{
  "id": "oai:papers.qnfo.org:adaptivity-closes-the-stabilizer-learning-gap-an-analytical-account-of-optimal-s",
  "slug": "adaptivity-closes-the-stabilizer-learning-gap-an-analytical-account-of-optimal-s",
  "title": "Adaptivity Closes the Stabilizer Learning Gap: An Analytical Account of Optimal Single-Copy Stabilizer State Learning",
  "authors": [
    "Rowan Brad Quni-Gudzinas"
  ],
  "abstract": "Stabilizer states underpin quantum error correction, device benchmarking, and the classical simulation of quantum circuits, yet their learnability shows a sample-complexity gap: an n-qubit stabilizer state can be identified from Θ(n) copies using joint two-copy Bell measurements, while non-adaptive single-copy measurements provably require Ω(n²) copies. The specific gap addressed here is explanatory: the recent proof that adaptivity alone closes this gap has not been accompanied by an explicit numerical account of how large the separation is at concrete system sizes, which assumptions the numbers rest on, and where the result could fail. We reconstruct the information-theoretic skeleton of the result with all arithmetic shown. We derive the hypothesis-class size N_n=2^n∏_i=1^n(2^i+1), giving log₂N_n=n²/2+3n/2+O(1) and N₁00≈ 2⁵150; an information-theoretic floor of $52$ copies at n=100 for any single-copy learner; the memory-versus-samples tradeoff Θ(n-k+1/ε) with the instance Θ(190) at (n,k,ε)=(100,10,0.01); and the nullity extension O(n 2^r), giving $3200$ copies at (n,r)=(100,5). Under stated constant assumptions, a workload of $1000$ characterizations at n=100 needs 5.0× 10⁵ adaptive versus 1.0× 10⁸ non-adaptive measurements, a factor of $200$. The account clarifies which experimental designs the result changes and what would falsify it.",
  "abstract_markdown": "Stabilizer states underpin quantum error correction, device benchmarking, and the classical simulation of quantum circuits, yet their learnability shows a sample-complexity gap: an $n$-qubit stabilizer state can be identified from $\\Theta(n)$ copies using joint two-copy Bell measurements, while non-adaptive single-copy measurements provably require $\\Omega(n^{2})$ copies. The specific gap addressed here is explanatory: the recent proof that adaptivity alone closes this gap has not been accompanied by an explicit numerical account of how large the separation is at concrete system sizes, which assumptions the numbers rest on, and where the result could fail. We reconstruct the information-theoretic skeleton of the result with all arithmetic shown. We derive the hypothesis-class size $N_{n}=2^{n}\\prod_{i=1}^{n}(2^{i}+1)$, giving $\\log_{2}N_{n}=n^{2}/2+3n/2+O(1)$ and $N_{100}\\approx 2^{5150}$; an information-theoretic floor of $52$ copies at $n=100$ for any single-copy learner; the memory-versus-samples tradeoff $\\Theta(n-k+1/\\varepsilon)$ with the instance $\\Theta(190)$ at $(n,k,\\varepsilon)=(100,10,0.01)$; and the nullity extension $O(n\\,2^{r})$, giving $3200$ copies at $(n,r)=(100,5)$. Under stated constant assumptions, a workload of $1000$ characterizations at $n=100$ needs $5.0\\times 10^{5}$ adaptive versus $1.0\\times 10^{8}$ non-adaptive measurements, a factor of $200$. The account clarifies which experimental designs the result changes and what would falsify it.",
  "date": "2026-10-08",
  "first_published": "2026-10-08",
  "last_revised": "2026-10-13T13:46:02Z",
  "modified": "2026-10-13T13:46:02Z",
  "version": "2.0.0",
  "version_count": 2,
  "language": "en",
  "keywords": [],
  "quality_tags": [
    "High-quality scholarly research",
    "open-access scholarly work",
    "versioned research publication"
  ],
  "quality_label": "High-quality scholarly research",
  "quality_basis": "QNFO publisher classification for open-access, versioned research with a public revision history and reader correction channel; this is not a claim of independent peer review.",
  "quality_evidence_url": "https://qnfo.org/about",
  "peer_review_status": "Not independently peer-reviewed",
  "resource_type": "ScholarlyArticle",
  "doi": null,
  "doi_url": null,
  "url": "https://papers.qnfo.org/papers/adaptivity-closes-the-stabilizer-learning-gap-an-analytical-account-of-optimal-s",
  "license_url": "https://legal.qnfo.org/",
  "license_spdx": null,
  "sha256": "eee824fdfbe93a1e55b3431c0b266abffddd5873564260ddbb73e23f2a208957",
  "version_url": "https://papers.qnfo.org/papers/adaptivity-closes-the-stabilizer-learning-gap-an-analytical-account-of-optimal-s/v/2.0.0",
  "datacite": {
    "schemaVersion": "http://datacite.org/schema/kernel-4",
    "identifiers": [
      {
        "identifier": "https://papers.qnfo.org/papers/adaptivity-closes-the-stabilizer-learning-gap-an-analytical-account-of-optimal-s",
        "identifierType": "URL"
      }
    ],
    "creators": [
      {
        "name": "Rowan Brad Quni-Gudzinas",
        "nameType": "Personal",
        "nameIdentifiers": [
          {
            "nameIdentifier": "https://orcid.org/0009-0002-4317-5604",
            "nameIdentifierScheme": "ORCID",
            "schemeUri": "https://orcid.org"
          }
        ]
      }
    ],
    "titles": [
      {
        "title": "Adaptivity Closes the Stabilizer Learning Gap: An Analytical Account of Optimal Single-Copy Stabilizer State Learning"
      }
    ],
    "publisher": "QNFO",
    "publicationYear": "2026",
    "resourceType": {
      "resourceTypeGeneral": "Text",
      "resourceType": "ScholarlyArticle"
    },
    "subjects": [
      {
        "subject": "High-quality scholarly research",
        "subjectScheme": "QNFO publisher classification",
        "schemeUri": "https://qnfo.org/about"
      },
      {
        "subject": "open-access scholarly work",
        "subjectScheme": "QNFO publisher classification",
        "schemeUri": "https://qnfo.org/about"
      },
      {
        "subject": "versioned research publication",
        "subjectScheme": "QNFO publisher classification",
        "schemeUri": "https://qnfo.org/about"
      }
    ],
    "dates": [
      {
        "date": "2026-10-08",
        "dateType": "Issued"
      },
      {
        "date": "2026-10-13",
        "dateType": "Updated"
      }
    ],
    "language": "en",
    "version": "2.0.0",
    "rightsList": [
      {
        "rightsUri": "https://legal.qnfo.org/"
      }
    ],
    "descriptions": [
      {
        "description": "Stabilizer states underpin quantum error correction, device benchmarking, and the classical simulation of quantum circuits, yet their learnability shows a sample-complexity gap: an n-qubit stabilizer state can be identified from Θ(n) copies using joint two-copy Bell measurements, while non-adaptive single-copy measurements provably require Ω(n²) copies. The specific gap addressed here is explanatory: the recent proof that adaptivity alone closes this gap has not been accompanied by an explicit numerical account of how large the separation is at concrete system sizes, which assumptions the numbers rest on, and where the result could fail. We reconstruct the information-theoretic skeleton of the result with all arithmetic shown. We derive the hypothesis-class size N_n=2^n∏_i=1^n(2^i+1), giving log₂N_n=n²/2+3n/2+O(1) and N₁00≈ 2⁵150; an information-theoretic floor of $52$ copies at n=100 for any single-copy learner; the memory-versus-samples tradeoff Θ(n-k+1/ε) with the instance Θ(190) at (n,k,ε)=(100,10,0.01); and the nullity extension O(n 2^r), giving $3200$ copies at (n,r)=(100,5). Under stated constant assumptions, a workload of $1000$ characterizations at n=100 needs 5.0× 10⁵ adaptive versus 1.0× 10⁸ non-adaptive measurements, a factor of $200$. The account clarifies which experimental designs the result changes and what would falsify it.",
        "descriptionType": "Abstract"
      },
      {
        "description": "Publisher quality classification: High-quality scholarly research. QNFO publisher classification for open-access, versioned research with a public revision history and reader correction channel; this is not a claim of independent peer review.",
        "descriptionType": "Other"
      },
      {
        "description": "Peer review status: Not independently peer-reviewed",
        "descriptionType": "Other"
      }
    ],
    "formats": [
      "text/html"
    ]
  }
}