Torsion detection in clique complexes is conditionally $QMA_1$-hard

Abstract

Quantum algorithms for topological data analysis compute Betti numbers, the ranks of the homology groups of a simplicial complex, which can be read off from the kernel of a combinatorial Laplacian. Deciding whether a Betti number of a clique complex is nonzero is $QMA_1$-hard, and remains so under a spectral gap promise on vertex-weighted graphs. Integral homology, however, contains information inaccessible to the Laplacian spectrum. A new part that appears in integral homology is torsion: cycles that become boundaries only after being traversed several times, as in a projective plane or a Klein bottle. We ask how hard it is to detect torsion, and we answer with a simple reduction. We attach to an arbitrary clique complex a fixed $31$-vertex triangulation of the projective plane. The $k$th mod-$2$ Betti number of the input then reappears as $2$-torsion two degrees up, while all rational homology disappears and every combinatorial Laplacian acquires a constant spectral gap. We conclude that detecting torsion in clique complexes of unweighted graphs is $NP$-hard, even under a constant gap promise, and that it is $QMA_1$-hard if mod-$2$ clique homology is.