The role of data embedding in equivariant quantum convolutional neural networks

TL;DR

The role of classical-to-quantum embedding on the performance of equivariant quantum convolutional neural networks (EQCNNs) for the classification of images is investigated and a clear dependence of classification accuracy on the underlying embedding, especially for initial training iterations is shown.

Abstract

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 a clear dependence of classification accuracy on the underlying embedding, especially for initial training iterations. The improvement in classification accuracy of EQCNN over non-equivariant QCNN may be present or absent depending on the particular embedding and dataset used. 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.

Figures (9)

• Figure 1: The structure of QCNN for 10 qubits. The orange and the cyan boxes represent respectively the parametrized convolutional and pooling ansatze. The input quantum state is $\vert \psi\rangle$ and the final measurement is $M$.
• Figure 2: A $2\times 2$ image and its transformations under reflection and rotation by $180^{\circ}$. The table shows group representations of these two symmetries with changing embedding of the image as a quantum state. $\mathcal{N}$ is the quantum state normalization factor.
• Figure 3: The relation between quantum-encoded image state $\vert \Psi\rangle$, basis-permuting matrix $A$, and the representations $R(g)$ and $R^{\prime}(g)$ before and after the basis permutation, respectively.
• Figure 4: (a) The convolutional ansatze and (b) the pooling ansatze used in this work. The specific use case of each ansatz is discussed in Sec. \ref{['our_symmetry']}.
• Figure 5: Test set accuracy obtained from equivariant and non-equivariant QCNNs when using maximum possible set of equivariant generators. The vertical axis shows the average accuracy obtained over 10 randomly initialized runs and the standard deviation is indicated with the shaded area. (a) Classification of classes $0$ and $1$ of Fashion MNIST dataset. The inset shows a magnified part of the same with the horizontal axis spanning a few hundreds of iterations. (b) Classification of classes $1$ and $2$ of Cifar10 dataset.