{
  "abstract": "We study the randomized classical query complexity of cyclic Forrelation, a cyclic analogue of $k$-Forrelation [AA15, STOC'15; BS21, STOC'21]. Given $k$ Boolean functions $f_1,\\ldots,f_k:\\mathbb{F}_2^n\\to\\{-1,1\\}$, the goal is to estimate $T_k=N^{-1}\\operatorname{Tr}(D_{f_1}H^{\\otimes n}\\cdots D_{f_k}H^{\\otimes n}),$ where $D_{f_i}=\\operatorname{diag}(f_i(x):x\\in\\mathbb{F}_2^n)$, $H$ is the Hadamard gate, and $N=2^n$. This normalized-trace problem can be solved with $k$ quantum queries in the DQC1 model. Its classical query complexity was left open in [JLS+26, FOCS'26], which conjectured a lower bound of $\\widetildeΩ(N^{1-2/k})$. We establish a nearly complete characterization of the nonadaptive classical query complexity of cyclic Forrelation. For even $k\\ge 4$, we prove a tight bound of $Θ_k(N^{1-2/k})$ for constant additive error. For odd $k\\ge 5$, we obtain a lower bound of $Ω_k(N^{1-2/(k-1)})$ and an upper bound of $O_k(N^{1-2/k})$. Surprisingly, $k=3$ admits a polylogarithmic-query nonadaptive classical algorithm, disproving the conjecture in this case. These results suggest an unexpected parity-dependent structure: the nonadaptive complexity may be governed by the independence number of the $k$-cycle, leading to the exponent $1-1/\\lfloor k/2\\rfloor$, rather than the previously conjectured $1-2/k$, which was motivated by a Fourier-growth heuristic. Finally, allowing adaptivity, we prove a lower bound of $Ω(N^{1/4})$ for every $k\\ge4$. Although formally close to $k$-Forrelation, the cyclic structure leads to rather different classical behavior and requires different techniques. Our lower bounds are based on hidden-shift collision constructions, and our upper bounds follow from randomized sampling together with cycle-specific conditional-moment estimates. Our results provide a systematic understanding of its query advantages over classical computation.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.41694",
  "author_total": 2,
  "authors": [
    {
      "name": "Yuxin Zhang"
    },
    {
      "name": "Changpeng Shao"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/1qt4s7vm8w4qb.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.41694",
  "fulltext_url": null,
  "id": "1qt4s7vm8w4qb",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.41694",
  "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",
      "quant-ph"
    ]
  ],
  "openalex": null,
  "pdf_url": "https://arxiv.org/pdf/2610.41694",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-30",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/1qt4s7vm8w4qb.ris",
  "sources": [
    {
      "first_seen": "2026-11-02T12:21:15Z",
      "native_id": "2610.41694",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.41694"
    }
  ],
  "subjects": [
    {
      "field": "physics",
      "origin": "arxiv:quant-ph",
      "subfield": "quantum"
    }
  ],
  "title": "Exponential quantum-classical query separations for cyclic Forrelation",
  "title_text": "Exponential quantum-classical query separations for cyclic Forrelation",
  "type": "preprint",
  "updated_at": "2026-11-04T15:20:51Z",
  "url": "https://research.draconitic.com/paper/1qt4s7vm8w4qb",
  "venue": "arXiv",
  "versions": []
}