{
  "abstract": "We prove that the spectrum of an unknown $d$-dimensional quantum state can be estimated to error $\\varepsilon$ in total variation distance using \\[ O\\!\\left(d^2\\min\\left\\{ \\frac{1}{(\\varepsilon\\log d)^4},\\; \\frac{1}{(\\varepsilon\\log d)^2} \\right\\}\\right) \\] copies. This matches the recent lower bound of Wang. When restricted to unentangled measurements, we give an algorithm with an additional factor of $d$ in copy complexity, which we conjecture to be optimal. We develop a framework for recovering the small eigenvalues of a quantum state by matching Chebyshev moments. We bound the variance of each Chebyshev moment estimate in terms of scalar derivatives of the corresponding polynomial, using classical and quantum Efron--Stein decompositions. Different rescalings of the Chebyshev polynomials balance approximation error and variance, yielding two regimes in our copy complexity bound.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.30171",
  "author_total": 3,
  "authors": [
    {
      "name": "Ainesh Bakshi"
    },
    {
      "name": "Apoorv Vikram Singh"
    },
    {
      "name": "Xinyu Tan"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/2b1r7qptwa6yb.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.30171",
  "fulltext_url": null,
  "id": "2b1r7qptwa6yb",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.30171",
  "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.DS"
    ],
    [
      "arxiv",
      "quant-ph"
    ],
    [
      "openalex",
      "1702|Quantum Computing Algorithms and Architecture"
    ]
  ],
  "openalex": "W7214327229",
  "pdf_url": "https://arxiv.org/pdf/2609.30171",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-24",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/2b1r7qptwa6yb.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:24:11Z",
      "native_id": "2609.30171",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.30171"
    }
  ],
  "subjects": [
    {
      "field": "physics",
      "origin": "arxiv:quant-ph",
      "subfield": "quantum"
    }
  ],
  "title": "Optimal spectrum estimation",
  "title_text": "Optimal spectrum estimation",
  "type": "preprint",
  "updated_at": "2026-10-30T12:24:11Z",
  "url": "https://research.draconitic.com/paper/2b1r7qptwa6yb",
  "venue": "arXiv",
  "versions": []
}