One hypothesis for the success of deep neural networks (DNNs) is that they are highly expressive, which enables them to be applied to many problems, and they have a strong inductive bias towards solutions that are simple, known as simplicity bias, which allows them to generalise well on unseen data because most real-world data is structured (i.e. simple). In this work, we explore the inductive bias and expressivity of quantum neural networks (QNNs), which gives us a way to compare their performance to those of DNNs. Our results show that it is possible to have simplicity bias with certain QNNs, but we prove that this type of QNN limits the expressivity of the QNN. We also show that it is possible to have QNNs with high expressivity, but they either have no inductive bias or a poor inductive bias and result in a worse generalisation performance compared to DNNs. We demonstrate that an artificial (restricted) inductive bias can be produced by intentionally restricting the expressivity of a QNN. Our results suggest a bias-expressivity tradeoff. Our conclusion is that the QNNs we studied can not generally offer an advantage over DNNs, because these QNNs either have a poor inductive bias or poor expressivity compared to DNNs.

通过比较深度神经网络（DNNs）和量子神经网络（QNNs）的性能，本研究探讨了量子神经网络的归纳偏置和表达能力，并发现了一个比较结果：无论是具有归纳偏置的限制型QNNs还是具有高表达能力但无归纳偏置的QNNs，它们与DNNs相比要么有表达能力较差要么在泛化性能上处于劣势。

量子神经网络是否存在简单性偏见？