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:从简单的座右铭学习量子机器,最近认识到了对称的重要性:如果任务显示对称(由一组 $mathfrak{G} 美元组成),学习模型应该尊重对称。这可以通过 $\ mathfrak{G}$-quiquivarative 量神经网络(QNNs) 即刻化量子电路的重要性,即由操作者用一个特定代表单位($\mathfrak{G}$ 生成的大门。但在实践中,对于可以使用的门类型可能存在额外的限制,例如能够在最多一k美元平方位上采取行动。在这项工作中,我们研究QNN的对称性和美元体积之间的相互作用如何影响它对于一个特殊案例($mathfqurak{G<unk> _S__n_n)的直观性能。我们的结果显示,如果QNNNN是一步制的,但是半制的对QNF$(美元),那么在一美元和二位平面的基中, 就会显示一个正统基质的对等的对等的对等的对等的对等(美元) 。</s>