{
  "id": "2410.19628",
  "version": "v1",
  "published": "2024-10-25T15:27:41.000Z",
  "updated": "2024-10-25T15:27:41.000Z",
  "title": "Design nearly optimal quantum algorithm for linear differential equations via Lindbladians",
  "authors": [
    "Zhong-Xia Shang",
    "Naixu Guo",
    "Dong An",
    "Qi Zhao"
  ],
  "comment": "9+10 pages, 1 figure, 1 table",
  "categories": [
    "quant-ph",
    "cond-mat.mes-hall"
  ],
  "abstract": "Solving linear ordinary differential equations (ODE) is one of the most promising applications for quantum computers to demonstrate exponential advantages. The challenge of designing a quantum ODE algorithm is how to embed non-unitary dynamics into intrinsically unitary quantum circuits. In this work, we propose a new quantum algorithm for ODEs by harnessing open quantum systems. Specifically, we utilize the natural non-unitary dynamics of Lindbladians with the aid of a new technique called the non-diagonal density matrix encoding to encode general linear ODEs into non-diagonal blocks of density matrices. This framework enables us to design a quantum algorithm that has both theoretical simplicity and good performance. Combined with the state-of-the-art quantum Lindbladian simulation algorithms, our algorithm, under a plausible input model, can outperform all existing quantum ODE algorithms and achieve near-optimal dependence on all parameters. We also give applications of our algorithm including the Gibbs state preparations and the partition function estimations.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2024-10-25T15:27:41.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "linear differential equations",
      "optimal quantum algorithm",
      "linear ordinary differential equations",
      "quantum ode algorithm",
      "state-of-the-art quantum lindbladian simulation algorithms"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 10,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}