{
  "id": "2410.21635",
  "version": "v1",
  "published": "2024-10-29T00:43:33.000Z",
  "updated": "2024-10-29T00:43:33.000Z",
  "title": "Learning the structure of any Hamiltonian from minimal assumptions",
  "authors": [
    "Andrew Zhao"
  ],
  "comment": "44 pages",
  "categories": [
    "quant-ph",
    "cs.DS",
    "cs.LG"
  ],
  "abstract": "We study the problem of learning an unknown quantum many-body Hamiltonian $H$ from black-box queries to its time evolution $e^{-\\mathrm{i} H t}$. Prior proposals for solving this task either impose some assumptions on $H$, such as its interaction structure or locality, or otherwise use an exponential amount of computational postprocessing. In this paper, we present efficient algorithms to learn any $n$-qubit Hamiltonian, assuming only a bound on the number of Hamiltonian terms, $m \\leq \\mathrm{poly}(n)$. Our algorithms do not need to know the terms in advance, nor are they restricted to local interactions. We consider two models of control over the time evolution: the first has access to time reversal ($t < 0$), enabling an algorithm that outputs an $\\epsilon$-accurate classical description of $H$ after querying its dynamics for a total of $\\widetilde{O}(m/\\epsilon)$ evolution time. The second access model is more conventional, allowing only forward-time evolutions; our algorithm requires $\\widetilde{O}(\\|H\\|^3/\\epsilon^4)$ evolution time in this setting. Central to our results is the recently introduced concept of a pseudo-Choi state of $H$. We extend the utility of this learning resource by showing how to use it to learn the Fourier spectrum of $H$, how to achieve nearly Heisenberg-limited scaling with it, and how to prepare it even under our more restricted access models.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2024-10-29T00:43:33.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "minimal assumptions",
      "unknown quantum many-body hamiltonian",
      "evolution time",
      "second access model",
      "fourier spectrum"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 44,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}