{
  "id": "1508.07473",
  "version": "v1",
  "published": "2015-08-29T16:22:18.000Z",
  "updated": "2015-08-29T16:22:18.000Z",
  "title": "Generator of an abstract quantum walk",
  "authors": [
    "Etuso Segawa",
    "Akito Suzuki"
  ],
  "categories": [
    "math-ph",
    "math.MP",
    "math.SP"
  ],
  "abstract": "We consider an abstract quantum walk defined by a unitary evolution operator $U$, which acts on a Hilbert space decomposed into a direct sum of Hilbert spaces $\\{\\mathcal{H}_v \\}_{v \\in V}$. We show that such $U$ naturally defines a directed graph $G_U$ and the probability of finding a quantum walker on $G_U$. The asymptotic property of an abstract quantum walker is governed by the generator $H$ of $U$ such that $U^n = e^{inH}$. We derive the generator of an evolution of the form $U = S(2d_A^* d_A -1)$, a generalization of the Szegedy evolution operator. Here $d_A$ is a boundary operator and $S$ a shift operator.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2015-08-29T16:22:18.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "81S25",
      "81P68",
      "82B41"
    ],
    "keywords": [
      "hilbert space",
      "szegedy evolution operator",
      "unitary evolution operator",
      "abstract quantum walker",
      "direct sum"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2015arXiv150807473S"
    }
  }
}