{
  "abstract": "We study the quantum query complexity of computing a semigroup product $x_1\\cdots x_n$, when one query reveals one input element and the multiplication table is given. For a finite aperiodic semigroup of size $N-1$, the argument of Aaronson, Grier, and Schaeffer gives an upper bound of $\\sqrt n\\,(N\\log(nN+2))^{O(N)}$ queries. To obtain query bounds that reflect algebraic structure, we study the product breadth $β$: the smallest bound such that every input word has a subsequence of at most $β$ letters with the same product. For nontrivial finite commutative aperiodic monoids, the bounded-error quantum query complexity is $Θ(\\min\\{n,\\sqrt{nβ}\\})$, and is thus characterized by product breadth. We further show that if such a monoid $M$ has aperiodicity index $k$ (the least positive integer satisfying $x^k=x^{k+1}$ for every $x\\in M$), then $β=O(k\\log(|M|+1)\\log\\log(|M|+2))$. For monoids with a stable partial order in which the identity is the minimum element, we prove that the bounded-error quantum query complexity is at most $\\sqrt{n+1}((β+2)\\log(n+2))^{O(\\log(β+2))}$. For arbitrary finite aperiodic semigroups of order $N-1$, we improve the bound of Aaronson, Grier, and Schaeffer, obtaining a bounded-error quantum query complexity of at most \\[ \\min\\left\\{n,\\sqrt n\\, \\log^{O((N\\log(N+2))^{1/3})}(n+2)\\right\\}. \\] The dependence on semigroup size is nearly tight: the bounded-depth Dyck lower bound of Ambainis et al. yields aperiodic monoids requiring $\\sqrt n\\,2^{Ω(N^{1/3})}$ quantum queries in the relevant parameter range.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.39974",
  "author_total": 2,
  "authors": [
    {
      "name": "Troy Lee"
    },
    {
      "name": "Miklos Santha"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/f2bdg3fpesfn2.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.39974",
  "fulltext_url": null,
  "id": "f2bdg3fpesfn2",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.39974",
  "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": "W7215093274",
  "pdf_url": "https://arxiv.org/pdf/2609.39974",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-30",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/f2bdg3fpesfn2.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:25:49Z",
      "native_id": "2609.39974",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.39974"
    }
  ],
  "subjects": [
    {
      "field": "physics",
      "origin": "arxiv:quant-ph",
      "subfield": "quantum"
    }
  ],
  "title": "The quantum query complexity of the semigroup product problem",
  "title_text": "The quantum query complexity of the semigroup product problem",
  "type": "preprint",
  "updated_at": "2026-10-30T12:25:49Z",
  "url": "https://research.draconitic.com/paper/f2bdg3fpesfn2",
  "venue": "arXiv",
  "versions": []
}