Theoretical Guarantees for Permutation-Equivariant Quantum Neural Networks

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 Sn), and show how

to build Sn-equivariant QNNs. We provide an analytical study of their performance, proving that

they do not suffer from barren plateaus, quickly reach overparametrization, and can 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.