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arXiv:2602.06381

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

Semin Park, Chae-Yeun Park

quant-phcs.LGSubmitted Feb 6, 2026• Announced Feb 9, 2026• 0 citations
v1 · Feb 6, 2026v2 · May 7, 2026
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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.

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8+28 pages; 2 figures

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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.
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Recent developments in the field of quantum machine learning have promoted the idea of incorporating physical symmetries in the structure of quantum circuits. A crucial milestone in this area is the realization of S n -permutation equivariant quantum neural networks (QNNs) that are equivariant under permutations of input objects. In this paper, we focus on encoding the rotational symmetry of point cloud datasets into the QNN. The key insight of the approach is that all rotationally invariant functions with vector inputs are equivalent to a function with inputs of vector inner products. We provide a structure of the QNN that is exactly invariant to both rotations and permutations, with its efficacy demonstrated numerically in the problems of two-dimensional image classifications and identifying high-energy particle decays, produced by proton-proton collisions, with the SO ( 1 , 3 ) Lorentz symmetry. Published by the American Physical Society 2024
Quantum neural network architectures that have little to no inductive biases are known to face trainability and generalization issues. Inspired by a similar problem, recent breakthroughs in machine learning address this challenge by creating models encoding the symmetries of the learning task. This is materialized through the usage of equivariant neural networks the action of which commutes with that of the symmetry. In this work, we import these ideas to the quantum realm by presenting a comprehensive theoretical framework to design equivariant quantum neural networks (EQNNs) for essentially any relevant symmetry group. We develop multiple methods to construct equivariant layers for EQNNs and analyze their advantages and drawbacks. Our methods can find unitary or general equivariant quantum channels efficiently even when the symmetry group is exponentially large or continuous. As a special implementation, we show how standard quantum convolutional neural networks (QCNNs) can be generalized to group-equivariant QCNNs where both the convolution and pooling layers are equivariant to the symmetry group. We then numerically demonstrate the effectiveness of a S U ( 2 ) -equivariant QCNN over symmetry-agnostic QCNN on a classification task of phases of matter in the bond-alternating Heisenberg model. Our framework can be readily applied to virtually all areas of quantum machine learning. Lastly, we discuss about how symmetry-informed models such as EQNNs provide hopes to alleviate central challenges such as barren plateaus, poor local minima, and sample complexity. Published by the American Physical Society 2024
Abstract The Symmetric group S n manifests itself in large classes of quantum systems as the invariance of certain characteristics of a quantum state with respect to permuting the qubits. Subgroups of S n arise, among many other contexts, to describe label symmetry of classical images with respect to spatial transformations, such as reflection or rotation. Equipped with the formalism of geometric quantum machine learning, in this study we propose the architectures of equivariant quantum convolutional neural networks (EQCNNs) adherent to S n and its subgroups. We demonstrate that a careful choice of pixel-to-qubit embedding order can facilitate easy construction of EQCNNs for small subgroups of S n . Our novel EQCNN architecture corresponding to the full permutation group S n is built by applying all possible QCNNs with equal probability, which can also be conceptualized as a dropout strategy in quantum neural networks. For subgroups of S n , our numerical results using MNIST datasets show better classification accuracy than non-equivariant QCNNs. The S n -equivariant QCNN architecture shows significantly improved training and test performance than non-equivariant QCNN for classification of connected and non-connected graphs. When trained with sufficiently large number of data, the S n -equivariant QCNN shows better average performance compared to S n -equivariant QNN . These results contribute towards building powerful quantum machine learning architectures in permutation-symmetric systems.
Face recognition systems struggle when faces appear at different orientations. Standard convolutional neural networks handle translation well but have no built-in way to deal with rotations or reflections. Quantum neural networks offer a different kind of expressiveness, but most existing designs ignore spatial symmetry altogether. This paper introduces Eq-MG-QCNN, a hybrid quantum-classical model that builds rotation symmetry directly into the quantum circuit. The quantum filter is designed to be exactly equivariant under the Klein four-group, which covers horizontal flips, vertical flips, and 180° rotations. This is done through two mechanisms: sharing rotation parameters across all qubits (as required by orbit analysis), and connecting all qubit pairs with symmetric CZ gates (forming a complete K₄ graph). The model is tested on the ORL and Yale face databases under four rotation angles (0°, 90°, 180°, 270°) and compared against a classical CNN, a classical equivariant CNN, and the MG-QCNN quantum baseline. All experiments use noiseless quantum simulation. Eq-MG-QCNN reaches 94.3% best accuracy on ORL and 89.9% on Yale with only six quantum parameters, two fewer than the baseline. The model also shows low rotational variation across all four test angles. These results suggest that embedding group symmetry into quantum circuits is a practical way to build orientation-stable feature extractors for face recognition.