We introduce an algorithm for computing geodesics on sampled manifolds that relies on simulation of quantum dynamics on a graph embedding of the sampled data. Our approach exploits classic results in semiclassical analysis and the quantum-classical correspondence, and forms a basis for techniques to learn the manifold from which a dataset is sampled, and subsequently for nonlinear dimensionality reduction of high-dimensional datasets. We illustrate the new algorithm with data sampled from model manifolds and also by a clustering demonstration based on COVID-19 mobility data. Finally, our method reveals interesting connections between the discretization provided by data sampling and quantization.

翻译：我们引入了一种基于量子动态模拟模型的计算抽样元件的大地测量算法,该算法依赖于在样本数据嵌入图中进行量子动态模拟。 我们的方法利用了半古典分析的经典结果和量子古典对应法,并形成了一种技术基础,用于学习作为抽样数据集的元件,并随后用于减少高维数据集的非线性维度。我们用模型元件和基于COVID-19流动数据的集群演示数据来展示新的算法。 最后,我们的方法揭示了数据取样提供的离散和量化之间的有趣联系。