{
  "abstract": "We study the Sachdev-Ye-Kitaev (SYK) Hamiltonian $H_{\\operatorname{SYK}}$ on $n$ Majorana modes with $k$-body interactions, and prove that $\\mathbb{E}\\|H_{\\operatorname{SYK}}\\|_{\\operatorname{op}} = (1 - o(1))\\cdot\\sqrt{2n}/k$ for super-constant $k\\leq o(\\sqrt{n})$, where the expectation is over the disorder variables in the Hamiltonian. This confirms predictions due to Garcia-Garcia, Jia and Verbaarschot and answers a question posed by Feng, Tian, and Wei. Our results extend to the sparse SYK Hamiltonian. As a corollary, we obtain that a dissipative quantum algorithm due to Basso, Chen, and Dalzell provably computes the ground state energy of the SYK Hamiltonian up to an $O(1)$-multiplicative factor for all $k < \\sqrt{n}/4$. Our key technical idea is identifying an explicit, deterministic linear operator $\\mathsf{x}$ such that a fixed quadratic form of $\\mathsf{x}^{2\\ell}$ exactly equals the expected trace moments of the SYK Hamiltonian for every $n$ and $k$. This linear operator can be naturally viewed as a twisted model of bosons on the space of hyperedges of a hypergraph. The problem thus reduces to identifying the spectral edge of $\\mathsf{x}$, which we show is dominated by the spectrum of a natural ${n \\choose k}$-dimensional matrix from the Johnson scheme and is straightforward to compute using known results. To show that our bound is sharp, we construct a witness state with a large quadratic form on $\\mathsf{x}$ and transform it into a certificate of a lower bound on the largest quadratic form on $H_{\\operatorname{SYK}}$. Beyond the SYK Hamiltonian, our techniques also apply to the $k-$local quantum spin glass (QSG), yielding $\\mathbb{E}\\|H_{k-\\operatorname{QSG}}\\|_{\\operatorname{op}}\\leq O(\\sqrt{n/k})$ for all $1\\leq k\\leq\\sqrt{n}/2$, thus improving the standard $O(\\sqrt{n})$ bound by a factor of $\\sqrt{k}$.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2607.27185",
  "author_total": 3,
  "authors": [
    {
      "name": "Arpon Basu"
    },
    {
      "name": "Pravesh K. Kothari"
    },
    {
      "name": "Siddhant Midha"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/7zt4e3wjfsjwb.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2607.27185",
  "fulltext_url": null,
  "id": "7zt4e3wjfsjwb",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2607.27185",
  "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",
      "math.PR"
    ],
    [
      "arxiv",
      "quant-ph"
    ],
    [
      "openalex",
      "3107|Quantum many-body systems"
    ]
  ],
  "openalex": "W7171828449",
  "pdf_url": "https://arxiv.org/pdf/2607.27185",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-05",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/7zt4e3wjfsjwb.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:27:01Z",
      "native_id": "2607.27185",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2607.27185"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.PR",
      "subfield": "probability"
    },
    {
      "field": "physics",
      "origin": "arxiv:quant-ph",
      "subfield": "quantum"
    }
  ],
  "title": "Sharp Bounds on Ground State Energy of the SYK Model",
  "title_text": "Sharp Bounds on Ground State Energy of the SYK Model",
  "type": "preprint",
  "updated_at": "2026-10-30T12:27:10Z",
  "url": "https://research.draconitic.com/paper/7zt4e3wjfsjwb",
  "venue": "arXiv",
  "versions": []
}