Clique Homology is QMA1-hard

TL;DR

This paper proves that determining homology groups of simplicial complexes, a key problem in computational topology, is computationally hard for quantum computers, specifically QMA1-hard, with implications for quantum advantage in topological data analysis.

Contribution

It establishes the QMA1-hardness of the homology decision problem, including for clique complexes, linking computational topology with quantum complexity theory.

Findings

Homology group determination is QMA1-hard.

The problem remains QMA1-hard for clique complexes.

Implications for quantum advantage in topological data analysis.

Abstract

We tackle 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 20 years ago. We show that this decision problem is QMA1-hard. Moreover, we show that a version of the problem satisfying a suitable promise and certain constraints is contained in QMA. This suggests that the seemingly classical problem may in fact be quantum mechanical. In fact, we are able to significantly strengthen this by showing that the problem remains QMA1-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 homological algebra. We discuss potential implications for the problem of quantum advantage in topological data analysis.