Geometric quantum machine learning based on equivariant quantum neural
networks (EQNN) recently appeared as a promising direction in quantum machine
learning. Despite the encouraging progress, the studies are still limited to
theory, and the role of hardware noise in EQNN training has never been
explored. This work studies the behavior of EQNN models in the presence of
noise. We show that certain EQNN models can preserve equivariance under Pauli
channels, while this is not possible under the amplitude damping channel. We
claim that the symmetry breaking grows linearly in the number of layers and
noise strength. We support our claims with numerical data from simulations as
well as hardware up to 64 qubits. Furthermore, we provide strategies to enhance
the symmetry protection of EQNN models in the presence of noise.

基于等变量量子神经网络（EQNN）的几何量子机器学习是量子机器学习中的一个有前景的方向，本文研究了 EQNN 模型在噪声存在下的行为，通过数值模拟和高达 64 比特的硬件实验支持实验的结果，并提出了在噪声存在时增强 EQNN 模型对称性保护的策略。

存在噪声下的几何量子机器学习中的对称性破缺

Symmetry breaking in geometric quantum machine learning in the presence  of noise

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。

数据嵌入在等变量量子卷积神经网络中的作用

The role of data embedding in equivariant quantum convolutional neural  networks

This paper presents a comprehensive comparative analysis of the performance
of Equivariant Quantum Neural Networks (EQNN) and Quantum Neural Networks
(QNN), juxtaposed against their classical counterparts: Equivariant Neural
Networks (ENN) and Deep Neural Networks (DNN). We evaluate the performance of
each network with two toy examples for a binary classification task, focusing
on model complexity (measured by the number of parameters) and the size of the
training data set. Our results show that the $\mathbb{Z}_2\times \mathbb{Z}_2$
EQNN and the QNN provide superior performance for smaller parameter sets and
modest training data samples.

该研究通过对比分析了等变量量子神经网络（EQNN）和量子神经网络（QNN）与它们的经典对应物等变量神经网络（ENN）和深度神经网络（DNN）的性能，在模型复杂度和训练数据集的大小方面，通过两个玩具示例的二分类任务对每个网络的性能进行了评估。我们的结果表明，Z2×Z2 EQNN 和 QNN 在较小参数集和适度训练数据样本的情况下提供了更好的性能。