{
  "abstract": "The quantum Fourier transform (QFT) is a fundamental primitive in quantum computation and quantum information. In this work, we extend the framework for the QFT from finite groups to finite-dimensional semisimple algebras, and give efficient QFTs for the partition algebra $P_n(d)$, Brauer algebra $B_n(d)$, and walled Brauer algebra $B_{r,s}(d)$. These algebras play important roles in generalized Schur-Weyl duality, statistical physics and many-body systems, and have recently found several applications in quantum algorithms. Unlike the group case, the Fourier transform over a semisimple algebra can be non-unitary. Nevertheless, we show that when the parameter $d$ is sufficiently large, the Fourier transform is well approximated by a unitary operator. Furthermore, we show that for each of the algebras $A$ from above, such an approximate Fourier transform can be implemented efficiently: we give a quantum algorithm with gate complexity $\\mathrm{poly}(n,\\log d,\\log(1/\\varepsilon))$ for approximating the Fourier transform to error $(d^{-1/2} + \\varepsilon) \\cdot \\mathrm{poly}(|A|)$. Along the way, we establish several properties of the Fourier basis of semisimple algebras that may be of independent interest.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2605.05337",
  "author_total": 3,
  "authors": [
    {
      "name": "Ben Foxman"
    },
    {
      "name": "Barak Nehoran"
    },
    {
      "name": "Yongshan Ding"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/a10xy8xn1ksdr.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2605.05337",
  "fulltext_url": null,
  "id": "a10xy8xn1ksdr",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2605.05337",
  "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": "W7160596565",
  "pdf_url": "https://arxiv.org/pdf/2605.05337",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-28",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/a10xy8xn1ksdr.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:25:12Z",
      "native_id": "2605.05337",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2605.05337"
    }
  ],
  "subjects": [
    {
      "field": "physics",
      "origin": "arxiv:quant-ph",
      "subfield": "quantum"
    }
  ],
  "title": "Efficient Quantum Fourier Transforms For Semisimple Algebras",
  "title_text": "Efficient Quantum Fourier Transforms For Semisimple Algebras",
  "type": "preprint",
  "updated_at": "2026-10-30T12:25:12Z",
  "url": "https://research.draconitic.com/paper/a10xy8xn1ksdr",
  "venue": "arXiv",
  "versions": []
}