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

We present new quantum algorithms for estimating homological invariants, specifically Betti and persistent Betti numbers, of a simplicial complex given through structured classical data. Our approach efficiently constructs block-encodings of (persistent) Laplacians, enabling estimation via stochastic rank methods with complexity polylogarithmic in the number of simplices across both sparse and dense regimes. Unlike prior spectral algorithms that suffer when Betti numbers are small, we introduce homology tracking and property testing techniques achieving exponential speedups under natural sparsity and structure assumptions. We also formulate homology triviality and equivalence testing as property testing problems, giving nearly linear-time quantum algorithms when the boundary rank is large. A cohomological formulation further yields rank-independent testing and polylog-time manipulation of $r$-cocycles via block-encoded projections. These results open a new direction in quantum topological data analysis and demonstrate provable quantum advantages in computing topological invariants.