  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.
