{
  "id": "2009.13309",
  "version": "v1",
  "published": "2020-09-28T13:26:53.000Z",
  "updated": "2020-09-28T13:26:53.000Z",
  "title": "Comment to Spatial Search by Quantum Walk is Optimal for Almost all Graphs",
  "authors": [
    "Ryszard Kukulski",
    "Adam Glos"
  ],
  "comment": "The comments applies for the paper found in https://journals.aps.org/prl/abstract/10.1103/PhysRevLett.116.100501",
  "categories": [
    "quant-ph"
  ],
  "abstract": "This comment is to correct the proof of optimality of quantum spatial search for Erd\\H{o}s-R\\'enyi graphs presented in `Spatial Search by Quantum Walk is Optimal for Almost all Graphs' (https://doi.org/10.1103/PhysRevLett.116.100501). The authors claim that if $p\\geq \\frac{\\log^{3/2}(n)}{n}$, then the CTQW-based search is optimal for almost all graphs. Below we point the issues found in the main paper, and propose corrections, which in fact improve the result to $p=\\omega(\\log(n)/n)$ in case of transition rate $\\gamma = 1/\\lambda_1$. In the case of the proof for simplified transition rate $1/(np)$ we pointed a possible issue with applying perturbation theory.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2020-09-28T13:26:53.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "quantum walk",
      "quantum spatial search",
      "main paper",
      "simplified transition rate",
      "applying perturbation theory"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}