{
  "abstract": "We study the one-shot and concurrent hitting for the discrete-time Grover-coined quantum walk on cubelike graphs $G=\\text{Cay}(\\mathbb Z_2^d,Ω)$ of degree $Δ=|Ω|$. Starting from the vertex labeled $0$, we identify $σ=\\bigoplus_{ω\\inΩ}ω$ as a natural target vertex; for the hypercube, $σ$ is precisely the antipodal vertex. For families with $Δ\\to\\infty$, let $T$ be an integer having the same parity as $Δ$ and satisfying $ \\left|T-\\frac{πΔ}{2}\\right|\\leq 1. $ We show that the probability $p_T(σ)$ of finding the walker at $σ$ when it is measured at time $T$ satisfies $$ p_T(σ)=1-O(Δ^{-1/5}). $$ Thus the target is found with probability tending to one after $Θ(Δ)$ steps. For the concurrently measured walk, let $H_T^{\\mathrm{Conc}}(σ)$ denote the probability that the target is detected at or before time $T$ when it is tested after every step. We prove $$ p_T(σ)\\leq T H_T^{\\mathrm{Conc}}(σ), $$ which implies $H_T^{\\mathrm{Conc}}(σ)=Ω(Δ^{-1})$ over the same time scale. The proof uses the Walsh-Fourier decomposition, an exact two-dimensional reduction of each Fourier mode, and a universal second-moment identity for the associated character sums. Our results extend Kempe's hypercube hitting phenomenon (J. Kempe, Probab. Theory Relat. Fields 133, 215-235, 2005) to arbitrary cubelike generating sets and establish the conjectured asymptotic hitting behavior for cublelike and augmented cubes in Mulherkar, Rajdeepak and Sunitha (Int. J. Quantum Inf. 20,2250020, 2022)",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.04503",
  "author_total": 1,
  "authors": [
    {
      "name": "Jaideep Mulherkar"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/1x9dn4cvcwf81.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.04503",
  "fulltext_url": null,
  "id": "1x9dn4cvcwf81",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.04503",
  "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": "W7211912147",
  "pdf_url": "https://arxiv.org/pdf/2609.04503",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-03",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/1x9dn4cvcwf81.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:19:14Z",
      "native_id": "2609.04503",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.04503"
    }
  ],
  "subjects": [
    {
      "field": "physics",
      "origin": "arxiv:quant-ph",
      "subfield": "quantum"
    }
  ],
  "title": "One-Shot and Concurrent Hitting Times for Grover-Coined Quantum Walks on Cubelike Graphs",
  "title_text": "One-Shot and Concurrent Hitting Times for Grover-Coined Quantum Walks on Cubelike Graphs",
  "type": "preprint",
  "updated_at": "2026-10-30T12:19:14Z",
  "url": "https://research.draconitic.com/paper/1x9dn4cvcwf81",
  "venue": "arXiv",
  "versions": []
}