{
  "abstract": "The commuting local-Hamiltonian (CLH) problem is a restriction of the local-Hamiltonian problem, in which the terms of the Hamiltonian are required to pairwise commute. A long line of work has shown that the problem lies in $\\mathsf{NP}$ for certain families of commuting local Hamiltonians. Nevertheless, there has been no formal evidence against the possibility that the general CLH problem is $\\mathsf{QMA}$-complete. CLH is complete for the complexity class $\\mathsf{QIMA}$, defined through quantum verifiers whose local gates commute. Therefore, CLH is $\\mathsf{QMA}$-complete if and only if $\\mathsf{QIMA}=\\mathsf{QMA}$. In this work, we introduce a classical-oracle analogue $\\mathsf{QIMA}^{\\mathcal{O}}$ and construct a classical oracle $\\mathcal{O}$ such that $\\mathsf{BQP}^{\\mathcal{O}}\\not\\subseteq\\mathsf{QIMA}^{\\mathcal{O}}$. Since $\\mathsf{BQP}^{\\mathcal{O}}\\subseteq\\mathsf{QMA}^{\\mathcal{O}}$ for any classical oracle $\\mathcal{O}$, this implies $\\mathsf{QIMA}^{\\mathcal{O}}\\neq\\mathsf{QMA}^{\\mathcal{O}}$ for our constructed oracle. Thus, our result provides relativized evidence against the possibility that the general commuting local-Hamiltonian problem is $\\mathsf{BQP}$-hard, and hence also against the possibility that it is $\\mathsf{QMA}$-complete.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.38957",
  "author_total": 2,
  "authors": [
    {
      "name": "Itay Shalit"
    },
    {
      "name": "Mark Zhandry"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/72zy3pvbx1ffg.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.38957",
  "fulltext_url": null,
  "id": "72zy3pvbx1ffg",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.38957",
  "language": null,
  "licence": "CC-BY-4.0",
  "licence_label": "CC BY 4.0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/licenses/by/4.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "cs.CC"
    ],
    [
      "arxiv",
      "quant-ph"
    ],
    [
      "openalex",
      "1702|Quantum Computing Algorithms and Architecture"
    ]
  ],
  "openalex": "W7215045470",
  "pdf_url": "https://arxiv.org/pdf/2609.38957",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-30",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/72zy3pvbx1ffg.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:25:49Z",
      "native_id": "2609.38957",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.38957"
    }
  ],
  "subjects": [
    {
      "field": "physics",
      "origin": "arxiv:quant-ph",
      "subfield": "quantum"
    }
  ],
  "title": "The Commuting Local Hamiltonian Problem: Relativized Evidence Against BQP-Hardness",
  "title_text": "The Commuting Local Hamiltonian Problem: Relativized Evidence Against BQP-Hardness",
  "type": "preprint",
  "updated_at": "2026-10-30T12:25:49Z",
  "url": "https://research.draconitic.com/paper/72zy3pvbx1ffg",
  "venue": "arXiv",
  "versions": []
}