{
  "abstract": "We study the transverse field Ising model, defined by the Hamiltonian $H =\\frac{1}{2}\\sum_{i, j\\in [n]} J_{ij} Z_i Z_j +\\sum_{i=1}^n h_i^z Z_i + η\\sum_{i} X_i$ where $J $ is the symmetric interaction matrix, and $η$ is the transverse field strength. Let $Δ(J)=λ_{\\max}(J)-λ_{\\min}(J)$ be the spectral width of $J.$ When the inverse temperature $β\\geq0$ satisfies $Δ(J)\\cdot\\frac{\\tanh(βη)}η\\leq1$, we give a randomized classical algorithm that approximates the partition function $Z(β)=\\operatorname{Tr}(e^{-βH})$ to a given relative error $ε\\in(0,1)$ in time polynomial in $n$, $β$, the model parameters, and $ε^{-1}$. When $ Δ(J) \\cdot \\frac{\\tanh(βη)}η > 1 ,$ we show that approximating $ Z(β)$ within an $\\exp(o(n))$-multiplicative factor is $\\textbf{NP}$-hard, and thus unlikely to admit an efficient classical or quantum algorithms under standard complexity theoretic assumptions. Furthermore, in the regime $Δ(J)\\cdot \\frac{\\tanh(βη)}η\\leq 1,$ we provide an efficient randomized classical algorithm that approximates Pauli string observables of the Gibbs state $ ρ_β= \\frac{e^{-βH}}{\\operatorname{Tr}(e^{-βH})}$ within an arbitrarily small additive error. In the special case when the observable is also diagonal in the $X$-basis, i.e. $P \\in \\{I, X\\}^{\\otimes n}$, the algorithm further achieves arbitrarily small relative error.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.02079",
  "author_total": 1,
  "authors": [
    {
      "name": "Thuy-Duong Vuong"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/80t004rrd489s.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.02079",
  "fulltext_url": null,
  "id": "80t004rrd489s",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.02079",
  "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.DS"
    ],
    [
      "arxiv",
      "math-ph"
    ],
    [
      "arxiv",
      "math.MP"
    ],
    [
      "arxiv",
      "math.PR"
    ],
    [
      "arxiv",
      "quant-ph"
    ],
    [
      "openalex",
      "1702|Quantum Computing Algorithms and Architecture"
    ]
  ],
  "openalex": "W7215920266",
  "pdf_url": "https://arxiv.org/pdf/2610.02079",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-01",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/80t004rrd489s.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:26:03Z",
      "native_id": "2610.02079",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.02079"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math-ph",
      "subfield": "mathematical-physics"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.PR",
      "subfield": "probability"
    },
    {
      "field": "physics",
      "origin": "arxiv:quant-ph",
      "subfield": "quantum"
    }
  ],
  "title": "A computational phase diagram for the transverse field Ising model",
  "title_text": "A computational phase diagram for the transverse field Ising model",
  "type": "preprint",
  "updated_at": "2026-10-30T12:26:07Z",
  "url": "https://research.draconitic.com/paper/80t004rrd489s",
  "venue": "arXiv",
  "versions": []
}