{
  "id": "1009.1340",
  "version": "v1",
  "published": "2010-09-07T16:54:40.000Z",
  "updated": "2010-09-07T16:54:40.000Z",
  "title": "Perfect state transfer, graph products and equitable partitions",
  "authors": [
    "Yang Ge",
    "Benjamin Greenberg",
    "Oscar Perez",
    "Christino Tamon"
  ],
  "comment": "18 pages, 6 figures",
  "journal": "International Journal of Quantum Information 9(3):823-842, 2011",
  "categories": [
    "quant-ph",
    "math.CO"
  ],
  "abstract": "We describe new constructions of graphs which exhibit perfect state transfer on continuous-time quantum walks. Our constructions are based on variants of the double cones [BCMS09,ANOPRT10,ANOPRT09] and the Cartesian graph products (which includes the n-cube) [CDDEKL05]. Some of our results include: (1) If $G$ is a graph with perfect state transfer at time $t_{G}$, where $t_{G}\\Spec(G) \\subseteq \\ZZ\\pi$, and $H$ is a circulant with odd eigenvalues, their weak product $G \\times H$ has perfect state transfer. Also, if $H$ is a regular graph with perfect state transfer at time $t_{H}$ and $G$ is a graph where $t_{H}|V_{H}|\\Spec(G) \\subseteq 2\\ZZ\\pi$, their lexicographic product $G[H]$ has perfect state transfer. (2) The double cone $\\overline{K}_{2} + G$ on any connected graph $G$, has perfect state transfer if the weights of the cone edges are proportional to the Perron eigenvector of $G$. This generalizes results for double cone on regular graphs studied in [BCMS09,ANOPRT10,ANOPRT09]. (3) For an infinite family $\\GG$ of regular graphs, there is a circulant connection so the graph $K_{1}+\\GG\\circ\\GG+K_{1}$ has perfect state transfer. In contrast, no perfect state transfer exists if a complete bipartite connection is used (even in the presence of weights) [ANOPRT09]. We also describe a generalization of the path collapsing argument [CCDFGS03,CDDEKL05], which reduces questions about perfect state transfer to simpler (weighted) multigraphs, for graphs with equitable distance partitions.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2010-09-07T16:54:40.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "perfect state transfer",
      "equitable partitions",
      "regular graph",
      "double cone",
      "cartesian graph products"
    ],
    "tags": [
      "journal article"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 18,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2010arXiv1009.1340G"
    }
  }
}