{
  "abstract": "Many quantum algorithms require coherent access to classical data, often modeled by quantum read-only memory (QROM). We initiate the study of the $\\mathrm{T}$ count of sparse QROM, in which only $s$ of the $2^n$ addresses store nonzero data. We prove that the optimal $\\mathrm{T}$ count is $Θ\\left(n+\\min\\left\\{s,\\sqrt{s\\left(m+\\log(2^{n+1}/s)\\right)}\\right\\}\\right)$. Our upper bounds use a multilevel hashing scheme, while our lower bounds reduce sparse QROM to state preparation and use counting arguments for adaptive Clifford+$\\mathrm{T}$ circuits. The lower bounds thus hold even when mid-circuit measurements and classically controlled operations are allowed. As applications, we obtain matching $\\mathrm{T}$-count bounds $Θ\\left(\\min\\left\\{s,\\sqrt{s\\log(2^{n+1}/s)}\\right\\} +\\sqrt{s\\log(1/\\varepsilon)}+\\log(1/\\varepsilon)\\right)$ for $s$-sparse state preparation and $Θ\\left(\\sqrt{2^n s\\left(n+\\log(1/\\varepsilon_{\\rm BE})\\right)} +\\log(1/\\varepsilon_{\\rm BE})\\right)$ for block encoding of $s$-sparse matrices, where $\\varepsilon$ and $\\varepsilon_{\\rm BE}$ are the precision of state preparation and block encoding, respectively.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2607.28260",
  "author_total": 6,
  "authors": [
    {
      "name": "Tongyang Li"
    },
    {
      "name": "Fengning Ou"
    },
    {
      "name": "Xinzhao Wang"
    },
    {
      "name": "Penghui Yao"
    },
    {
      "name": "Pei Yuan"
    },
    {
      "name": "Shengyu Zhang"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/0z19v0cs5jaxw.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2607.28260",
  "fulltext_url": null,
  "id": "0z19v0cs5jaxw",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2607.28260",
  "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": "W7171941899",
  "pdf_url": "https://arxiv.org/pdf/2607.28260",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-01",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/0z19v0cs5jaxw.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:26:03Z",
      "native_id": "2607.28260",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2607.28260"
    }
  ],
  "subjects": [
    {
      "field": "physics",
      "origin": "arxiv:quant-ph",
      "subfield": "quantum"
    }
  ],
  "title": "Optimal T Counts under Sparsity: from QROM to State Preparation and Block Encoding",
  "title_text": "Optimal T Counts under Sparsity: from QROM to State Preparation and Block Encoding",
  "type": "preprint",
  "updated_at": "2026-10-30T12:26:03Z",
  "url": "https://research.draconitic.com/paper/0z19v0cs5jaxw",
  "venue": "arXiv",
  "versions": []
}