{
  "id": "1009.1306",
  "version": "v2",
  "published": "2010-09-07T14:29:47.000Z",
  "updated": "2012-01-11T03:04:51.000Z",
  "title": "Limit Theorems for the Discrete-Time Quantum Walk on a Graph with Joined Half Lines",
  "authors": [
    "Kota Chisaki",
    "Norio Konno",
    "Etsuo Segawa"
  ],
  "comment": "18 pages, 7 figures",
  "journal": "Quantum Information and Computation 12, (2012) 0314--0333",
  "categories": [
    "quant-ph"
  ],
  "abstract": "We consider a discrete-time quantum walk $W_{t,\\kappa}$ at time $t$ on a graph with joined half lines $\\mathbb{J}_\\kappa$, which is composed of $\\kappa$ half lines with the same origin. Our analysis is based on a reduction of the walk on a half line. The idea plays an important role to analyze the walks on some class of graphs with \\textit{symmetric} initial states. In this paper, we introduce a quantum walk with an enlarged basis and show that $W_{t,\\kappa}$ can be reduced to the walk on a half line even if the initial state is \\textit{asymmetric}. For $W_{t,\\kappa}$, we obtain two types of limit theorems. The first one is an asymptotic behavior of $W_{t,\\kappa}$ which corresponds to localization. For some conditions, we find that the asymptotic behavior oscillates. The second one is the weak convergence theorem for $W_{t,\\kappa}$. On each half line, $W_{t,\\kappa}$ converges to a density function like the case of the one-dimensional lattice with a scaling order of $t$. The results contain the cases of quantum walks starting from the general initial state on a half line with the general coin and homogeneous trees with the Grover coin.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2012-01-11T03:04:51.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "discrete-time quantum walk",
      "joined half lines",
      "limit theorems",
      "weak convergence theorem",
      "general initial state"
    ],
    "tags": [
      "journal article"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 18,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2010arXiv1009.1306C"
    }
  }
}