{
  "id": "2209.11220",
  "version": "v1",
  "published": "2022-09-22T17:58:57.000Z",
  "updated": "2022-09-22T17:58:57.000Z",
  "title": "Quantum algorithms for uncertainty quantification: application to partial differential equations",
  "authors": [
    "Francois Golse",
    "Shi Jin",
    "Nana Liu"
  ],
  "categories": [
    "quant-ph",
    "math-ph",
    "math.MP"
  ],
  "abstract": "Most problems in uncertainty quantification, despite its ubiquitousness in scientific computing, applied mathematics and data science, remain formidable on a classical computer. For uncertainties that arise in partial differential equations (PDEs), large numbers M>>1 of samples are required to obtain accurate ensemble averages. This usually involves solving the PDE M times. In addition, to characterise the stochasticity in a PDE, the dimension L of the random input variables is high in most cases, and classical algorithms suffer from curse-of-dimensionality. We propose new quantum algorithms for PDEs with uncertain coefficients that are more efficient in M and L in various important regimes, compared to their classical counterparts. We introduce transformations that transfer the original d-dimensional equation (with uncertain coefficients) into d+L (for dissipative equations) or d+2L (for wave type equations) dimensional equations (with certain coefficients) in which the uncertainties appear only in the initial data. These transformations also allow one to superimpose the M different initial data, so the computational cost for the quantum algorithm to obtain the ensemble average from M different samples is then independent of M, while also showing potential advantage in d, L and precision in computing ensemble averaged solutions or physical observables.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2022-09-22T17:58:57.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "partial differential equations",
      "quantum algorithm",
      "uncertainty quantification",
      "initial data",
      "application"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}