Clique Homology is $${{\mathsf{QMA}}}_{1}$$ QMA 1 -hard

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 $${{\mathsf{QMA}}}_{1}$$ QMA 1 -hard and the exact counting version is $$\#{\mathsf{BQP}}$$ # BQP -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 $${\mathsf{QMA}}$$ QMA and $$\#{\mathsf{BQP}}$$ # BQP , 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.