Finding Solutions of the Navier-Stokes Equations through Quantum Computing-Recent Progress, a Generalization, and Next Steps Forward

Efficient simulation of a quantum system's dynamics is expected to be an important application area for quantum computers as existing classical computers cannot do this. However, quantum systems are not unique in being hard to simulate. For example, classical nonlinear continuum systems and fields are governed by nonlinear partial differential equations whose solution is also hard for classical computers. Solving such equations is essential for many economically important industries/applications such as the aerospace industry, weather-forecasting, fiber-optics communication, and plasma magneto-hydrodynamics. This raises the question: can a quantum computer speed-up solving these equations? In this Review, a new quantum algorithm is described for solving nonlinear partial differential equations for which the answer is yes. First, a new quantum algorithm is discussed for solving the Navier-Stokes nonlinear partial differential equations which govern the flow of a viscous fluid. Its construction, verification, and computational cost are described, and it is shown that a significant quantum speed-up is possible. Its generalization to a quantum algorithm for solving nonlinear partial differential equations is described. The Review closes with a discussion of next steps forward. These new quantum algorithms open up a large new application area for quantum computing with substantial economic impact, including the trillion-dollar aerospace industry.

Automated summary

This article introduces a new quantum algorithm for solving nonlinear partial differential equations that cannot be efficiently simulated by classical computers. Specifically, it focuses on the Navier-Stokes equations governing viscous fluid flow. The construction and verification of this algorithm show a significant quantum speed-up, opening up new opportunities for quantum computing applications in the economy.