Towards quantum topological data analysis: torsion detection

Abstract: Topological data analysis (TDA) has become an attractive area for the application of quantum computing. Recent advances have uncovered many interesting connections between the two fields. On one hand, complexity theoretic results show that estimating Betti numbers, a central task in TDA, is NP hard, indicating that a generic quantum speedup is unlikely. On the other hand, several recent studies have explored structured, less generic settings and demonstrated that quantum algorithms can still achieve significant speedups under certain conditions. To date, most of these efforts have focused on Betti numbers, which are topological invariants capturing the intrinsic connectivity and holes in a dataset. However, there is another important feature of topological spaces: torsion. Torsion represents a distinct component of homology that can reveal richer structural information. In this work, we introduce a quantum algorithm for torsion detection, that is, determining whether a given simplicial complex contains torsion. Our algorithm, assisted by a low complexity classical procedure, can succeed with high probability and potentially offer exponential speedup over the classical counterpart.