{
  "abstract": "The Travelling Salesman Problem (TSP) is a classic combinatorial optimisation problem which is NP-hard in optimisation form and NP-complete as a decision problem. The standard question asks for the shortest tour that visits every city exactly once and returns to the origin. This paper studies a threshold-fraction variant, which we call the robust TSP, formulated as a promise problem: given a distance threshold $K$ and a promised fraction $p$, decide whether at least a fraction $p$ of all tours have total length at most $K$, or whether no such tour exists. We show that Grover's quantum search algorithm decides this problem using $O(1/\\sqrt{p})$ oracle queries, independent of the number of cities $N$, with each query implemented by a quantum circuit of polynomial size in $N$. For any fixed robustness level $p$, the total runtime is therefore polynomial in $N$. Measured in oracle queries, the quantum algorithm offers a quadratic improvement in the dependence on $1/p$ over classical random sampling, which needs $Θ(1/p)$ samples; because each quantum query is costly, however, it is faster in total running time only when $p$ is very small (roughly below $N^{-4}$), where the number of queries is no longer constant. We describe the problem formulation, detail the oracle construction (edge-length loading, sequential quantum addition, and threshold comparison), and analyse the query complexity to achieve the required result.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.14062",
  "author_total": 1,
  "authors": [
    {
      "name": "Aayra Dhindsa Sawkmie"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/2tepswcm0zmkz.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.14062",
  "fulltext_url": null,
  "id": "2tepswcm0zmkz",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.14062",
  "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.DS"
    ],
    [
      "arxiv",
      "quant-ph"
    ],
    [
      "openalex",
      "1702|Quantum Computing Algorithms and Architecture"
    ]
  ],
  "openalex": "W7221835996",
  "pdf_url": "https://arxiv.org/pdf/2610.14062",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-07",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/2tepswcm0zmkz.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:29:06Z",
      "native_id": "2610.14062",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.14062"
    }
  ],
  "subjects": [
    {
      "field": "physics",
      "origin": "arxiv:quant-ph",
      "subfield": "quantum"
    }
  ],
  "title": "Grover Search for a Threshold-Fraction Variant of the Travelling Salesman Problem: A Circuit-Level Construction",
  "title_text": "Grover Search for a Threshold-Fraction Variant of the Travelling Salesman Problem: A Circuit-Level Construction",
  "type": "preprint",
  "updated_at": "2026-10-30T12:29:06Z",
  "url": "https://research.draconitic.com/paper/2tepswcm0zmkz",
  "venue": "arXiv",
  "versions": []
}