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：对称性的重要性近期在量子机器学习领域受到关注，其核心理念可概括为：若任务呈现某种对称性（由群$\mathfrak{G}$描述），则学习模型应保持该对称性。这可通过$\mathfrak{G}$-等变量子神经网络（QNNs）实现，即其门操作由与$\mathfrak{G}$的给定表示对易的算子生成的参数化量子电路。然而在实际应用中，可用门的类型可能存在额外限制，例如仅能作用于最多$k$个量子比特。本文研究在$\mathfrak{G}=S_n$（对称群）这一特殊情形下，QNN生成元中对称性与$k$-体性质的相互作用如何影响其表达能力。我们的结果表明：若QNN由一维和二维$S_n$-等变门生成，则QNN是半普适而非普适的。这意味着QNN能在不变子空间中生成任意特殊酉矩阵，但无法控制子空间之间的相对相位。进而我们证明：为实现普适性，需要引入$n$体生成元（当$n$为偶数时）或$(n-1)$体生成元（当$n$为奇数时）。因此，本研究使我们朝着深入理解等变QNN的能力与局限性迈进一步。