Quantum algorithms achieve polylogarithmic complexity for Betti number estimation and homology testing via block-encoded Laplacians and cohomological projections, claiming exponential speedups under sparsity assumptions.
Testing the presence of balanced and bipartite components in a sparse graph is qma1-hard.arXiv preprint arXiv:2412.14932
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New aspects of quantum topological data analysis: Betti number estimation, and testing and tracking of homology and cohomology classes
Quantum algorithms achieve polylogarithmic complexity for Betti number estimation and homology testing via block-encoded Laplacians and cohomological projections, claiming exponential speedups under sparsity assumptions.