{
  "id": "1009.2131",
  "version": "v2",
  "published": "2010-09-11T01:59:31.000Z",
  "updated": "2011-07-28T07:26:58.000Z",
  "title": "Crossovers induced by discrete-time quantum walks",
  "authors": [
    "Kota Chisaki",
    "Norio Konno",
    "Etsuo Segawa",
    "Yutaka Shikano"
  ],
  "comment": "14 pages, 1 figure",
  "journal": "Quantum Information and Computation 11 (2011) pp.0741-0760",
  "categories": [
    "quant-ph",
    "cond-mat.stat-mech",
    "math-ph",
    "math.MP",
    "math.PR"
  ],
  "abstract": "We consider crossovers with respect to the weak convergence theorems from a discrete-time quantum walk (DTQW). We show that a continuous-time quantum walk (CTQW) and discrete- and continuous-time random walks can be expressed as DTQWs in some limit. At first we generalize our previous study [Phys. Rev. A \\textbf{81}, 062129 (2010)] on the DTQW with position measurements. We show that the position measurements per each step with probability $p \\sim 1/n^\\beta$ can be evaluated, where $n$ is the final time and $0<\\beta<1$. We also give a corresponding continuous-time case. As a consequence, crossovers from the diffusive spreading (random walk) to the ballistic spreading (quantum walk) can be seen as the parameter $\\beta$ shifts from 0 to 1 in both discrete- and continuous-time cases of the weak convergence theorems. Secondly, we introduce a new class of the DTQW, in which the absolute value of the diagonal parts of the quantum coin is proportional to a power of the inverse of the final time $n$. This is called a final-time-dependent DTQW (FTD-DTQW). The CTQW is obtained in a limit of the FTD-DTQW. We also obtain the weak convergence theorem for the FTD-DTQW which shows a variety of spreading properties. Finally, we consider the FTD-DTQW with periodic position measurements. This weak convergence theorem gives a phase diagram which maps sufficiently long-time behaviors of the discrete- and continuous-time quantum and random walks.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2011-07-28T07:26:58.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "discrete-time quantum walk",
      "weak convergence theorem",
      "random walk",
      "crossovers",
      "continuous-time case"
    ],
    "tags": [
      "journal article"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 14,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2010arXiv1009.2131C"
    }
  }
}