{
  "id": "2505.01602",
  "version": "v1",
  "published": "2025-05-02T21:53:19.000Z",
  "updated": "2025-05-02T21:53:19.000Z",
  "title": "Schrödingerization based quantum algorithms for the fractional Poisson equation",
  "authors": [
    "Shi Jin",
    "Nana Liu",
    "Yue Yu"
  ],
  "comment": "quantum algorithms for fractional differential equations",
  "categories": [
    "math.NA",
    "cs.NA"
  ],
  "abstract": "We develop a quantum algorithm for solving high-dimensional fractional Poisson equations. By applying the Caffarelli-Silvestre extension, the $d$-dimensional fractional equation is reformulated as a local partial differential equation in $d+1$ dimensions. We propose a quantum algorithm for the finite element discretization of this local problem, by capturing the steady-state of the corresponding differential equations using the Schr\\\"odingerization approach from \\cite{JLY22SchrShort, JLY22SchrLong, analogPDE}. The Schr\\\"odingerization technique transforms general linear partial and ordinary differential equations into Schr\\\"odinger-type systems, making them suitable for quantum simulation. This is achieved through the warped phase transformation, which maps the equation into a higher-dimensional space. We provide detailed implementations of the method and conduct a comprehensive complexity analysis, which can show up to exponential advantage -- with respect to the inverse of the mesh size in high dimensions -- compared to its classical counterpart. Specifically, while the classical method requires $\\widetilde{\\mathcal{O}}(d^{1/2} 3^{3d/2} h^{-d-2})$ operations, the quantum counterpart requires $\\widetilde{\\mathcal{O}}(d 3^{3d/2} h^{-2.5})$ queries to the block-encoding input models, with the quantum complexity being independent of the dimension $d$ in terms of the inverse mesh size $h^{-1}$. Numerical experiments are conducted to verify the validity of our formulation.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2025-05-02T21:53:19.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "quantum algorithm",
      "technique transforms general linear partial",
      "solving high-dimensional fractional poisson equations",
      "schrödingerization",
      "local partial differential equation"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}