QKAN：量子柯尔莫戈罗夫-阿诺尔德网络及其在机器学习和多变量态制备中的应用
QKAN: quantum Kolmogorov-Arnold networks with applications in machine learning and multivariate state preparation
AI总结：
QKAN是一种基于柯尔莫戈罗夫-阿诺尔德网络的量子算法框架，通过块编码递归构建，用于量子学习和多变量态制备，其宽而浅的结构可高效训练。
摘要：
我们介绍了量子柯尔莫戈罗夫-阿诺尔德网络（QKAN），一种受最近提出的柯尔莫戈罗夫-阿诺尔德网络（KAN）启发的量子算法框架。QKAN继承了KAN的组合结构，并基于块编码，通过递归从单层构建。我们展示了QKAN在两个应用中的算法用途。首先，我们介绍并分析QKAN作为量子学习模型，将块编码矩阵的特征值视为神经元，并在网络边缘应用参数化激活函数。我们证明QKAN是一种宽而浅的神经架构，浅层深度可通过高效块编码输入时的指数级宽层来补偿。我们进一步讨论如何使用参数化量子电路和量子线性代数子程序对QKAN进行参数化和训练。其次，我们展示QKAN可以作为具有浅层组合结构函数的多变量量子态制备协议。我们通过高效制备多变量高斯量子态来演示这一点，使用两层QKAN。展望未来，我们预计QKAN的组合和模块化设计将使量子机器学习和量子态制备的新应用成为可能。
英文摘要：
We introduce quantum Kolmogorov-Arnold networks (QKAN), a quantum algorithmic framework inspired by the recently proposed Kolmogorov-Arnold Networks (KAN). QKAN inherits the compositional structure of KAN and is based on block-encodings, constructed recursively from a single layer using quantum singular value transformation. We demonstrate the algorithmic utility of QKAN in two applications. First, we introduce and analyze QKAN as a quantum learning model, treating the eigenvalues of block-encoded matrices as neurons and applying parametrized activation functions on the edges of the network. We show that QKAN is a wide-and-shallow neural architecture, where shallow depth is compensated by exponentially wide layers whenever efficient block-encodings of inputs are available. We further discuss how to parametrize and train QKAN using parametrized quantum circuits and quantum linear algebra subroutines. Second, we demonstrate that QKAN can serve as a multivariate quantum state-preparation protocol for functions with shallow compositional structure. We demonstrate this by efficiently preparing a multivariate Gaussian quantum state using a two-layer QKAN. Looking forward, we anticipate that QKAN's compositional and modular design will enable new applications in quantum machine learning and quantum state preparation.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
