Given a data set with a notion of distance, such as a point cloud in Euclidean space, topological data analysis (TDA) uses techniques from algebraic topology and metric geometry to infer the topology of a hypothetical manifold from which the data are sampled. This inference is achieved by calculating topological invariants, some of which are difficult to compute classically. Meanwhile, quantum TDA utilizes quantum processes to extract the invariants used in making such inferences in an attempt to speed up the computations. Because applying transformations to the original classical dataset could alter the associated topological invariants, we investigate which quantum encodings would best preserve the invariants of the original dataset. This line of inquiry is distinct from standard approaches in quantum TDA, whose typical starting point is not from the classical dataset directly, but rather from the associated combinatorial objects, such as simplicial complexes, which typically demand a lot of resources to construct. We take the first step at a more direct approach by focusing on which quantum encodings acting directly on the data are admissible for applying quantum algorithms to extract topological features from classical datasets.

翻译：给定一个具有距离概念的数据集（例如欧几里得空间中的点云），拓扑数据分析（TDA）利用代数拓扑和度量几何的技术来推断数据采样自的假设流形的拓扑结构。这种推断通过计算拓扑不变量实现，而某些拓扑不变量在经典计算中难以求解。与此同时，量子TDA利用量子过程来提取用于此类推断的不变量，以期加速计算。由于对原始经典数据集应用变换可能会改变相关的拓扑不变量，我们研究了哪些量子编码能最好地保留原始数据集的这些不变量。这一研究思路不同于量子TDA中的标准方法，后者的典型起点并非直接来自经典数据集，而是来自相关的组合对象（如单纯复形），这些对象的构建通常需要大量资源。我们采取更直接的方法迈出第一步，重点研究哪些直接作用于数据的量子编码可被允许用于应用量子算法从经典数据集中提取拓扑特征。