{
  "annotation_id": "e6917bba-ba9c-47cb-ac43-95788d9b5944",
  "date_created": "2026-03-02T18:01:53.157000Z",
  "date_modified": "2026-03-02T18:01:53.157000Z",
  "file_hash": "3329bfcdaebdbb43ce53f2b846b44ce40187d6749951a9b0082781f84c02eaa2",
  "private": false,
  "record": {
    "abstract": "The controlled-not gate and the single qubit gates are considered elementary\ngates in quantum computing. It is natural to ask how many such elementary gates\nare needed to implement more elaborate gates or circuits. Recall that a\ncontrolled-U gate can be realized with two controlled-not gates and four single\nqubit gates. We prove that this implementation is optimal if and only if the\nmatrix U satisfies the conditions tr U != 0, tr UX != 0, and det U != 1. We\nalso derive optimal implementations in the non-generic cases.",
    "arxiv_id": "quant-ph/0207157",
    "authors": [
      "Guang Song",
      "Andreas Klappenecker"
    ],
    "categories": [
      "quant-ph"
    ],
    "journal_ref": "Journal of Quantum Information and Computation, 3(2), pages\n  139-155, 2003",
    "title": "Optimal Realizations of Controlled Unitary Gates",
    "url": "https://arxiv.org/abs/quant-ph/0207157"
  },
  "schema_id": "dorsal/arxiv",
  "source": {
    "execution_id": "1588561f-1010-4ec3-8ec8-51c982504489",
    "id": "arXiv Dataset IDs",
    "type": "Model",
    "variant": "snapshot-2026-03-01",
    "version": "0.1.0"
  },
  "user_id": 1000002
}