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Efficient quantum algorithm for dissipative nonlinear differential equations

Liu, Jin-Peng ; Kolden, Herman Øie ; Krovi, Hari K ; Loureiro, Nuno F ; Trivisa, Konstantina ; Childs, Andrew M

Proceedings of the National Academy of Sciences - PNAS, 2021-08, Vol.118 (35), p.1 [Periódico revisado por pares]

United States: National Academy of Sciences

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  • Título:
    Efficient quantum algorithm for dissipative nonlinear differential equations
  • Autor: Liu, Jin-Peng ; Kolden, Herman Øie ; Krovi, Hari K ; Loureiro, Nuno F ; Trivisa, Konstantina ; Childs, Andrew M
  • Assuntos: Algorithms ; Carleman linearization ; CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSICS ; Complexity ; Computational fluid dynamics ; Decay ; Differential equations ; Epidemic models ; Epidemiology ; Error analysis ; Fluid dynamics ; Hydrodynamics ; Linearity ; Lower bounds ; Mathematical models ; MATHEMATICS AND COMPUTING ; Navier–Stokes equation ; Nonlinear differential equations ; Nonlinear systems ; Nonlinearity ; Ordinary differential equations ; Physical Sciences ; plasma dynamics ; Quadratic equations ; quantum algorithm ; Quantum mechanics ; Science & Technology - Other Topics
  • É parte de: Proceedings of the National Academy of Sciences - PNAS, 2021-08, Vol.118 (35), p.1
  • Notas: ObjectType-Article-1
    SourceType-Scholarly Journals-1
    ObjectType-Feature-2
    content type line 23
    SC0020312; SC0020264; CCF-1813814
    USDOE Office of Science (SC), Advanced Scientific Computing Research (ASCR)
    National Science Foundation (NSF)
    Edited by Anthony Leggett, University of Illinois at Urbana–Champaign, Urbana, IL, and approved July 19, 2021 (received for review March 6, 2021)
    Author contributions: J.-P.L., H.Ø.K., H.K.K., N.F.L., K.T., and A.M.C. designed research; J.-P.L., H.Ø.K., H.K.K., N.F.L., K.T., and A.M.C. performed research; J.P.L. led the theoretical analysis; H.Ø.K. led the numerical experiments; and J.-P.L., H.Ø.K., H.K.K., N.F.L., K.T., and A.M.C. wrote the paper.
  • Descrição: Nonlinear differential equations model diverse phenomena but are notoriously difficult to solve. While there has been extensive previous work on efficient quantum algorithms for linear differential equations, the linearity of quantum mechanics has limited analogous progress for the nonlinear case. Despite this obstacle, we develop a quantum algorithm for dissipative quadratic n-dimensional ordinary differential equations. Assuming [Formula: see text], where R is a parameter characterizing the ratio of the nonlinearity and forcing to the linear dissipation, this algorithm has complexity [Formula: see text], where T is the evolution time, ϵ is the allowed error, and q measures decay of the solution. This is an exponential improvement over the best previous quantum algorithms, whose complexity is exponential in T. While exponential decay precludes efficiency, driven equations can avoid this issue despite the presence of dissipation. Our algorithm uses the method of Carleman linearization, for which we give a convergence theorem. This method maps a system of nonlinear differential equations to an infinite-dimensional system of linear differential equations, which we discretize, truncate, and solve using the forward Euler method and the quantum linear system algorithm. We also provide a lower bound on the worst-case complexity of quantum algorithms for general quadratic differential equations, showing that the problem is intractable for [Formula: see text] Finally, we discuss potential applications, showing that the [Formula: see text] condition can be satisfied in realistic epidemiological models and giving numerical evidence that the method may describe a model of fluid dynamics even for larger values of R.
  • Editor: United States: National Academy of Sciences
  • Idioma: Inglês
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