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

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.

💡 Research Summary

HyQuRP (Hybrid Quantum‑Classical Neural Network with Rotational and Permutational equivariance) is a novel architecture designed to process three‑dimensional point clouds while rigorously respecting the two fundamental symmetries of such data: global rigid rotations (the SO(3) group) and arbitrary permutations of the point order (the symmetric group Sₙ). Existing quantum machine‑learning (QML) approaches either ignore these symmetries or rely on ad‑hoc constructions that break equivariance, largely because Schur‑Weyl duality makes it difficult to build a non‑trivial unitary that is simultaneously equivariant under both SU(2) (the double cover of SO(3)) and Sₙ.

HyQuRP overcomes this limitation through a carefully engineered pipeline that combines representation‑theoretic principles with practical quantum circuit design. The main components are:

1. Singlet‑State Initialization – For a point cloud containing N points, a register of 2 N qubits is prepared, grouped into N adjacent Bell‑singlet pairs (|01⟩ − |10⟩)/√2. Each singlet is invariant under any collective SU(2) rotation, providing a natural SU(2)‑invariant substrate.

2. Per‑Pair Geometric Encoding – Each 3‑D coordinate p ∈ ℝ³ is encoded on the even‑indexed qubit of its pair via the unitary
     E(p) = exp(i Θ p·σ),
   where σ = (X, Y, Z) are Pauli matrices and Θ is a fixed scaling hyper‑parameter. Because the Pauli vector transforms as a 3‑vector under SU(2), the encoding satisfies E(Rp) = U_R E(p) U_R† for any rotation R ∈ SO(3), guaranteeing rotational equivariance.

3. Group‑Twirling Quantum Network – Arbitrary quantum operations are projected onto the subspace of operators that commute with the joint action of SO(3) and Sₙ by applying the twirling map
     T_G