  It is well known that artificial neural networks initialized from independent
and identically distributed priors converge to Gaussian processes in the limit
of large number of neurons per hidden layer. In this work we prove an analogous
result for Quantum Neural Networks (QNNs). Namely, we show that the outputs of
certain models based on Haar random unitary or orthogonal deep QNNs converge to
Gaussian processes in the limit of large Hilbert space dimension $d$. The
derivation of this result is more nuanced than in the classical case due to the
role played by the input states, the measurement observable, and the fact that
the entries of unitary matrices are not independent. An important consequence
of our analysis is that the ensuing Gaussian processes cannot be used to
efficiently predict the outputs of the QNN via Bayesian statistics.
Furthermore, our theorems imply that the concentration of measure phenomenon in
Haar random QNNs is worse than previously thought, as we prove that expectation
values and gradients concentrate as $\mathcal{O}\left(\frac{1}{e^d
\sqrt{d}}\right)$. Finally, we discuss how our results improve our
understanding of concentration in $t$-designs.
