{
  "id": "2012.05259",
  "version": "v1",
  "published": "2020-12-09T19:00:50.000Z",
  "updated": "2020-12-09T19:00:50.000Z",
  "title": "Improved spectral gaps for random quantum circuits: large local dimensions and all-to-all interactions",
  "authors": [
    "Jonas Haferkamp",
    "Nicholas Hunter-Jones"
  ],
  "comment": "27 pages, 2 figures",
  "journal": "Phys. Rev. A 104, 022417 (2021)",
  "doi": "10.1103/PhysRevA.104.022417",
  "categories": [
    "quant-ph",
    "cond-mat.stat-mech",
    "cond-mat.str-el"
  ],
  "abstract": "Random quantum circuits are a central concept in quantum information theory with applications ranging from demonstrations of quantum computational advantage to descriptions of scrambling in strongly-interacting systems and black holes. The utility of random quantum circuits in these settings stems from their ability to rapidly generate quantum pseudo-randomness. In a seminal paper by Brand\\~ao, Harrow, and Horodecki, it was proven that the $t$-th moment operator of local random quantum circuits on $n$ qudits with local dimension $q$ has a spectral gap of at least $\\Omega(n^{-1}t^{-5-3.1/\\log(q)})$, which implies that they are efficient constructions of approximate unitary designs. As a first result, we use Knabe bounds for the spectral gaps of frustration-free Hamiltonians to show that $1D$ random quantum circuits have a spectral gap scaling as $\\Omega(n^{-1})$, provided that $t$ is small compared to the local dimension: $t^2\\leq O(q)$. This implies a (nearly) linear scaling of the circuit depth in the design order $t$. Our second result is an unconditional spectral gap bounded below by $\\Omega(n^{-1}\\log^{-1}(n) t^{-\\alpha(q)})$ for random quantum circuits with all-to-all interactions. This improves both the $n$ and $t$ scaling in design depth for the non-local model. We show this by proving a recursion relation for the spectral gaps involving an auxiliary random walk. Lastly, we solve the smallest non-trivial case exactly and combine with numerics and Knabe bounds to improve the constants involved in the spectral gap for small values of $t$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2020-12-09T19:00:50.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "spectral gap",
      "large local dimensions",
      "all-to-all interactions",
      "knabe bounds",
      "local random quantum circuits"
    ],
    "tags": [
      "journal article"
    ],
    "publication": {
      "publisher": "APS",
      "journal": "Phys. Rev. A"
    },
    "note": {
      "typesetting": "TeX",
      "pages": 27,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}