论文标题
量子持续的同源性
Quantum Persistent Homology
论文作者
论文摘要
持久性同源性是一种强大的数学工具，它汇总了有关数据形状的有用信息，允许人们在调整分辨率时检测持久性拓扑特征。但是，这种拓扑特征的计算通常是一项相当艰巨的任务，需要进行基础数据进行下采样。为了解决这个问题，我们开发了持久性betti数字的有效量子计算，该计算在不同尺度上跟踪数据的拓扑特征。我们的方法采用了一个持久的狄拉克操作员，其正方形产生了持久的组合拉普拉斯式，进而捕获了数据的持久性特征。我们还测试了点云数据上的算法。
Persistent homology is a powerful mathematical tool that summarizes useful information about the shape of data allowing one to detect persistent topological features while one adjusts the resolution. However, the computation of such topological features is often a rather formidable task necessitating the subsampling the underlying data. To remedy this, we develop an efficient quantum computation of persistent Betti numbers, which track topological features of data across different scales. Our approach employs a persistent Dirac operator whose square yields the persistent combinatorial Laplacian, and in turn the underlying persistent Betti numbers which capture the persistent features of data. We also test our algorithm on point cloud data.
