HyQuRP: Hybrid quantum-classical neural network with rotational and permutational equivariance for 3D point clouds

TL;DR

HyQuRP introduces a representation-theoretic, rotation- and permutation-equivariant hybrid quantum–classical network for 3D point clouds. It combines singlet-state initialization, per-pair rotation-equivariant encoding, and a twirled, pair-permutation–invariant quantum network with a permutation-invariant classical head to achieve end-to-end invariance. Across sparse-point benchmarks, HyQuRP outperforms strong classical and quantum baselines, illustrating data efficiency and the potential of symmetry-guided quantum learning for geometric data. The work highlights a principled approach to integrating group theory into QML architectures and outlines pathways toward scalable quantum implementations.

Abstract

We introduce HyQuRP, a hybrid quantum-classical neural network equivariant to rotational and permutational symmetries. While existing equivariant quantum machine learning models often rely on ad hoc constructions, HyQuRP is built upon the formal foundations of group representation theory. In the sparse-point regime, HyQuRP consistently outperforms strong classical and quantum baselines across multiple benchmarks. For example, when six subsampled points are used, HyQuRP ($\sim$1.5K parameters) achieves 76.13% accuracy on the 5-class ModelNet benchmark, compared to approximately 71% for PointNet, PointMamba, and PointTransformer with similar parameter counts. These results highlight HyQuRP's exceptional data efficiency and suggest the potential of quantum machine learning models for processing 3D point cloud data.

Figures (2)

• Figure 1: HyQuRP Pipeline. A hybrid pipeline maps a set of $N$ 3D points to a $K\times1$ score vector, where $K$ is the number of classes. The quantum circuit is constructed as follows: (i) Initialize each qubit pair in the singlet state. (ii) Encode points on the even-indexed qubits (0, 2, 4, …) while preserving the pair structure. (iii) Apply twirling over cycles of length $2$ through $N$, under appropriate constraints, resulting in two effective gate types for each cycle length; we then repeat this block $B$ times to form a quantum network. (iv) For every unordered pair $\langle i,j \rangle$, measure two types of Hamiltonians ($H^+_{\langle i, j\rangle}$ and $H^-_{\langle i, j\rangle}$), giving a total $2\tbinom{N}{2}$ expectation values. (v) Send the features to a classical head (pre-pooling neural layers, permutation-invariant pooling, post-pooling classifier) to produce the score vector. Each color indicates how each component behaves under group actions on the input point set: (Purple) rotation- and permutation-invariant, (Blue) rotation- and permutation-equivariant, (Gray) rotation-invariant and permutation-equivariant, (Green) permutation-invariant readout and classifier. Dashed rounded boxes indicate repetition by $B$.
• Figure 2: Overall Results. Solid curves represent the mean accuracy across all seeds (per dataset and point budget) and shaded regions indicate $\pm\sigma$. The upper-right table reports trainable parameter counts (Light/Mid) and the average ranks aggregated over all settings (lower is better). The lower-right table reports cosine similarity and $\ell_2$-norm ratio between the logits produced from a single point-set and its rotated and permuted counterparts after 10 epochs; definitions and details are provided in Appendix \ref{['app:additional_details']}.