Clique Homology is <math xmlns="http://www.w3.org/1998/Math/MathML"> <msub><mrow><mi>QMA</mi></mrow> <mrow><mn>1</mn></mrow> </msub> </math> -hard.

Abstract

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 <math xmlns="http://www.w3.org/1998/Math/MathML"> <msub><mrow><mi>QMA</mi></mrow> <mrow><mn>1</mn></mrow> </msub> </math> -hard and the exact counting version is <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>#</mi> <mi>BQP</mi></math> -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 <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>QMA</mi></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>#</mi> <mi>BQP</mi></math> , 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.