{
  "id": "1212.6523",
  "version": "v5",
  "published": "2012-12-28T14:33:36.000Z",
  "updated": "2014-09-17T16:27:33.000Z",
  "title": "A quantum algorithm for obtaining the lowest eigenstate of a Hamiltonian assisted with an ancillary qubit system",
  "authors": [
    "Jeongho Bang",
    "Seung-Woo Lee",
    "Chang-Woo Lee",
    "Hyunseok Jeong"
  ],
  "comment": "11 pages, 3 figures. To be appeared in Quantum Information Processing",
  "categories": [
    "quant-ph",
    "math-ph",
    "math.MP"
  ],
  "abstract": "We propose a quantum algorithm to obtain the lowest eigenstate of any Hamiltonian simulated by a quantum computer. The proposed algorithm begins with an arbitrary initial state of the simulated system. A finite series of transforms is iteratively applied to the initial state assisted with an ancillary qubit. The fraction of the lowest eigenstate in the initial state is then amplified up to $\\simeq 1$. We prove that our algorithm can faithfully work for any arbitrary Hamiltonian in the theoretical analysis. Numerical analyses are also carried out. We firstly provide a numerical proof-of-principle demonstration with a simple Hamiltonian in order to compare our scheme with the so-called \"Demon-like algorithmic cooling (DLAC)\", recently proposed in [Nature Photonics 8, 113 (2014)]. The result shows a good agreement with our theoretical analysis, exhibiting the comparable behavior to the best \"cooling\" with the DLAC method. We then consider a random Hamiltonian model for further analysis of our algorithm. By numerical simulations, we show that the total number $n_c$ of iterations is proportional to $\\simeq {\\cal O}(D^{-1}\\epsilon^{-0.19})$, where $D$ is the difference between the two lowest eigenvalues, and $\\epsilon$ is an error defined as the probability that the finally obtained system state is in an unexpected (i.e. not the lowest) eigenstate.",
  "revisions": [
    {
      "version": "v4",
      "updated": "2014-02-26T11:28:21.000Z",
      "title": "Quantum algorithm for obtaining the lowest eigenstate of a Hamiltonian from arbitrary initial state assisted with an ancillary qubit",
      "abstract": "We propose a quantum algorithm for obtaining the lowest eigenstate of a Hamiltonian. The proposed algorithm starts with an arbitrary initial state. A finite series of transforms iteratively applied to the initial state, assisted with an ancillary qubit, amplifies the fraction of the lowest eigenstate in the initial state so that it approaches unity. We analyze the process of the algorithm, and prove that the algorithm successfully works for an arbitrary Hamiltonian unless the initial state is completely orthogonal to the lowest eigenstate. Numerical analysis is also carried out comparing with so-called imaginary time propagation (ITP). We firstly perform numerical simulations using a simple Hamiltonian model where the eigenvalues are equally spaced. We then consider a random Hamiltonian model for more general analysis. By numerical simulations, we find that the convergences of the two different methods are similar, but our algorithm is slightly faster.",
      "comment": null,
      "journal": null,
      "doi": null
    },
    {
      "version": "v5",
      "updated": "2014-09-17T16:27:33.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "arbitrary initial state",
      "lowest eigenstate",
      "ancillary qubit",
      "quantum algorithm",
      "imaginary time propagation"
    ],
    "tags": [
      "journal article"
    ],
    "publication": {
      "doi": "10.1007/s11128-014-0836-5",
      "journal": "Quantum Information Processing",
      "year": 2015,
      "month": "Jan",
      "volume": 14,
      "number": 1,
      "pages": 103
    },
    "note": {
      "typesetting": "TeX",
      "pages": 11,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2015QuIP...14..103B"
    }
  }
}