In theoretical machine learning, the statistical complexity is a notion that measures the richness of a hypothesis space. In this work, we apply a particular measure of statistical complexity, namely the Rademacher complexity, to the quantum circuit model in quantum computation and study how the statistical complexity depends on various quantum circuit parameters. In particular, we investigate the dependence of the statistical complexity on the resources, depth, width, and the number of input and output registers of a quantum circuit. To study how the statistical complexity scales with resources in the circuit, we introduce a resource measure of magic based on the $(p,q)$ group norm, which quantifies the amount of magic in the quantum channels associated with the circuit. These dependencies are investigated in the following two settings: (i) where the entire quantum circuit is treated as a single quantum channel, and (ii) where each layer of the quantum circuit is treated as a separate quantum channel. The bounds we obtain can be used to constrain the capacity of quantum neural networks in terms of their depths and widths as well as the resources in the network.

翻译：在理论机器学习中,统计的复杂性是一个测量假设空间丰富度的概念。在这项工作中,我们对量子计算中的量子电路模型应用了某种统计复杂性的测量,即Rademacher复杂度,对量子计算中的量子电路模型进行了一定的统计复杂性测量,并研究统计复杂性如何取决于量子电路的资源、深度、宽度、输入量子电路的输入和输出登记册的数量。为了研究统计复杂性规模如何与电路中的资源,我们引入了一种基于$(p)q)组规范的神奇资源测量法,该标准测量了与电路相关的量子信道中的魔法数量。这些依赖性在以下两种情况下调查:(一) 整个量子电路被当作单一量子电道,以及(二) 量子电路的每一层被当作单独的量子信道。我们获得的界限可以用来限制量子神经网络的深度和宽度以及网络资源的能力。