{
  "annotation_id": "0ead0177-6234-4875-aa18-4968a037c50d",
  "date_created": "2026-03-02T18:02:34.492000Z",
  "date_modified": "2026-03-02T18:02:34.492000Z",
  "file_hash": "f2a788b7c2500447fb9d5224804e03e4e5397a358653192e094bd1754b2e6170",
  "private": false,
  "record": {
    "abstract": "In this Letter we make progress on a longstanding open problem of Aaronson\nand Ambainis [Theory of Computing 1, 47 (2005)]: we show that if A is the\nadjacency matrix of a sufficiently sparse low-dimensional graph then the\nunitary operator e^{itA} can be approximated by a unitary operator U(t) whose\nsparsity pattern is exactly that of a low-dimensional graph which gets more\ndense as |t| increases. Secondly, we show that if U is a sparse unitary\noperator with a gap \\Delta in its spectrum, then there exists an approximate\nlogarithm H of U which is also sparse. The sparsity pattern of H gets more\ndense as 1/\\Delta increases. These two results can be interpreted as a way to\nconvert between local continuous-time and local discrete-time processes. As an\nexample we show that the discrete-time coined quantum walk can be realised as\nan approximately local continuous-time quantum walk. Finally, we use our\nconstruction to provide a definition for a fractional quantum fourier\ntransform.",
    "arxiv_id": "quant-ph/0611231",
    "authors": [
      "Tobias J. Osborne"
    ],
    "categories": [
      "quant-ph"
    ],
    "doi": "10.1103/PhysRevLett.101.140503",
    "journal_ref": "Phys. Rev. Lett. 101, 140503 (2008)",
    "title": "Approximate locality for quantum systems on graphs",
    "url": "https://arxiv.org/abs/quant-ph/0611231"
  },
  "schema_id": "dorsal/arxiv",
  "source": {
    "execution_id": "73feba0e-a4d8-412b-80aa-432fdfb5e0c4",
    "id": "arXiv Dataset IDs",
    "type": "Model",
    "variant": "snapshot-2026-03-01",
    "version": "0.1.0"
  },
  "user_id": 1000002
}