Efficient quantum algorithm for dissipative nonlinear differential equations

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.