The importance of symmetries has recently been recognized in quantum machine learning from the simple motto: if a task exhibits a symmetry (given by a group $\mathfrak{G}$), the learning model should respect said symmetry. This can be instantiated via $\mathfrak{G}$-equivariant Quantum Neural Networks (QNNs), i.e., parametrized quantum circuits whose gates are generated by operators commuting with a given representation of $\mathfrak{G}$. In practice, however, there might be additional restrictions to the types of gates one can use, such as being able to act on at most $k$ qubits. In this work we study how the interplay between symmetry and $k$-bodyness in the QNN generators affect its expressiveness for the special case of $\mathfrak{G}=S_n$, the symmetric group. Our results show that if the QNN is generated by one- and two-body $S_n$-equivariant gates, the QNN is semi-universal but not universal. That is, the QNN can generate any arbitrary special unitary matrix in the invariant subspaces, but has no control over the relative phases between them. Then, we show that in order to reach universality one needs to include $n$-body generators (if $n$ is even) or $(n-1)$-body generators (if $n$ is odd). As such, our results brings us a step closer to better understanding the capabilities and limitations of equivariant QNNs.

翻译：对称性的重要性近来在量子机器学习中得到了认可，其核心理念为：若某任务具有对称性（由群 $\mathfrak{G}$ 描述），则学习模型应尊重该对称性。这可通过 $\mathfrak{G}$ 等变量子神经网络（QNN）实现，即参数化量子电路，其量子门由与 $\mathfrak{G}$ 的给定表示对易的算符生成。然而在实践中，可使用的量子门类型可能受到额外限制，例如最多仅能作用于 $k$ 个量子比特。本文研究在 $\mathfrak{G}=S_n$（对称群）的特殊情形下，QNN 生成器中的对称性与 $k$ 体性之间的相互作用如何影响其表达能力。结果表明：若 QNN 由单体和双体 $S_n$ 等变门生成，则该 QNN 为半普适性而非普适性，即 QNN 能生成不变子空间内的任意特殊酉矩阵，但无法控制这些子空间之间的相对相位。进一步，为达到普适性，需引入 $n$ 体生成器（若 $n$ 为偶数）或 $(n-1)$ 体生成器（若 $n$ 为奇数）。因此，我们的研究结果使我们更深入理解等变 QNN 的能力与局限性。