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.

翻译：量子机器学习（QML）领域的一个重大挑战是建立量子计算在加速机器学习常见任务（如神经网络相关任务）中的应用。脊波变换一直是神经网络理论研究中基础性的数学工具，但由于其传统经典计算的数值实现随着数据维度$D$增加需要指数级运行时间$\exp(O(D))$，脊波变换在学习任务中的实际适用性受到限制。为解决此问题，我们提出量子脊波变换（QRT），它能在量子计算的线性运行时间$O(D)$内实现量子态的脊波变换。作为应用，我们还证明了可将QRT用作QML的基本子程序，在不需对原始网络进行大规模优化的条件下，高效地找到大型浅宽神经网络的稀疏可训练子网络。该应用发现了在这一领域证明中奖彩票假设（即找到此类稀疏可训练神经网络）的高效途径。这些结果为利用常见经典神经网络加速学习任务的QML开辟了新道路。