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.

翻译：几何深度学习指的是利用数据集的对称性来约束神经网络的参数空间，从而提升其可训练性和泛化能力的场景。最近，这一思想被引入量子机器学习领域，催生了等变量子神经网络（EQNNs）。本文研究了经典到量子数据嵌入方式对等变量子卷积神经网络（EQCNNs）用于图像分类性能的影响。我们探讨了数据嵌入方法与对称群表示之间的关系，并分析了表示方式的改变如何影响EQCNN的表达能力。我们通过数值实验，将采用三种不同基置换幅度嵌入的EQCNN与非等变量子卷积神经网络（QCNN）的分类准确率进行了比较。结果表明，在训练迭代次数较少时，所有EQCNN的分类准确率均高于非等变QCNN；而随着迭代次数增加，这种提升效果关键取决于所用的嵌入方式。本工作有望帮助学界更好地理解几何量子机器学习中数据嵌入选择的重要性。