Clique Homology is $${{\mathsf{QMA}}}_{1}$$-hard
Marcos Crichigno, Tamara Kohler · Nature Communications · 2024
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 $${{\mathsf{QMA}}}_{1}$$ -hard and the exact counting version is $$\#{\mathsf{BQP}}$$ -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 $${\mathsf{QMA}}$$ and $$\#{\mathsf{BQP}}$$ , 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. The question of the computational complexity of the homology problem is a long-standing open problem in computational topology. Here, the authors show that the decision version of the problem is QMA1-hard and the exact counting version of the problem is #BPQ-hard, and discuss implications for topological data analysis.