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Despite the great promise of quantum machine learning models, there areseveral challenges one must overcome before unlocking their full potential. Forinstance, models based on quantum neural networks (QNNs) can suffer fromexcessive local minima and barren plateaus in their training landscapes.Recently, the nascent field of geometric quantum machine learning (GQML) hasemerged as a potential solution to some of those issues. The key insight ofGQML is that one should design architectures, such as equivariant QNNs,encoding the symmetries of the problem at hand. Here, we focus on problems withpermutation symmetry (i.e., the group of symmetry $S_n$), and show how to build$S_n$-equivariant QNNs. We provide an analytical study of their performance,proving that they do not suffer from barren plateaus, quickly reachoverparametrization, and generalize well from small amounts of data. To verifyour results, we perform numerical simulations for a graph state classificationtask. Our work provides the first theoretical guarantees for equivariant QNNs,thus indicating the extreme power and potential of GQML.

Theoretical Guarantees for Permutation-Equivariant Quantum Neural Networks