Geometric deep learning refers to the scenario in which the symmetries of a
dataset are used to constrain the parameter space of a neural network and thus,
improve their trainability and generalization. Recently this idea has been
incorporated into the field of quantum machine learning, which has given rise
to equivariant quantum neural networks (EQNNs). In this work, we investigate
the role of classical-to-quantum embedding on the performance of equivariant
quantum convolutional neural networks (EQCNNs) for the classification of
images. We discuss the connection between the data embedding method and the
resulting representation of a symmetry group and analyze how changing
representation affects the expressibility of an EQCNN. We numerically compare
the classification accuracy of EQCNNs with three different basis-permuted
amplitude embeddings to the one obtained from a non-equivariant quantum
convolutional neural network (QCNN). Our results show that all the EQCNNs
achieve higher classification accuracy than the non-equivariant QCNN for small
numbers of training iterations, while for large iterations this improvement
crucially depends on the used embedding. It is expected that the results of
this work can be useful to the community for a better understanding of the
importance of data embedding choice in the context of geometric quantum machine
learning.

对使用数据集的对称性来约束神经网络的参数空间以提高其可训练性和泛化能力的几何深度学习方法在量子机器学习领域得到了应用，其中包括了等变量量子神经网络 (EQNNs)。本研究探讨了经典到量子嵌入对等变量量子卷积神经网络（EQCNNs）图像分类性能的影响，分析了数据嵌入方法与对称群表示的关联，以及不同表示对 EQCNN 表达能力的影响。我们数值比较了基于三种不同基佯波函数嵌入的 EQCNN 与非等变量量子卷积神经网络（QCNN）的分类准确性。结果显示，在训练迭代次数较少时，所有的 EQCNN 都比非等变量 QCNN 具有更高的分类准确性，而在迭代次数较多时，这种改进明显依赖于所使用的嵌入方法。预计本研究的结果对于几何量子机器学习中数据嵌入选择的重要性有着更好的了解，对学术界具有实用意义。