{
  "id": "1410.7034",
  "version": "v1",
  "published": "2014-10-26T14:35:24.000Z",
  "updated": "2014-10-26T14:35:24.000Z",
  "title": "On the Relation between Random Walks and Quantum Walks",
  "authors": [
    "Stefan Boettcher",
    "Stefan Falkner",
    "Renato Portugal"
  ],
  "comment": "8 pages, RevTex4-1; for related information, see http://www.physics.emory.edu/faculty/boettcher/",
  "categories": [
    "cond-mat.stat-mech",
    "quant-ph"
  ],
  "abstract": "We propose a general relation between the walk dimensions $d_{w}$ of random walks and quantum walks, which illuminates fundamental aspects of the walk dynamics, such as its mean-square displacement. To this end, we extend the renormalization group analysis (RG) of the stochastic master equation to one with a unitary propagator. As in the classical case, the solution $\\rho(x,t)$ in space and time of this quantum walk equation exhibits a scaling collapse for the variable $x^{d_{w}}/t$. In all cases studied, we find that $d_{w}$ of the discrete-time quantum walk with the Grover coin takes on exactly half the value found for the random walk on the same geometry, irrespective of whether it is a homogeneous lattice or a heterogeneous network. We demonstrate this circumstance for several networks with distinct properties using exact RG and confirm the collapse in each case with extensive numerical simulation.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2014-10-26T14:35:24.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "03.67.Ac",
      "05.10.Cc",
      "05.40.Fb"
    ],
    "keywords": [
      "random walk",
      "stochastic master equation",
      "renormalization group analysis",
      "illuminates fundamental aspects",
      "quantum walk equation"
    ],
    "publication": {
      "doi": "10.1103/PhysRevA.91.052330",
      "journal": "Physical Review A",
      "year": 2015,
      "month": "May",
      "volume": 91,
      "number": 5,
      "pages": "052330"
    },
    "note": {
      "typesetting": "RevTeX",
      "pages": 8,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2015PhRvA..91e2330B"
    }
  }
}