{
  "id": "0810.4968",
  "version": "v2",
  "published": "2008-10-28T02:53:26.000Z",
  "updated": "2009-03-25T22:33:33.000Z",
  "title": "Quantum Algorithms Using the Curvelet Transform",
  "authors": [
    "Yi-Kai Liu"
  ],
  "comment": "64 pages, 4 figures; improved algorithm and lower bound for finding the center of a radial function; revised presentation; to appear in STOC 2009",
  "journal": "Proc. ACM Symposium on Theory of Computing (STOC), pp.391-400, 2009.",
  "categories": [
    "quant-ph"
  ],
  "abstract": "The curvelet transform is a directional wavelet transform over R^n, which is used to analyze functions that have singularities along smooth surfaces (Candes and Donoho, 2002). I demonstrate how this can lead to new quantum algorithms. I give an efficient implementation of a quantum curvelet transform, together with two applications: a single-shot measurement procedure for approximately finding the center of a ball in R^n, given a quantum-sample over the ball; and, a quantum algorithm for finding the center of a radial function over R^n, given oracle access to the function. I conjecture that these algorithms succeed with constant probability, using one quantum-sample and O(1) oracle queries, respectively, independent of the dimension n -- this can be interpreted as a quantum speed-up. To support this conjecture, I prove rigorous bounds on the distribution of probability mass for the continuous curvelet transform. This shows that the above algorithms work in an idealized \"continuous\" model.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2009-03-25T22:33:33.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "quantum algorithm",
      "directional wavelet transform",
      "quantum curvelet transform",
      "single-shot measurement procedure",
      "analyze functions"
    ],
    "tags": [
      "journal article"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 64,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2008arXiv0810.4968L"
    }
  }
}