{
  "annotation_id": "7c6c6776-8e5d-40f2-8c2f-aedaba1f7496",
  "date_created": "2026-03-02T18:01:58.991000Z",
  "date_modified": "2026-03-02T18:01:58.991000Z",
  "file_hash": "e4714a684b6dd717180fe0698d85186b42eb5a160769010bed6a24a997cde71f",
  "private": false,
  "record": {
    "abstract": "We give a general method for proving quantum lower bounds for problems with\nsmall range. Namely, we show that, for any symmetric problem defined on\nfunctions $f:\\{1, ..., N\\}\\to\\{1, ..., M\\}$, its polynomial degree is the same\nfor all $M\\geq N$. Therefore, if we have a quantum lower bound for some\n(possibly, quite large) range $M$ which is shown using polynomials method, we\nimmediately get the same lower bound for all ranges $M\\geq N$. In particular,\nwe get $\\Omega(N^{1/3})$ and $\\Omega(N^{2/3})$ quantum lower bounds for\ncollision and element distinctness with small range.",
    "arxiv_id": "quant-ph/0305179",
    "authors": [
      "Andris Ambainis"
    ],
    "categories": [
      "quant-ph",
      "cs.CC"
    ],
    "journal_ref": "Theory of Computing, 1:37-46, 2005",
    "title": "Polynomial Degree and Lower Bounds in Quantum Complexity: Collision and   Element Distinctness with Small Range",
    "url": "https://arxiv.org/abs/quant-ph/0305179"
  },
  "schema_id": "dorsal/arxiv",
  "source": {
    "execution_id": "e6def447-7b24-4233-a510-fb7a409f16fd",
    "id": "arXiv Dataset IDs",
    "type": "Model",
    "variant": "snapshot-2026-03-01",
    "version": "0.1.0"
  },
  "user_id": 1000002
}