{
  "id": "2210.06796",
  "version": "v1",
  "published": "2022-10-13T07:21:30.000Z",
  "updated": "2022-10-13T07:21:30.000Z",
  "title": "Circuit depth versus energy in topologically ordered systems",
  "authors": [
    "Arkin Tikku",
    "Isaac H. Kim"
  ],
  "comment": "26 pages, 4 figures",
  "categories": [
    "quant-ph",
    "cond-mat.str-el"
  ],
  "abstract": "We prove a nontrivial circuit-depth lower bound for preparing a low-energy state of a locally interacting quantum many-body system in two dimensions, assuming the circuit is geometrically local. For preparing any state which has an energy density of at most $\\epsilon$ with respect to Kitaev's toric code Hamiltonian on a two dimensional lattice $\\Lambda$, we prove a lower bound of $\\Omega\\left(\\min\\left(1/\\epsilon^{\\frac{1-\\alpha}{2}}, \\sqrt{|\\Lambda|}\\right)\\right)$ for any $\\alpha >0$. We discuss two implications. First, our bound implies that the lowest energy density obtainable from a large class of existing variational circuits (e.g., Hamiltonian variational ansatz) cannot, in general, decay exponentially with the circuit depth. Second, if long-range entanglement is present in the ground state, this can lead to a nontrivial circuit-depth lower bound even at nonzero energy density. Unlike previous approaches to prove circuit-depth lower bounds for preparing low energy states, our proof technique does not rely on the ground state to be degenerate.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2022-10-13T07:21:30.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "circuit depth",
      "topologically ordered systems",
      "nontrivial circuit-depth lower bound",
      "interacting quantum many-body system",
      "energy density"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 26,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}