{
  "abstract": "Topological data analysis (TDA) is a machine learning technique that uses topology to extract patterns from data and has shown the potential to exhibit quantum advantage. A key concept in TDA is persistent homology, which measures the robustness of topological information at different lengthscales. In this paper, we introduce and study the problem of normalized persistence, a practically motivated and easily interpretable version of persistent homology that counts the fraction of holes that persist at different lengthscales. We prove that a variant of normalized persistence is $\\mathsf{DQC}_1$-hard and contained in $\\mathsf{BQP}$, giving evidence of an exponential quantum speedup for TDA under the standard assumption that $\\mathsf{DQC}_1 \\not\\subseteq \\mathsf{BPP}$. These are the first $\\mathsf{DQC}_1$-hardness results for clique complexes, making them directly applicable to TDA instances. We also find a close connection between normalized persistence and the complexity of estimating spectral quantities in the low-energy subspace of local Hamiltonians. We study a family of such problems, including a low-energy normalized subtrace and spectral density. We show that these are $\\mathsf{DQC}_1$-hard for $O(1)$-local Hamiltonians, strengthening previous results that required log-local interactions. We also introduce a variant of $\\mathsf{DQC}_1$ with perfect completeness ($\\mathsf{SDQC}_1$) to characterize the hardness of problems normalized by an exact kernel. This includes normalized persistence for $O(1)$-local Hamiltonians, which we show is $\\mathsf{SDQC}_1$-hard.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2607.03278",
  "author_total": 4,
  "authors": [
    {
      "name": "Dominic Lowe"
    },
    {
      "name": "M. S. Kim"
    },
    {
      "name": "Roberto Bondesan"
    },
    {
      "name": "Ryu Hayakawa"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/84rw1m839rcqa.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2607.03278",
  "fulltext_url": null,
  "id": "84rw1m839rcqa",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2607.03278",
  "language": null,
  "licence": "LicenseRef-arXiv-nonexclusive-1.0",
  "licence_label": "arXiv licence",
  "licence_source": "article",
  "licence_url": "http://arxiv.org/licenses/nonexclusive-distrib/1.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "cs.CC"
    ],
    [
      "arxiv",
      "cs.LG"
    ],
    [
      "arxiv",
      "quant-ph"
    ],
    [
      "openalex",
      "1702|Quantum Computing Algorithms and Architecture"
    ]
  ],
  "openalex": "W7167579568",
  "pdf_url": "https://arxiv.org/pdf/2607.03278",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-29",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/84rw1m839rcqa.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:25:31Z",
      "native_id": "2607.03278",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2607.03278"
    }
  ],
  "subjects": [
    {
      "field": "physics",
      "origin": "arxiv:quant-ph",
      "subfield": "quantum"
    }
  ],
  "title": "Complexity of Normalized Persistence Problems for Topological Data Analysis and Local Hamiltonians",
  "title_text": "Complexity of Normalized Persistence Problems for Topological Data Analysis and Local Hamiltonians",
  "type": "preprint",
  "updated_at": "2026-10-30T12:25:31Z",
  "url": "https://research.draconitic.com/paper/84rw1m839rcqa",
  "venue": "arXiv",
  "versions": []
}