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Efficient quantum algorithm for dissipative nonlinear differential equations.

Jin-Peng Liu1,2,3, Herman Øie Kolden4,5, Hari K Krovi6

  • 1Joint Center for Quantum Information and Computer Science, University of Maryland, College Park, MD 20742.

Proceedings of the National Academy of Sciences of the United States of America
|August 27, 2021
PubMed
Summary
This summary is machine-generated.

We developed a quantum algorithm for solving nonlinear differential equations, offering an exponential speedup for certain problems. This breakthrough advances quantum computing applications in science and engineering.

Keywords:
Carleman linearizationNavier–Stokes equationnonlinear differential equationsplasma dynamicsquantum algorithm

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Area of Science:

  • Quantum computing
  • Computational mathematics
  • Applied physics

Background:

  • Nonlinear differential equations are crucial for modeling complex systems but are difficult to solve computationally.
  • Existing quantum algorithms excel at linear differential equations, but progress for nonlinear cases has been limited by quantum mechanics' linearity.
  • Dissipative quadratic ordinary differential equations present a significant challenge in computational science.

Purpose of the Study:

  • To develop a novel quantum algorithm for solving dissipative quadratic n-dimensional ordinary differential equations.
  • To achieve an exponential improvement in computational complexity compared to existing quantum algorithms.
  • To explore the applicability of the algorithm to real-world scientific models.

Main Methods:

  • The study employs the method of Carleman linearization to transform nonlinear equations into an infinite-dimensional linear system.
  • This linear system is then discretized, truncated, and solved using the forward Euler method and a quantum linear system algorithm.
  • A convergence theorem for Carleman linearization is provided, alongside a lower bound on the complexity for general quadratic differential equations.

Main Results:

  • A quantum algorithm with complexity [Formula: see text] is presented for dissipative quadratic ordinary differential equations under the condition [Formula: see text].
  • This represents an exponential improvement over previous quantum algorithms, whose complexity scales exponentially with evolution time T.
  • The algorithm demonstrates efficiency for driven equations, even with dissipation, and shows potential for models in epidemiology and fluid dynamics.

Conclusions:

  • The developed quantum algorithm offers a significant advancement for solving a class of nonlinear differential equations.
  • The [Formula: see text] condition is achievable in practical epidemiological models, suggesting broad applicability.
  • Numerical evidence indicates the method's potential for fluid dynamics modeling, even beyond the strict [Formula: see text] regime.