{
  "abstract": "In the quest to identify further quantum algorithms exhibiting superpolynomial speed-ups, a recurring theme is that the complexity of a problem is largely shaped by the input access model. Here, we study this phenomenon for the state hidden subgroup problem (StateHSP), a quantum generalization of the hidden subgroup problem in which the goal is to identify the symmetries of an unknown quantum state. For finite abelian groups, existing Fourier-sampling algorithms use $O(\\log(|G|)/ε)$ copies of the state, but whether this scaling is optimal has remained open. We settle the complexity of the abelian StateHSP in both the previously studied sample model and a new query model, which is a stronger and operationally natural generalization that provides access to the state-preparation unitary and its inverse. In the query model, we give a time-efficient quantum algorithm using $O(\\log(|G/H|)/\\sqrtε)$ forward and inverse queries, and prove a matching $Ω(\\log(|G/H|)/\\sqrtε)$ lower bound which holds even in the stronger conjugate-query and controlled-query settings. By contrast, we show that in the sample model, $Θ(\\log(|G/H|)/ε)$ copies are both sufficient and information-theoretically necessary, even if one allows for arbitrary collective measurements. Thus, the quadratic improvement in $ε$ genuinely arises from coherent access to the preparation circuit. As applications, we obtain faster algorithms for learning stabilizer groups, locating unentanglement, and identifying hidden translation symmetries.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.35656",
  "author_total": 4,
  "authors": [
    {
      "name": "Yuhan Liu"
    },
    {
      "name": "Jose Carrasco"
    },
    {
      "name": "Jens Eisert"
    },
    {
      "name": "Armando Bellante"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/7yxk1fvy06dn2.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.35656",
  "fulltext_url": null,
  "id": "7yxk1fvy06dn2",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.35656",
  "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": "W7214761125",
  "pdf_url": "https://arxiv.org/pdf/2609.35656",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-28",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/7yxk1fvy06dn2.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:25:12Z",
      "native_id": "2609.35656",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.35656"
    }
  ],
  "subjects": [
    {
      "field": "physics",
      "origin": "arxiv:quant-ph",
      "subfield": "quantum"
    }
  ],
  "title": "The power of oracle access: Optimal sample and query complexity of the abelian state hidden subgroup problem",
  "title_text": "The power of oracle access: Optimal sample and query complexity of the abelian state hidden subgroup problem",
  "type": "preprint",
  "updated_at": "2026-10-30T12:25:12Z",
  "url": "https://research.draconitic.com/paper/7yxk1fvy06dn2",
  "venue": "arXiv",
  "versions": []
}