{
  "abstract": "Full tomography of an unknown quantum evolution is resource-intensive and often unnecessary when the goal is only to predict selected properties. This motivates the study of classical shadow estimation of unitary channels (CSEU), a task in which one queries an unknown $d$-dimensional unitary $U$ and stores classical data that can later be used to predict expectation values $\\mathrm{tr}[O UρU^\\dagger]$ for arbitrary input states $ρ$ and observables $O$. Here we propose a parallel and non-adaptive CSEU protocol that uses $\\mathcal{O}(\\min\\{d\\varepsilon^{-2},d^{3/2}\\varepsilon^{-1}\\})$ queries to $U$ when the observables have constant effective size. This achieves Heisenberg scaling with respect to the additive error $\\varepsilon$ and is query-optimal, as we prove a matching lower bound $Ω(\\min\\{d\\varepsilon^{-2},d^{3/2}\\varepsilon^{-1}\\})$ for the query complexity, which remains valid even when stronger access to the unknown unitary is available. Our CSEU protocol is a versatile primitive with applications to several quantum learning tasks. In particular, it yields a near-optimal algorithm for learning an arbitrary Hamiltonian from its real-time evolution in the high-precision regime and recovers the best-known bounds for several tasks in a unified way.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2606.13638",
  "author_total": 5,
  "authors": [
    {
      "name": "Entong He"
    },
    {
      "name": "Zihao Li"
    },
    {
      "name": "Noam Scully"
    },
    {
      "name": "Sisi Zhou"
    },
    {
      "name": "Yuxiang Yang"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/4bqk8njw6ykvg.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2606.13638",
  "fulltext_url": null,
  "id": "4bqk8njw6ykvg",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2606.13638",
  "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",
      "quant-ph"
    ],
    [
      "openalex",
      "1702|Quantum Computing Algorithms and Architecture"
    ]
  ],
  "openalex": "W7164481982",
  "pdf_url": "https://arxiv.org/pdf/2606.13638",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-29",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/4bqk8njw6ykvg.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:25:31Z",
      "native_id": "2606.13638",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2606.13638"
    }
  ],
  "subjects": [
    {
      "field": "physics",
      "origin": "arxiv:quant-ph",
      "subfield": "quantum"
    }
  ],
  "title": "Optimal classical shadow estimation of unitary channels at Heisenberg limit",
  "title_text": "Optimal classical shadow estimation of unitary channels at Heisenberg limit",
  "type": "preprint",
  "updated_at": "2026-10-30T12:25:31Z",
  "url": "https://research.draconitic.com/paper/4bqk8njw6ykvg",
  "venue": "arXiv",
  "versions": []
}