{
  "abstract": "Random quantum objects are powerful resources for quantum information processing, yet exact Haar randomness is costly and typically unnecessary. We introduce an explicit sparse commuting circuit ensemble on $n$ qubits that reproduces low-order Haar moments in the stringent relative-error sense. The circuit consists of a sparse Clifford phase layer followed by independent single-qubit Clifford gates. Acting on a simple product state, the resulting ensemble forms $ε$-approximate projective $2$- and $3$-designs in relative error, with the required logarithmic interaction degree being asymptotically optimal within this circuit family. It admits an ancilla-free implementation of quantum depth $O(\\log(n/ε))$ on an all-to-all architecture, as well as an adaptive constant-depth implementation---in fact, depth seven---using $O(n\\log(n/ε))$ ancilla qubits. Departing from existing shallow-design paradigms, our analysis exploits the intrinsic moment structure of commuting phase circuits; at third order, this requires a new block decomposition and combinatorial analysis that also suggests a route toward higher-order shallow designs. Our results show that precise Haar-like statistics can emerge from sparse commuting dynamics with remarkably low quantum resources, with applications to randomized characterization, quantum metrology, quantum algorithms, and many-body physics.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.03925",
  "author_total": 4,
  "authors": [
    {
      "name": "Qingyue Zhang"
    },
    {
      "name": "Junjie Chen"
    },
    {
      "name": "Zhou You"
    },
    {
      "name": "You Zhou"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/8p41zd3k7tz0b.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.03925",
  "fulltext_url": null,
  "id": "8p41zd3k7tz0b",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.03925",
  "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": "W7208785483",
  "pdf_url": "https://arxiv.org/pdf/2609.03925",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-03",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/8p41zd3k7tz0b.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:18:31Z",
      "native_id": "2609.03925",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.03925"
    }
  ],
  "subjects": [
    {
      "field": "physics",
      "origin": "arxiv:quant-ph",
      "subfield": "quantum"
    }
  ],
  "title": "Ultra-Precise Quantum Projective Designs in Constant Depth",
  "title_text": "Ultra-Precise Quantum Projective Designs in Constant Depth",
  "type": "preprint",
  "updated_at": "2026-10-30T12:18:31Z",
  "url": "https://research.draconitic.com/paper/8p41zd3k7tz0b",
  "venue": "arXiv",
  "versions": []
}