Permutation-equivariant quantum convolutional neural networks

TL;DR

This paper introduces permutation-equivariant quantum convolutional neural networks (EQCNNs) that leverage symmetry properties of quantum systems and classical image transformations, demonstrating improved accuracy and training performance on various datasets.

Contribution

The paper proposes novel EQCNN architectures for the full permutation group and its subgroups, utilizing symmetry principles to enhance quantum machine learning models.

Findings

EQCNNs outperform non-equivariant QCNNs on MNIST classification.

S_n-equivariant QCNNs show better graph classification accuracy.

Large data training improves average performance of S_n-equivariant QNNs.

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. The subgroups of $S_{n}$ arise, among many other contexts, to describe label symmetry of classical images with respect to spatial transformations, e.g. reflection or rotation. Equipped with the formalism of geometric quantum machine learning, in this work 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.