Journal Article

Efficient Quantum Algorithm for Nonlinear Reaction–Diffusion Equations and Energy Estimation

Liu, Jin-Peng, Wang, Jiasu, Low, Guang Hao, Jordan, Stephen, Fang, Di, An, Dong

2023

Overview

Nonlinear differential equations exhibit rich phenomena in many fields but are notoriously challenging to solve. Recently, Liu et al. (in: Proceedings of the National Academy of Sciences 118(35), 2021) demonstrated the first efficient quantum algorithm for dissipative quadratic differential equations under the condition R < 1 , where R measures the ratio of nonlinearity to dissipation using the ℓ 2 norm. Here we develop an efficient quantum algorithm based on Liu et al. (2021) for reaction–diffusion equations, a class of nonlinear partial differential equations (PDEs). To achieve this, we improve upon the Carleman linearization approach introduced in Liu et al. (2021) to obtain a faster convergence rate under the condition R D < 1 , where R D measures the ratio of nonlinearity to dissipation using the ℓ ∞ norm. Since R D is independent of the number of spatial grid points n while R increases with n , the criterion R D < 1 is significantly milder than R < 1 for high-dimensional systems and can stay convergent under grid refinement for approximating PDEs. As applications of our quantum algorithm we consider the Fisher-KPP and Allen-Cahn equations, which have interpretations in classical physics. In particular, we show how to estimate the mean square kinetic energy in the solution by postprocessing the quantum state that encodes it to extract derivative information.

• Publisher: Springer Berlin Heidelberg,Springer Nature B.V,Springer
• Subject: Algorithms / Classical and Quantum Gravitation / Complex Systems / Differential equations / Dissipation / Grid refinement (mathematics) / Kinetic energy / Mathematical and Computational Physics / Mathematical Physics / Nonlinear differential equations / Nonlinearity / Ordinary differential equations / Partial differential equations / Physics / Physics and Astronomy / Quantum computing / Quantum Physics / Reaction-diffusion equations / Relativity Theory / Theoretical