{
  "abstract": "Alice and Bob hold complementary halves of the columns of an $n\\times n$ matrix of $k$-bit integers and decide whether it is singular over $\\mathbb Q$. Up to constant-factor dimension growth, this partitioned problem is equivalent to additive singularity, where Alice and Bob hold $n\\times n$ matrices $A,B$ of $k$-bit integers and test the singularity of $A+B$ over $\\mathbb Q$. We prove a bounded-error quantum lower bound $Ω(\\min\\{n^{3/2}\\sqrt{k\\log n},n^2\\log(nk)\\})$ and a randomized upper bound $O(n^2\\min\\{k,\\log(k+\\log n)\\})$. The bounds match at $Θ(n^2\\log(nk))$ for $k\\ge n\\log n$. The lower bound holds even under the promise of rank $n-1$ versus $n$. Its proof combines an integer witness with a local digit encoding, while the upper-bound protocol lifts modular kernel relations using short messages. Bitwise sharing changes how the communication complexity depends on the entry bit length $k$. If Alice and Bob instead hold $n\\times n$ matrices $A,B$ of $k$-bit integers and test whether their entrywise bitwise XOR $A\\oplus B$ is singular over $\\mathbb Q$, we prove a quantum lower bound $Ω(n^2k)$, matching the trivial upper bound. This lower bound holds even under the promise of rank $n-1$ versus $n$. Our proof uses a dense weighted equation and a signed Fourier witness whose exceptional dyadic frequencies can be removed at small variation cost.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.33521",
  "author_total": 1,
  "authors": [
    {
      "name": "Chengu Wang"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/2gvvejgzee3cf.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.33521",
  "fulltext_url": null,
  "id": "2gvvejgzee3cf",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.33521",
  "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",
      "1703|Complexity and Algorithms in Graphs"
    ]
  ],
  "openalex": "W7223917098",
  "pdf_url": "https://arxiv.org/pdf/2610.33521",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-24",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/2gvvejgzee3cf.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:35:35Z",
      "native_id": "2610.33521",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.33521"
    }
  ],
  "subjects": [
    {
      "field": "physics",
      "origin": "arxiv:quant-ph",
      "subfield": "quantum"
    }
  ],
  "title": "Communication Complexity of Integer Matrix Singularity",
  "title_text": "Communication Complexity of Integer Matrix Singularity",
  "type": "preprint",
  "updated_at": "2026-10-30T12:35:35Z",
  "url": "https://research.draconitic.com/paper/2gvvejgzee3cf",
  "venue": "arXiv",
  "versions": []
}