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.

翻译：众所周知，从独立同分布先验初始化的人工神经网络在每隐藏层神经元数量趋于无穷时收敛于高斯过程。本文证明量子神经网络（QNNs）具有类似的结论。具体而言，我们证明基于 Haar 随机酉或正交深度 QNNs 的某些模型在希尔伯特空间维度 $d$ 趋于无穷时输出收敛于高斯过程。由于输入态、测量可观测量以及酉矩阵元素非独立性的影响，该结论的推导比经典情形更为复杂。我们分析的一个重要结果是：由此产生的高斯过程无法通过贝叶斯统计有效预测 QNNs 的输出。此外，我们的定理表明，Haar 随机 QNNs 中的测度集中现象比此前认知更为严重，因为期望值与梯度的集中速度被证明为 $\mathcal{O}\left(\frac{1}{e^d \sqrt{d}}\right)$。最后，我们讨论这些结论如何深化对 $t$-设计中集中现象的理解。