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.

翻译：尽管量子机器学习模型前景广阔，但在充分发挥其潜力之前仍需克服若干挑战。例如，基于量子神经网络（QNN）的模型可能会在训练过程中遭遇大量局部极小值和贫瘠高原问题。近年来，新兴的几何量子机器学习（GQML）领域成为解决这些问题的潜在方案。GQML的核心思想是设计编码问题对称性的架构（如等变QNN）。本文聚焦于具有置换对称性（即$S_n$对称群）的问题，展示了如何构建$S_n$等变QNN。我们对其性能进行了分析研究，证明了该类模型不会出现贫瘠高原现象、能快速达到过参数化状态，并且可从少量数据中实现良好泛化。为验证理论结果，我们在图态分类任务中开展了数值模拟。本研究为等变QNN提供了首个理论保证，充分展现了GQML的强大潜能与价值。