{
  "id": "2103.05235",
  "version": "v1",
  "published": "2021-03-09T05:45:25.000Z",
  "updated": "2021-03-09T05:45:25.000Z",
  "title": "A new type of spectral mapping theorem for quantum walks with a moving shift on graphs",
  "authors": [
    "Sho Kubota",
    "Kei Saito",
    "Yusuke Yoshie"
  ],
  "comment": "26 pages",
  "categories": [
    "quant-ph",
    "math.CO"
  ],
  "abstract": "The conventional spectral mapping theorem for quantum walks can only be applied for walks employing a shift operator whose square is the identity. This theorem gives most of the eigenvalues of the time evolution $U$ by lifting the eigenvalues of an induced self-adjoint matrix $T$ onto the unit circle on the complex plane. We acquire a new spectral mapping theorem for the Grover walk with a shift operator whose cube is the identity on finite graphs. Moreover, graphs we can consider for a quantum walk with such a shift operator is characterized by a triangulation. We call these graphs triangulable graphs in this paper. One of the differences between our spectral mapping theorem and the conventional one is that lifting the eigenvalues of $T-1/2$ onto the unit circle gives most of the eigenvalues of $U$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2021-03-09T05:45:25.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "05C50",
      "05C81",
      "81Q99"
    ],
    "keywords": [
      "quantum walk",
      "moving shift",
      "shift operator",
      "unit circle",
      "eigenvalues"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 26,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}