{
  "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 \\emph{torsion}: cycles that become boundaries only after being traversed several times, as in a projective plane or a Klein bottle. We provide a simple reduction that allows us to establish the first hardness results for the problem of detecting torsion. 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 \\emph{unweighted} graphs is $NP$-hard, even under a constant gap promise, and that it is $QMA_1$-hard if mod-$2$ clique homology is.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.14110",
  "author_total": 1,
  "authors": [
    {
      "name": "Adam Wesołowski"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/d18jmkt1a8he5.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.14110",
  "fulltext_url": null,
  "id": "d18jmkt1a8he5",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.14110",
  "language": null,
  "licence": "CC-BY-SA-4.0",
  "licence_label": "CC BY-SA 4.0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/licenses/by-sa/4.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "cs.CC"
    ],
    [
      "arxiv",
      "quant-ph"
    ],
    [
      "openalex",
      "1703|Topological and Geometric Data Analysis"
    ]
  ],
  "openalex": "W7213222986",
  "pdf_url": "https://arxiv.org/pdf/2609.14110",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-22",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/d18jmkt1a8he5.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:23:39Z",
      "native_id": "2609.14110",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.14110"
    }
  ],
  "subjects": [
    {
      "field": "physics",
      "origin": "arxiv:quant-ph",
      "subfield": "quantum"
    }
  ],
  "title": "Torsion detection in clique complexes is conditionally $QMA_1$-hard",
  "title_text": "Torsion detection in clique complexes is conditionally QMA_1-hard",
  "type": "preprint",
  "updated_at": "2026-10-30T12:23:39Z",
  "url": "https://research.draconitic.com/paper/d18jmkt1a8he5",
  "venue": "arXiv",
  "versions": []
}