  Despite the great promise of quantum machine learning models, there are
several challenges one must overcome before unlocking their full potential. For
instance, models based on quantum neural networks (QNNs) can suffer from
excessive local minima and barren plateaus in their training landscapes.
Recently, the nascent field of geometric quantum machine learning (GQML) has
emerged as a potential solution to some of those issues. The key insight of
GQML is that one should design architectures, such as equivariant QNNs,
encoding the symmetries of the problem at hand. Here, we focus on problems with
permutation 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 reach
overparametrization, and generalize well from small amounts of data. To verify
our results, we perform numerical simulations for a graph state classification
task. Our work provides the first theoretical guarantees for equivariant QNNs,
thus indicating the extreme power and potential of GQML.
