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 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 much worse than previously thought, as we prove that
expectation values and gradients concentrate as $\mathcal{O}\left(\frac{1}{e^d
\sqrt{d}}\right)$ -- exponentially in the Hilbert space dimension. Finally, we
discuss how our results improve our understanding of concentration in
$t$-designs.

研究证明在量子神经网络中，基于 Haar 随机酉或正交深度 QNN 的输出会在大 Hilbert 空间维度 d 下收敛到高斯过程，但无法通过贝叶斯统计有效预测其输出，且浓缩度现象比以前预想的更糟，因为期望值和梯度浓缩为 O（1/e^d * sqrt（d））-- 在希尔伯特空间维度上呈指数级。