  Quantum machine learning models based on parametrized quantum circuits, also
called quantum neural networks (QNNs), are considered to be among the most
promising candidates for applications on near-term quantum devices. Here we
explore the expressivity and inductive bias of QNNs by exploiting an exact
mapping from QNNs with inputs $x$ to classical perceptrons acting on $x \otimes
x$ (generalised to complex inputs). The simplicity of the perceptron
architecture allows us to provide clear examples of the shortcomings of current
QNN models, and the many barriers they face to becoming useful general-purpose
learning algorithms. For example, a QNN with amplitude encoding cannot express
the Boolean parity function for $n\geq 3$, which is but one of an exponential
number of data structures that such a QNN is unable to express. Mapping a QNN
to a classical perceptron simplifies training, allowing us to systematically
study the inductive biases of other, more expressive embeddings on Boolean
data. Several popular embeddings primarily produce an inductive bias towards
functions with low class balance, reducing their generalisation performance
compared to deep neural network architectures which exhibit much richer
inductive biases. We explore two alternate strategies that move beyond standard
QNNs. In the first, we use a QNN to help generate a classical DNN-inspired
kernel. In the second we draw an analogy to the hierarchical structure of deep
neural networks and construct a layered non-linear QNN that is provably fully
expressive on Boolean data, while also exhibiting a richer inductive bias than
simple QNNs. Finally, we discuss characteristics of the QNN literature that may
obscure how hard it is to achieve quantum advantage over deep learning
algorithms on classical data.
