Gapped Clique Homology on Weighted Graphs is QMA₁-Hard and Contained in QMA

We study the complexity of a classic problem in computational topology, the homology problem: given a description of some space <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$X$</tex> and an integer <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$k$</tex>, decide if <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$X$</tex> contains a k-dimensional hole. The setting and statement of the homology problem are completely classical, yet we find that the complexity is characterized by quantum complexity classes. Our result can be seen as an aspect of a connection between homology and supersymmetric quantum mechanics [1]. We consider clique complexes, motivated by the practical application of topological data analysis (TDA). The clique complex of a graph is the simplicial complex formed by declaring every <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$k+1$</tex> -clique in the graph to be a k-simplex. Our main result is that deciding whether the clique complex of a weighted graph has a hole or not, given a suitable promise on the gap, is QMA1-hard and contained in QMA. Our main innovation is a technique to lower bound the eigenvalues of the combinatorial Laplacian operator. For this, we invoke a tool from algebraic topology known as spectral sequences. In particular, we exploit a connection between spectral sequences and Hodge theory [2]. Spectral sequences will play a role analogous to perturbation theory for combinatorial Lapla-cians. In addition, we develop the simplicial surgery technique used in prior work [3]. Our result provides some suggestion that the quantum TDA algorithm [4] cannot be dequantized. More broadly, we hope that our results will open up new possibilities for quantum advantage in topological data analysis.