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

For any subgroup $G$ of the symmetric group $\mathcal{S}_n$ on $n$ symbols,
we present results for the uniform $\mathcal{C}^k$ approximation of
$G$-invariant functions by $G$-invariant polynomials. For the case of totally
symmetric functions ($G = \mathcal{S}_n$), we show that this gives rise to the
sum-decomposition Deep Sets ansatz of Zaheer et al. (2018), where both the
inner and outer functions can be chosen to be smooth, and moreover, the inner
function can be chosen to be independent of the target function being
approximated. In particular, we show that the embedding dimension required is
independent of the regularity of the target function, the accuracy of the
desired approximation, as well as $k$. Next, we show that a similar procedure
allows us to obtain a uniform $\mathcal{C}^k$ approximation of antisymmetric
functions as a sum of $K$ terms, where each term is a product of a smooth
totally symmetric function and a smooth antisymmetric homogeneous polynomial of
degree at most $\binom{n}{2}$. We also provide upper and lower bounds on $K$
and show that $K$ is independent of the regularity of the target function, the
desired approximation accuracy, and $k$.

给定对称群 $S_n$ 的任何子群 $G$，我们研究了 $G$ 的不变函数对 $G$ 的不变多项式的均匀 $C^k$ 逼近，特别是对于完全对称函数的情况，我们表明这导致了 Zaheer 等人（2018）关于和分解 Deep Sets 的假设，其中内部和外部函数都可以选择为光滑函数，并且内部函数可以选择与所逼近的目标函数无关。

G - 不变和反对称函数的统一 C^k 逼近、嵌入维度和多项式表示

Uniform $\mathcal{C}^k$ Approximation of $G$-Invariant and Antisymmetric  Functions, Embedding Dimensions, and Polynomial Representations

The learnable, linear neural network layers between tensor power spaces of
$\mathbb{R}^{n}$ that are equivariant to the orthogonal group, $O(n)$, the
special orthogonal group, $SO(n)$, and the symplectic group, $Sp(n)$, were
characterised in arXiv:2212.08630. We present an algorithm for multiplying a
vector by any weight matrix for each of these groups, using category theoretic
constructions to implement the procedure. We achieve a significant reduction in
computational cost compared with a naive implementation by making use of
Kronecker product matrices to perform the multiplication. We show that our
approach extends to the symmetric group, $S_n$, recovering the algorithm of
arXiv:2303.06208 in the process.

使用范畴论构建算法实现可变换于正交群、特殊正交群、辛群及对称群的张量空间间的学习可线性神经网络层，运用克罗内克积矩阵大幅降低运算成本。

Brauer 群不变神经网络层计算算法

An Algorithm for Computing with Brauer's Group Equivariant Neural  Network Layers

The symmetric group $\mathsf{S}_n$ and the partition algebra
$\mathsf{P}_k(n)$ centralize one another in their actions on the $k$-fold
tensor power $\mathsf{M}_n^{\otimes k}$ of the $n$-dimensional permutation
module $\mathsf{M}_n$ of $\mathsf{S}_n$. The duality afforded by the commuting
actions determines an algebra homomorphism $\Phi_{k,n}: \mathsf{P}_k(n) \to
\mathsf{End}_{\mathsf{S}_n}(\mathsf{M}_n^{\otimes k})$ from the partition
algebra to the centralizer algebra
$\mathsf{End}_{\mathsf{S}_n}(\mathsf{M}_n^{\otimes k})$, which is a surjection
for all $k, n \in \mathbb{Z}_{\ge 1}$, and an isomorphism when $n \ge 2k$. We
present results that can be derived from the duality between $\mathsf{S}_n$ and
$\mathsf{P}_k(n)$; for example, (i) expressions for the multiplicities of the
irreducible $\mathsf{S}_n$-summands of $\mathsf{M}_n^{\otimes k}$, (ii)
formulas for the dimensions of the irreducible modules for the centralizer
algebra $\mathsf{End}_{\mathsf{S}_n}(\mathsf{M}_n^{\otimes k})$, (iii) a
bijection between vacillating tableaux and set-partition tableaux, (iv)
identities relating Stirling numbers of the second kind and the number of fixed
points of permutations, and (v) character values for the partition algebra
$\mathsf{P}_k(n)$. When $2k >n$, the map $\Phi_{k,n}$ has a nontrivial kernel
which is generated as a two-sided ideal by a single idempotent. We describe the
kernel and image of $\Phi_{k,n}$ in terms of the orbit basis of
$\mathsf{P}_k(n)$ and explain how the surjection $\Phi_{k,n}$ can also be used
to obtain the fundamental theorems of invariant theory for the symmetric group.

本文研究了对称群和划分代数在张量幂次、中心化张量幂次、定点数等方面的关系，并讨论了它们之间的双重作用及其相关应用。