{
  "abstract": "We introduce the quantum Černý complexity $\\mathrm{qc}(w)$ of a binary word $w$: the least dimension $d$ for which there exist quantum channels $A_0,A_1$ on $d\\times d$ density matrices and a start state $ρ_0$ such that $w$ is the unique shortest word whose associated channel is constant on the reachable set. We show that $2\\le\\mathrm{qc}(w)\\le\\lceil\\sqrt{|w|+1}\\,\\rceil$ for every nonempty $w$, a quadratic saving over the classical analogue, and that constant words are extremal: $\\mathrm{qc}(0^m)=\\lceil\\sqrt{m+1}\\,\\rceil$. In contrast, $\\mathrm{qc}(01^n0)=2$ for every $n\\ge 1$, realized by a single qubit whose rotation angle acts as a counter; consequently there is no quantum analogue of the Černý function, and $\\mathrm{qc}$ is strongly anti-correlated with intuitive notions of descriptive complexity. We further study the variant $\\mathrm{qcp}$ in which the synchronization target is required to be a pure state. We prove that in dimension $2$ no word of length at least $2$ can be a unique shortest synchronizing word with pure target, and we exhibit an explicit qutrit instance, combining a coherent rotation with a measure-and-funnel channel, achieving $\\mathrm{qcp}(01^n0)=3$ with target a computational basis state and with synchronization holding universally over all input states. Thus purity of the reset state costs exactly one dimension on this family. We also observe that $\\mathrm{qc}$ is computable, by reduction to the first-order theory of the reals.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.40154",
  "author_total": 3,
  "authors": [
    {
      "name": "Pui Hang Lee"
    },
    {
      "name": "Pui-Yee Lee"
    },
    {
      "name": "Bjørn Kjos-Hanssen"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/b153vk31vkm0j.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.40154",
  "fulltext_url": null,
  "id": "b153vk31vkm0j",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.40154",
  "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",
      "cs.DM"
    ],
    [
      "arxiv",
      "quant-ph"
    ],
    [
      "openalex",
      "1703|semigroups and automata theory"
    ]
  ],
  "openalex": "W7215084694",
  "pdf_url": "https://arxiv.org/pdf/2609.40154",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-30",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/b153vk31vkm0j.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:25:49Z",
      "native_id": "2609.40154",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.40154"
    }
  ],
  "subjects": [
    {
      "field": "physics",
      "origin": "arxiv:quant-ph",
      "subfield": "quantum"
    }
  ],
  "title": "Quantum Černý complexity of binary words",
  "title_text": "Quantum Černý complexity of binary words",
  "type": "preprint",
  "updated_at": "2026-10-30T12:25:49Z",
  "url": "https://research.draconitic.com/paper/b153vk31vkm0j",
  "venue": "arXiv",
  "versions": []
}