On the statistical complexity of quantum circuits
This work addresses the theoretical understanding of quantum circuit complexity for researchers in quantum machine learning, but it is incremental as it applies existing statistical measures to quantum settings.
The authors tackled the problem of quantifying the statistical complexity of quantum circuits using Rademacher complexity, deriving bounds that depend on parameters like depth, width, and a new magic resource measure, with results applicable to constraining quantum neural network capacities.
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.