{
  "id": "2203.11137",
  "version": "v1",
  "published": "2022-03-21T17:12:31.000Z",
  "updated": "2022-03-21T17:12:31.000Z",
  "title": "Adiabatic paths of Hamiltonians, symmetries of topological order, and automorphism codes",
  "authors": [
    "David Aasen",
    "Zhenghan Wang",
    "Matthew B. Hastings"
  ],
  "comment": "36 pages, 10 figures",
  "categories": [
    "quant-ph",
    "cond-mat.str-el",
    "hep-th",
    "math-ph",
    "math.MP"
  ],
  "abstract": "The recent \"honeycomb code\" is a fault-tolerant quantum memory defined by a sequence of checks which implements a nontrivial automorphism of the toric code. We argue that a general framework to understand this code is to consider continuous adiabatic paths of gapped Hamiltonians and we give a conjectured description of the fundamental group and second and third homotopy groups of this space in two spatial dimensions. A single cycle of such a path can implement some automorphism of the topological order of that Hamiltonian. We construct such paths for arbitrary automorphisms of two-dimensional doubled topological order. Then, realizing this in the case of the toric code, we turn this path back into a sequence of checks, constructing an automorphism code closely related to the honeycomb code.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2022-03-21T17:12:31.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "topological order",
      "automorphism code",
      "adiabatic paths",
      "hamiltonian",
      "honeycomb code"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 36,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}