{
  "abstract": "We construct a constant-degree and constant-gap quantum expander on $n$ qubits where each unitary can be implemented by a depth-$1$ and 1D circuit of Pauli or CNOT gates. We provide two applications of this expander. First, we use it to construct a family of frustration-free 1D Hamiltonians whose ground states obey the entanglement-gap relation $S = Θ(Δ^{-1/2})$; this is believed to be optimal, but achieving it had been open. Second, we use it to provide a streaming protocol that tests for closeness to a class of 1D volume-law entangled states. Moreover, we extend our quantum expander to a constant-degree and constant-gap expander on the unitary group where each unitary is a single $T$ gate, a single $T^{\\dagger}$ gate, or a depth-$1$ Clifford circuit. This implies that a random sequence of unitaries from the expander yields a gapped walk on a dense subgroup of the unitary group. This improves upon previous work by Bourgain and Gamburd which did not control the dependence of the gap on the dimension.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.01605",
  "author_total": 5,
  "authors": [
    {
      "name": "Anurag Anshu"
    },
    {
      "name": "Shankar Balasubramanian"
    },
    {
      "name": "Jonas Haferkamp"
    },
    {
      "name": "Aram W. Harrow"
    },
    {
      "name": "Xinyu Tan"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/7f0xn8ykwb88s.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.01605",
  "fulltext_url": null,
  "id": "7f0xn8ykwb88s",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.01605",
  "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",
      "cond-mat.str-el"
    ],
    [
      "arxiv",
      "cs.CC"
    ],
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      "arxiv",
      "cs.IT"
    ],
    [
      "arxiv",
      "math.GR"
    ],
    [
      "arxiv",
      "math.IT"
    ],
    [
      "arxiv",
      "quant-ph"
    ],
    [
      "openalex",
      "1702|Quantum Computing Algorithms and Architecture"
    ]
  ],
  "openalex": "W7206167968",
  "pdf_url": "https://arxiv.org/pdf/2609.01605",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-01",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/7f0xn8ykwb88s.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:18:31Z",
      "native_id": "2609.01605",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.01605"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.GR",
      "subfield": "algebra-number-theory"
    },
    {
      "field": "physics",
      "origin": "arxiv:cond-mat.str-el",
      "subfield": "condensed-matter"
    },
    {
      "field": "physics",
      "origin": "arxiv:quant-ph",
      "subfield": "quantum"
    }
  ],
  "title": "Depth-1 expanders on the unitary group and applications",
  "title_text": "Depth-1 expanders on the unitary group and applications",
  "type": "preprint",
  "updated_at": "2026-10-30T12:18:36Z",
  "url": "https://research.draconitic.com/paper/7f0xn8ykwb88s",
  "venue": "arXiv",
  "versions": []
}