{
  "id": "quant-ph/0511096",
  "version": "v2",
  "published": "2005-11-09T20:08:32.000Z",
  "updated": "2006-04-10T15:51:48.000Z",
  "title": "A Polynomial Quantum Algorithm for Approximating the Jones Polynomial",
  "authors": [
    "Dorit Aharonov",
    "Vaughan Jones",
    "Zeph Landau"
  ],
  "comment": "19 pages. This is a revised version of our paper quant-ph/0511096. The main change is a much more intuitive and less technical exposition of the Path model representation, which makes the paper much shorter. In addition, we somewhat updated the introduction, corrected various technical errors, and hopefully improved the writing",
  "journal": "STOC 2006",
  "categories": [
    "quant-ph"
  ],
  "abstract": "The Jones polynomial, discovered in 1984, is an important knot invariant in topology. Among its many connections to various mathematical and physical areas, it is known (due to Witten) to be intimately connected to Topological Quantum Field Theory (TQFT). The works of Freedman, Kitaev, Larsen and Wang provide an efficient simulation of TQFT by a quantum computer, and vice versa. These results implicitly imply the existence of an efficient quantum algorithm that provides a certain additive approximation of the Jones polynomial at the fifth root of unity, e^{2\\pi i/5}, and moreover, that this problem is BQP-complete. Unfortunately, this important algorithm was never explicitly formulated. Moreover, the results mentioned are heavily based on TQFT, which makes the algorithm essentially inaccessible to computer scientists. We provide an explicit and simple polynomial quantum algorithm to approximate the Jones polynomial of an n-strands braid with m crossings at any primitive root of unity e^{2\\pi i/k}, where the running time of the algorithm is polynomial in m,n and k. Our algorithm is based, rather than on TQFT, on well known mathematical results. By the results of Freedman et. al., our algorithm solves a BQP complete problem. The algorithm we provide exhibits a structure which we hope is generalizable to other quantum algorithmic problems. Candidates of particular interest are the approximations of other #P-hard problems, most notably, the partition function of the Potts model, a model which is known to be tightly connected to the Jones polynomial.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2006-04-10T15:51:48.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "jones polynomial",
      "simple polynomial quantum algorithm",
      "quantum algorithmic problems",
      "bqp complete problem",
      "efficient quantum algorithm"
    ],
    "tags": [
      "journal article"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 19,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2005quant.ph.11096A"
    }
  }
}