{
  "annotation_id": "e31285db-4f7b-4adc-a4cc-a729a26e7734",
  "date_created": "2026-03-02T18:02:37.908000Z",
  "date_modified": "2026-03-02T18:02:37.908000Z",
  "file_hash": "1b89903f196a9f75a4ab6f5aaa6842a25331dee6988044faff84e244d4ede184",
  "private": false,
  "record": {
    "abstract": "A quantum error-correcting code is defined to be a unitary mapping (encoding)\nof k qubits (2-state quantum systems) into a subspace of the quantum state\nspace of n qubits such that if any t of the qubits undergo arbitrary\ndecoherence, not necessarily independently, the resulting n qubits can be used\nto faithfully reconstruct the original quantum state of the k encoded qubits.\nQuantum error-correcting codes are shown to exist with asymptotic rate k/n = 1\n- 2H(2t/n) where H(p) is the binary entropy function -p log p - (1-p) log\n(1-p). Upper bounds on this asymptotic rate are given.",
    "arxiv_id": "quant-ph/9512032",
    "authors": [
      "A. R. Calderbank",
      "Peter W. Shor"
    ],
    "categories": [
      "quant-ph"
    ],
    "doi": "10.1103/PhysRevA.54.1098",
    "journal_ref": "Phys. Rev. A, Vol. 54, No. 2, pp. 1098-1106, 1996",
    "title": "Good Quantum Error-Correcting Codes Exist",
    "url": "https://arxiv.org/abs/quant-ph/9512032"
  },
  "schema_id": "dorsal/arxiv",
  "source": {
    "execution_id": "2c23a2f4-81da-4ad7-b104-913a7a9c91d3",
    "id": "arXiv Dataset IDs",
    "type": "Model",
    "variant": "snapshot-2026-03-01",
    "version": "0.1.0"
  },
  "user_id": 1000002
}