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.

量子机器学习中，基于置换对称系统的强大算法被提出，利用对称群、等价量子卷积神经网络等关键词建立了置换对称系统的强大算法。

置换等变量量子卷积神经网络

Permutation-equivariant quantum convolutional neural networks

Quantum Neural Networks (QNNs) are suggested as one of the quantum algorithms
which can be efficiently simulated with a low depth on near-term quantum
hardware in the presence of noises. However, their performance highly relies on
choosing the most suitable architecture of Variational Quantum Algorithms
(VQAs), and the problem-agnostic models often suffer issues regarding
trainability and generalization power. As a solution, the most recent works
explore Geometric Quantum Machine Learning (GQML) using QNNs equivariant with
respect to the underlying symmetry of the dataset. GQML adds an inductive bias
to the model by incorporating the prior knowledge on the given dataset and
leads to enhancing the optimization performance while constraining the search
space. This work proposes equivariant Quantum Convolutional Neural Networks
(EquivQCNNs) for image classification under planar $p4m$ symmetry, including
reflectional and $90^\circ$ rotational symmetry. We present the results tested
in different use cases, such as phase detection of the 2D Ising model and
classification of the extended MNIST dataset, and compare them with those
obtained with the non-equivariant model, proving that the equivariance fosters
better generalization of the model.

这项研究提出了与平面 $p4m$ 对称性相等的量子卷积神经网络（EquivQCNNs），用于图像分类，并在不同的用例中进行测试，证明了等变性促进了模型更好的泛化能力。