{
  "id": "1911.06060",
  "version": "v1",
  "published": "2019-11-14T12:22:46.000Z",
  "updated": "2019-11-14T12:22:46.000Z",
  "title": "A zeta function related to the transition matrix of the discrete-time quantum walk on a graph",
  "authors": [
    "Norio Konno",
    "Iwao Sato",
    "Etsuo Segawa"
  ],
  "categories": [
    "math.CO"
  ],
  "abstract": "We present the structure theorem for the positive support of the cube of the Grover transition matrix of the discrete-time quantum walk (the Grover walk) on a general graph $G$ under same condition. Thus, we introduce a zeta function on the positive support of the cube of the Grover transition matrix of $G$, and present its Euler product and its determinant expression. As a corollary, we give the characteristic polynomial for the positive support of the cube of the Grover transition matrix of a regular graph, and so obtain its spectra. Finally, we present the poles and the radius of the convergence of this zeta function.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2019-11-14T12:22:46.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "60F50",
      "05C50",
      "15A15",
      "05C60"
    ],
    "keywords": [
      "discrete-time quantum walk",
      "zeta function",
      "grover transition matrix",
      "positive support",
      "euler product"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}