Quantum algorithms for viscosity solutions to nonlinear Hamilton-Jacobi equations based on an entropy penalization...

Area of Science:

• Applied Mathematics
• Quantum Computing
• Computational Science

Background:

• Nonlinear Hamilton-Jacobi equations are crucial in diverse fields like optimal control, mean-field games, and machine learning.
• Efficiently solving these equations, especially for convex Hamiltonians, presents significant computational challenges.

Purpose of the Study:

• To develop a novel framework for the efficient extraction of viscosity solutions for nonlinear Hamilton-Jacobi equations with convex Hamiltonians.
• To enable quantum simulations of these complex dynamics for broader applicability.

Main Methods:

• An entropy penalization method is employed, generalizing the Cole-Hopf transform to convex Hamiltonians.
• Viscous Hamilton-Jacobi dynamics are approximated by discrete-time linear dynamics, which in turn approximate a linear heat-like parabolic equation.
• This approach extends to continuous-time dynamics and is suitable for quantum simulation.

Main Results:

• The framework provides a method for approximating viscosity solutions of nonlinear Hamilton-Jacobi equations using linear dynamics.
• The method's validity extends to arbitrary nonlinearities with convex Hamiltonians and for arbitrarily long times.
• Quantum algorithms (analog and digital) are presented for extracting key properties of the viscosity solution without full state reconstruction.

Conclusions:

• The developed framework offers an efficient and robust approach for solving nonlinear Hamilton-Jacobi equations using quantum computation.
• This work overcomes a key obstacle in quantum algorithms for nonlinear partial differential equations, paving the way for advanced simulations.
• The method's applicability spans various scientific and engineering domains requiring solutions to these types of equations.