{
  "id": "1911.07495",
  "version": "v1",
  "published": "2019-11-18T09:19:58.000Z",
  "updated": "2019-11-18T09:19:58.000Z",
  "title": "Uniform mixing on abelian Cayley graphs",
  "authors": [
    "Xiwang Cao",
    "Keqin Feng"
  ],
  "categories": [
    "math.CO"
  ],
  "abstract": "In the past few decades, quantum algorithms have become a popular research area of both mathematicians and engineers. In 2003, Childs et al. found a graph in which the continuous-time quantum walk spreads exponentially faster than any classical algorithm for a certain black-box problem. Since then, there are extensive explorations on quantum walks on graphs. Among them, uniform mixing provides a uniform probability distribution of information over time which attracts a special attention. However, there are only a few known examples of graphs which admit uniform mixing. In this paper, we give a characterization of abelian Cayley graphs having uniform mixing. Some concrete constructions of such graphs are provided. Particularly, for cubelike graphs, it is shown that the Cayley graph ${\\rm Cay}(\\mathbb{F}_2^{2k};S)$ having uniform mixing if and only if $f$ is a bent function, where $f$ is the characteristic function of $S$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2019-11-18T09:19:58.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "05C25",
      "81P45",
      "81Q35"
    ],
    "keywords": [
      "abelian cayley graphs",
      "uniform mixing",
      "continuous-time quantum walk spreads",
      "quantum walk spreads exponentially faster",
      "popular research area"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}