{
  "id": "2212.00020",
  "version": "v1",
  "published": "2022-11-30T12:21:51.000Z",
  "updated": "2022-11-30T12:21:51.000Z",
  "title": "Abstract Model of Continuous-Time Quantum Walk Based on Bernoulli Functionals and Perfect State Transfer",
  "authors": [
    "Ce Wang"
  ],
  "categories": [
    "quant-ph",
    "math-ph",
    "math.FA",
    "math.MP",
    "math.PR"
  ],
  "abstract": "In this paper, we present an abstract model of continuous-time quantum walk (CTQW) based on Bernoulli functionals and show that the model has perfect state transfer (PST), among others. Let $\\mathfrak{h}$ be the space of square integrable complex-valued Bernoulli functionals, which is infinitely dimensional. First, we construct on a given subspace $\\mathfrak{h}_L \\subset \\mathfrak{h}$ a self-adjoint operator $\\Delta_L$ via the canonical unitary involutions on $\\mathfrak{h}$, and by analyzing its spectral structure we find out all its eigenvalues. Then, we introduce an abstract model of CTQW with $\\mathfrak{h}_L$ as its state space, which is governed by the Schr\\\"{o}dinger equation with $\\Delta_L$ as the Hamiltonian. We define the time-average probability distribution of the model, obtain an explicit expression of the distribution, and, especially, we find the distribution admits a symmetry property. We also justify the model by offering a graph-theoretic interpretation to the operator $\\Delta_L$ as well as to the model itself. Finally, we prove that the model has PST at time $t=\\frac{\\pi}{2}$. Some other interesting results are also proven of the model.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2022-11-30T12:21:51.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "continuous-time quantum walk",
      "perfect state transfer",
      "abstract model",
      "square integrable complex-valued bernoulli functionals",
      "time-average probability distribution"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}