{
  "abstract": "Estimating nonlinear properties of an unknown quantum state with restrictive experimental accessibility is a fundamental problem in quantum learning. We study the sample complexity of estimating the state moments $\\operatorname{Tr}(ρ^t)$, allowing arbitrary adaptive single-copy measurements. While purity estimation has optimal sample complexity $Θ(\\sqrt d)$ at constant additive error, the optimal dimension dependence for higher moments has remained unclear. We resolve this question for every fixed integer $t\\ge 2$: for any sufficiently small constant additive error, the optimal sample complexity is \\[ Θ\\!\\left(d^{\\frac{\\lceil\\log_2 t\\rceil}{1+\\lceil\\log_2 t\\rceil}}\\right) \\] with constant success probability. The complexity thus follows a dyadic hierarchy: for each integer $h\\geq1$, all moment orders $2^{h-1}<t\\leq2^h$ share the same exponent. In particular, the third and fourth moments both require $Θ(d^{2/3})$ copies. Our results also apply to multistate overlaps, weighted and partial-transpose moments, fixed-rank spectrum estimation, and distributed estimation of nonlinear quantities. We also establish an $Ω(d^{1-1/t})$ lower bound for arbitrary nonadaptive single-copy protocols, matching the known upper bound and demonstrating an advantage from adaptivity for every $t\\geq4$. Our upper bound uses adaptive state filtering to reduce higher-order moment estimation to lower-order moment estimation in a smaller state space. For the lower bound, we directly compare moment-matched ensembles through a smooth interpolation, bypassing the maximally mixed state as an intermediate hypothesis and yielding matching bounds beyond $\\sqrt d$.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.37781",
  "author_total": 1,
  "authors": [
    {
      "name": "Zhenhuan Liu"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/3rbb2vqd5s9v2.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.37781",
  "fulltext_url": null,
  "id": "3rbb2vqd5s9v2",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.37781",
  "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",
      "1702|Quantum Computing Algorithms and Architecture"
    ]
  ],
  "openalex": "W7214932427",
  "pdf_url": "https://arxiv.org/pdf/2609.37781",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-15",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/3rbb2vqd5s9v2.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:30:32Z",
      "native_id": "2609.37781",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.37781"
    }
  ],
  "subjects": [
    {
      "field": "physics",
      "origin": "arxiv:quant-ph",
      "subfield": "quantum"
    }
  ],
  "title": "Optimal single-copy estimation of quantum state moments: why $\\operatorname{Tr}(ρ^3)$ and $\\operatorname{Tr}(ρ^4)$ are equally hard",
  "title_text": "Optimal single-copy estimation of quantum state moments: why Tr(ρ^3) and Tr(ρ^4) are equally hard",
  "type": "preprint",
  "updated_at": "2026-10-30T12:33:28Z",
  "url": "https://research.draconitic.com/paper/3rbb2vqd5s9v2",
  "venue": "arXiv",
  "versions": []
}