On the universality of Sn-equivariant k-body gates
Sujay Kazi, Martín Larocca, Marco Cerezo · New Journal of Physics · 2024
Abstract 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 G ), the learning model should respect said symmetry. This can be instantiated via G -equivariant quantum neural networks (QNNs), i.e. parametrized quantum circuits whose gates are generated by operators commuting with a given representation of 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 mostkqubits. In this work we study how the interplay between symmetry andk-bodyness in the QNN generators affect its expressiveness for the special case of G=Sn , the symmetric group. Our results show that if the QNN is generated by one- and two-bodySn-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 includen-body generators (ifnis even) or (n−1) -body generators (ifnis odd). As such, our results brings us a step closer to better understanding the capabilities and limitations of equivariant QNNs.