{
  "abstract": "We argue that the exponential quantum advantage in Shor's algorithm is broken under one-layer depolarizing noise of arbitrary small error rate, by analyzing a resonance breaking phenomenon in noisy quantum circuits. First, we express the distribution of measurements on $n$-qubit strings as the superposition of the $4^n$ wave functions in the Pauli path integral. In the noiseless case, the bit-strings achieving resonant peaks guaranteed a constant rate of successful measurements to factor a large number. Secondly, when the middle layer of the quantum circuit has independent depolarizing noise of rate $λ$, for any Hamming weight $d$, we obtain corresponding measurement rate bounded by $2(1-λ)^d$ for higher frequency terms and by $O(n^d)/2^{n/2}$ for low frequency terms. The rate approaches to zero as $n$ and $d$ approaches to infinity. Resonance breaking destroys the exponential quantum advantage. Furthermore, we design a classical factoring algorithm in polynomial time $O(n^d)$ to substitute the low frequency contribution.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.37636",
  "author_total": 1,
  "authors": [
    {
      "name": "Zhengwei Liu"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/15vfx6w7jv103.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.37636",
  "fulltext_url": null,
  "id": "15vfx6w7jv103",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.37636",
  "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"
    ],
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      "arxiv",
      "cs.IT"
    ],
    [
      "arxiv",
      "math-ph"
    ],
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      "arxiv",
      "math.IT"
    ],
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      "arxiv",
      "math.MP"
    ],
    [
      "arxiv",
      "math.OA"
    ],
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      "arxiv",
      "quant-ph"
    ],
    [
      "openalex",
      "1702|Quantum Computing Algorithms and Architecture"
    ]
  ],
  "openalex": "W7214975912",
  "pdf_url": "https://arxiv.org/pdf/2609.37636",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-29",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/15vfx6w7jv103.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:25:31Z",
      "native_id": "2609.37636",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.37636"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.OA",
      "subfield": "analysis-pdes"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math-ph",
      "subfield": "mathematical-physics"
    },
    {
      "field": "physics",
      "origin": "arxiv:quant-ph",
      "subfield": "quantum"
    }
  ],
  "title": "Resonance Breaking in Noisy Shor's Algorithm",
  "title_text": "Resonance Breaking in Noisy Shor's Algorithm",
  "type": "preprint",
  "updated_at": "2026-10-30T12:25:35Z",
  "url": "https://research.draconitic.com/paper/15vfx6w7jv103",
  "venue": "arXiv",
  "versions": []
}