{
  "id": "1905.03682",
  "version": "v1",
  "published": "2019-05-09T15:17:19.000Z",
  "updated": "2019-05-09T15:17:19.000Z",
  "title": "Operator growth bounds from graph theory",
  "authors": [
    "Chi-Fang Chen",
    "Andrew Lucas"
  ],
  "comment": "47 pages, 14 figures",
  "categories": [
    "math-ph",
    "hep-th",
    "math.MP",
    "quant-ph"
  ],
  "abstract": "Let $A$ and $B$ be local operators in Hamiltonian quantum systems with $N $ degrees of freedom and finite-dimensional Hilbert space. We prove that the commutator norm $\\lVert [A(t),B]\\rVert$ is upper bounded by a topological combinatorial problem: counting irreducible weighted paths between two points on the Hamiltonian's factor graph. We show that the Lieb-Robinson velocity always overestimates the ballistic growth of operators. In quantum systems drawn from zero-mean random ensembles with few-body interactions, we prove stronger bounds on the ensemble-averaged $\\mathbb{E}\\left[ \\lVert [A(t),B]\\rVert^2\\right]$. In such quantum systems on Erd\\\"os-R\\'enyi factor graphs, we prove that the scrambling time $t_{\\mathrm{s}}$, at which $\\lVert [A(t_{\\mathrm{s}}),B]\\rVert=\\mathrm{\\Theta}(1)$, is almost surely $t_{\\mathrm{s}}=\\mathrm{\\Omega}(\\sqrt{\\log N})$; we further prove $t_{\\mathrm{s}}=\\mathrm{\\Omega}(\\log N) $ to high order in perturbation theory in $1/N$. We constrain infinite temperature quantum chaos in the $q$-local Sachdev-Ye-Kitaev model at any order in $1/N$; at leading order, our upper bound on the Lyapunov exponent is within a factor of 2 of the known result at any $q>2$. We also speculate on the implications of our theorems for conjectured holographic descriptions of quantum gravity.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2019-05-09T15:17:19.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "operator growth bounds",
      "graph theory",
      "constrain infinite temperature quantum chaos",
      "hamiltonian quantum systems",
      "local sachdev-ye-kitaev model"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 47,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}