  A significant challenge in the field of quantum machine learning (QML) is to
establish applications of quantum computation to accelerate common tasks in
machine learning such as those for neural networks. Ridgelet transform has been
a fundamental mathematical tool in the theoretical studies of neural networks,
but the practical applicability of ridgelet transform to conducting learning
tasks was limited since its numerical implementation by conventional classical
computation requires an exponential runtime $\exp(O(D))$ as data dimension $D$
increases. To address this problem, we develop a quantum ridgelet transform
(QRT), which implements the ridgelet transform of a quantum state within a
linear runtime $O(D)$ of quantum computation. As an application, we also show
that one can use QRT as a fundamental subroutine for QML to efficiently find a
sparse trainable subnetwork of large shallow wide neural networks without
conducting large-scale optimization of the original network. This application
discovers an efficient way in this regime to demonstrate the lottery ticket
hypothesis on finding such a sparse trainable neural network. These results
open an avenue of QML for accelerating learning tasks with commonly used
classical neural networks.
