{
  "annotation_id": "661c887f-da3d-4478-8297-bbb9379b9bda",
  "date_created": "2026-03-02T18:02:37.942000Z",
  "date_modified": "2026-03-02T18:02:37.942000Z",
  "file_hash": "623655188317ea1806abb500c05cd83f3cc8c38fca6d3a920d1d82dea0adc988",
  "private": false,
  "record": {
    "abstract": "One of the main problems for the future of practical quantum computing is to\nstabilize the computation against unwanted interactions with the environment\nand imperfections in the applied operations. Existing proposals for quantum\nmemories and quantum channels require gates with asymptotically zero error to\nstore or transmit an input quantum state for arbitrarily long times or\ndistances with fixed error. In this report a method is given which has the\nproperty that to store or transmit a qubit with maximum error $\\epsilon$\nrequires gates with error at most $c\\epsilon$ and storage or channel elements\nwith error at most $\\epsilon$, independent of how long we wish to store the\nstate or how far we wish to transmit it. The method relies on using\nconcatenated quantum codes with hierarchically implemented recovery operations.\nThe overhead of the method is polynomial in the time of storage or the distance\nof the transmission. Rigorous and heuristic lower bounds for the constant $c$\nare given.",
    "arxiv_id": "quant-ph/9608012",
    "authors": [
      "Emanuel Knill",
      "Raymond Laflamme"
    ],
    "categories": [
      "quant-ph"
    ],
    "title": "Concatenated Quantum Codes",
    "url": "https://arxiv.org/abs/quant-ph/9608012"
  },
  "schema_id": "dorsal/arxiv",
  "source": {
    "execution_id": "8cc953df-b9eb-4787-b1a9-0bcd3b457852",
    "id": "arXiv Dataset IDs",
    "type": "Model",
    "variant": "snapshot-2026-03-01",
    "version": "0.1.0"
  },
  "user_id": 1000002
}