{
  "id": "2312.13892",
  "version": "v1",
  "published": "2023-12-21T14:39:24.000Z",
  "updated": "2023-12-21T14:39:24.000Z",
  "title": "Efficient Quantum Algorithm for Filtering Product States",
  "authors": [
    "Reinis Irmejs",
    "Mari Carmen Bañuls",
    "J. Ignacio Cirac"
  ],
  "comment": "10 pages, 8 figures",
  "categories": [
    "quant-ph"
  ],
  "abstract": "We introduce a quantum algorithm to efficiently prepare states with an arbitrarily small energy variance at the target energy. We achieve it by filtering a product state at the given energy with a Lorentzian filter of width $\\delta$. Given a local Hamiltonian on $N$ qubits, we construct a parent Hamiltonian whose ground state corresponds to the filtered product state with variable energy variance proportional to $\\delta\\sqrt{N}$. We prove that the parent Hamiltonian is gapped and its ground state can be efficiently implemented in $\\mathrm{poly}(N,1/\\delta)$ time via adiabatic evolution. We numerically benchmark the algorithm for a particular non-integrable model and find that the adiabatic evolution time to prepare the filtered state with a width $\\delta$ is independent of the system size $N$. Furthermore, the adiabatic evolution can be implemented with circuit depth $\\mathcal{O}(N^2\\delta^{-4})$. Our algorithm provides a way to study the finite energy regime of many body systems in quantum simulators by directly preparing a finite energy state, providing access to an approximation of the microcanonical properties at an arbitrary energy.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2023-12-21T14:39:24.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "efficient quantum algorithm",
      "filtering product states",
      "adiabatic evolution",
      "parent hamiltonian",
      "variable energy variance proportional"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 10,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}