{
  "abstract": "We give quantum algorithms that estimate the log partition function of an $n$-qubit quantum Hamiltonian to additive error $ε$ with Heisenberg scaling, while minimizing calls to thermal-state purification preparation. The input is a block encoding of $H$, unitaries preparing thermofield-double states at chosen inverse temperatures, with their inverses and controlled versions, and an adaptive slowly-varying cooling schedule. The system-size bounds below assume $0\\preceq H\\preceq hI$ with $h=O(n)$ and block-encoding normalization $α=Θ(h)$. For constant inverse temperature, we show that such a schedule of length $O(\\sqrt n)$ always exists and can be generated from overlap estimates with $\\widetilde O(\\sqrt n)$ further preparations. Given such a schedule and classically chosen temperatures, we estimate the log partition function with $\\widetilde O(n^{5/4}/ε)$ expected calls to state preparation and to the block encoding, improving by $n^{1/4}$ on non-adaptive strategies. To achieve our result, we use a recursive-doubling identity that expresses each schedule increment as a weighted sum of logarithms of thermofield-double overlaps and of short imaginary-time steps implemented by quantum singular value transformation combined with previously developed variance reduction techniques. If thermofield doubles can be prepared controlled on a temperature register, a quantum inverse-binomial log estimator combined with quantum mean estimation reduces both query counts to $\\widetilde O(n/ε)$. We show that the latter scaling is optimal in block-encoding queries up to polylogarithmic factors and discuss end-to-end complexities for one-dimensional partition-function estimation.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.06197",
  "author_total": 3,
  "authors": [
    {
      "name": "Yufei Wang"
    },
    {
      "name": "Daniel Stilck França"
    },
    {
      "name": "Samuel Slezak"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/5hz48ea326h9e.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.06197",
  "fulltext_url": null,
  "id": "5hz48ea326h9e",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.06197",
  "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",
      "quant-ph"
    ]
  ],
  "openalex": null,
  "pdf_url": "https://arxiv.org/pdf/2610.06197",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-05",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/5hz48ea326h9e.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:27:01Z",
      "native_id": "2610.06197",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.06197"
    }
  ],
  "subjects": [
    {
      "field": "physics",
      "origin": "arxiv:quant-ph",
      "subfield": "quantum"
    }
  ],
  "title": "Improved Adaptive Estimation of Quantum Partition Functions with Heisenberg Scaling",
  "title_text": "Improved Adaptive Estimation of Quantum Partition Functions with Heisenberg Scaling",
  "type": "preprint",
  "updated_at": "2026-10-30T12:27:01Z",
  "url": "https://research.draconitic.com/paper/5hz48ea326h9e",
  "venue": "arXiv",
  "versions": []
}