标题： 量子拓扑数据分析的新方面：贝蒂数估计，以及同调类和上同调类的检测与追踪 显示英文标题

标题： New aspects of quantum topological data analysis: Betti number estimation, and testing and tracking of homology and cohomology classes

摘要： 近期，量子计算在拓扑数据分析（TDA）中的应用引起了越来越多的关注。特别是，已经提出了几种用于估计（归一化的）贝蒂数的量子算法，这是TDA中的一个核心挑战。然而，最近证明了估计贝蒂数是一个NP难问题，揭示了在这一任务上实现通用量子优势的复杂性理论限制。受此限制的启发，并受到先前进展的激励，我们探索了更广泛的量子方法来处理TDA。首先，我们考虑了一种单纯复形以更具信息量的形式指定的场景，这使得可以使用替代的量子算法来估计贝蒂数和持久贝蒂数。然后，我们超越了贝蒂数，研究了测试给定循环的同调类以及区分同调类的问题。我们还为这些问题引入了上同调技术以及相应的量子算法。随后，我们讨论了这些方法在测试和跟踪同调类方面的潜在应用，这对TDA应用可能很有用。我们的结果显示，尽管一般贝蒂数估计问题是困难的，但在结构化设置下，量子算法仍然可以提供加速。 显示英文摘要

摘要： Recently, the application of quantum computation to topological data analysis (TDA) has received increasing attention. In particular, several quantum algorithms have been proposed for estimating (normalized) Betti numbers, a central challenge in TDA. However, it was recently proven that estimating Betti numbers is an NP-hard problem, revealing a complexity-theoretic limitation to achieving a generic quantum advantage for this task. Motivated by this limitation and inspired by previous progress, we explore broader quantum approaches to TDA. First, we consider scenarios in which a simplicial complex is specified in a more informative form, enabling alternative quantum algorithms to estimate Betti numbers and persistent Betti numbers. We then move beyond Betti numbers and study the problem of testing the homology class of a given cycle, as well as distinguishing between homology classes. We also introduce cohomological techniques for these problems, along with a quantum algorithm. We then discuss their potential use in the testing and tracking of homology classes, which can be useful for TDA applications. Our results show that, despite the hardness of general Betti number estimation, quantum algorithms can still offer speed-ups in structured settings.