning](https://mlsanity.com/paper/2205.02261/) (2022)

## Abstract (arXiv)

Group-equivariant quantum machine learning has emerged as a promising paradigm by incorporating symmetry into quantum models. However, constructing models simultaneously equivariant to both rotational and permutational symmetries in a principled manner remains a bottleneck. In this work, we develop a general framework for dual-equivariant gates under rotations and permutations and analyze the dimension of the resulting gate space using group representation theory. Based on this, we introduce HyQuRP, a hybrid quantum-classical neural network with dual equivariance. On 3D point cloud classification benchmarks in the sparse-point regime, HyQuRP outperforms strong classical and quantum baselines. For example, when six subsampled points are used, HyQuRP (~1.5K parameters) achieves 76.13% accuracy on the 5-class ModelNet benchmark, compared with 72.54%, 71.09%, and 71.03% for Tensor Field Network, PointNet, and PointMamba with similar parameter counts. These results highlight HyQuRP's strong data efficiency and suggest the potential of equivariant quantum machine learning approaches in symmetry-sensitive tasks.