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

Abstract: The application of quantum computation to topological data analysis (TDA) has received growing attention. While estimating Betti numbers is a central task in TDA, general complexity theoretic limitations restrict the possibility of quantum speedups. To address this, we explore quantum algorithms under a more structured input model. We show that access to additional topological information enables improved quantum algorithms for estimating Betti and persistent Betti numbers. Building on this, we introduce a new approach based on homology tracking, which avoids computing the kernel of combinatorial Laplacians used in prior methods. This yields a framework that remains efficient even when Betti numbers are small, offering substantial and sometimes exponential speedups. Beyond Betti number estimation, we formulate and study the homology property testing problem, and extend our approach to the cohomological setting. We present quantum algorithms for testing triviality and distinguishing homology classes, revealing new avenues for quantum advantage in TDA.