Overparametrization is one of the most surprising and notorious phenomena in machine learning. Recently, there have been several efforts to study if, and how, Quantum Neural Networks (QNNs) acting in the absence of hardware noise can be overparametrized. In particular, it has been proposed that a QNN can be defined as overparametrized if it has enough parameters to explore all available directions in state space. That is, if the rank of the Quantum Fisher Information Matrix (QFIM) for the QNN's output state is saturated. Here, we explore how the presence of noise affects the overparametrization phenomenon. Our results show that noise can "turn on" previously-zero eigenvalues of the QFIM. This enables the parametrized state to explore directions that were otherwise inaccessible, thus potentially turning an overparametrized QNN into an underparametrized one. For small noise levels, the QNN is quasi-overparametrized, as large eigenvalues coexists with small ones. Then, we prove that as the magnitude of noise increases all the eigenvalues of the QFIM become exponentially suppressed, indicating that the state becomes insensitive to any change in the parameters. As such, there is a pull-and-tug effect where noise can enable new directions, but also suppress the sensitivity to parameter updates. Finally, our results imply that current QNN capacity measures are ill-defined when hardware noise is present.

翻译：过参数化是机器学习中最令人惊讶且臭名昭著的现象之一。近期，已有若干研究致力于探讨在无硬件噪声环境下运行的量子神经网络（QNNs）是否及如何实现过参数化。特别地，有研究提出，若QNN拥有足够参数以探索状态空间中所有可及方向，则可定义为过参数化，即QNN输出态的量子Fisher信息矩阵（QFIM）的秩达到饱和。本文探究了噪声存在对过参数化现象的影响。我们的结果表明，噪声能够“激活”QFIM中原本为零的特征值，使参数化状态得以探索原本不可及的方向，从而可能将过参数化的QNN转化为欠参数化状态。在低噪声水平下，QNN呈现准过参数化特征，大特征值与小特征值并存。随后我们证明，随着噪声强度增大，QFIM所有特征值均呈指数级抑制，表明状态对参数的任何变化均变得不敏感。因此存在一种拉锯效应：噪声既能开启新方向，也会抑制对参数更新的敏感度。最终，我们的结果暗示，当前QNN容量度量在存在硬件噪声时缺乏明确定义。