{
  "id": "2312.03083",
  "version": "v1",
  "published": "2023-12-05T19:02:19.000Z",
  "updated": "2023-12-05T19:02:19.000Z",
  "title": "Dual-VQE: A quantum algorithm to lower bound the ground-state energy",
  "authors": [
    "Hanna Westerheim",
    "Jingxuan Chen",
    "Zoë Holmes",
    "Ivy Luo",
    "Theshani Nuradha",
    "Dhrumil Patel",
    "Soorya Rethinasamy",
    "Kathie Wang",
    "Mark M. Wilde"
  ],
  "comment": "8 pages, 1 figure",
  "categories": [
    "quant-ph"
  ],
  "abstract": "The variational quantum eigensolver (VQE) is a hybrid quantum--classical variational algorithm that produces an upper-bound estimate of the ground-state energy of a Hamiltonian. As quantum computers become more powerful and go beyond the reach of classical brute-force simulation, it is important to assess the quality of solutions produced by them. Here we propose a dual variational quantum eigensolver (dual-VQE) that produces a lower-bound estimate of the ground-state energy. As such, VQE and dual-VQE can serve as quality checks on their solutions; in the ideal case, the VQE upper bound and the dual-VQE lower bound form an interval containing the true optimal value of the ground-state energy. The idea behind dual-VQE is to employ semi-definite programming duality to rewrite the ground-state optimization problem as a constrained maximization problem, which itself can be bounded from below by an unconstrained optimization problem to be solved by a variational quantum algorithm. When using a convex combination ansatz in conjunction with a classical generative model, the quantum computational resources needed to evaluate the objective function of dual-VQE are no greater than those needed for that of VQE. We simulated the performance of dual-VQE on the transverse-field Ising model, and found that, for the example considered, while dual-VQE training is slower and noisier than VQE, it approaches the true value with error of order $10^{-2}$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2023-12-05T19:02:19.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "ground-state energy",
      "quantum algorithm",
      "dual-vqe lower bound form",
      "dual variational quantum eigensolver",
      "quantum computational resources"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 8,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}