We address the long-standing question of the computational complexity of determining homology groups of simplicial complexes, a fundamental task in computational topology, posed by Kaibel and Pfetsch over twenty years ago. We show that decision problem is -hard and the exact counting version is -hard. In fact, we strengthen this by showing that the problems remains hard in the case of clique complexes, a family of simplicial complexes specified by a graph which is relevant to the problem of topological data analysis. The proof combines a number of techniques from Hamiltonian complexity and algebraic topology. As we discuss, a version of the problems satisfying a suitable promise and certain constraints is contained in and , respectively. This suggests that the seemingly classical problem may in fact be quantum mechanical. We discuss potential implications for the problem of quantum advantage in topological data analysis.
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http://www.ncbi.nlm.nih.gov/pmc/articles/PMC11561080 | PMC |
http://dx.doi.org/10.1038/s41467-024-54118-z | DOI Listing |
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