Quantum Solvable Nonlinear Differential Equations
Quantum computers have the potential to efficiently solve nonlinear ordinary differential equations (ODEs), which play a crucial role in various industries and scientific fields. However, it remains unclear which nonlinear ODEs, and under what assumptions, can achieve exponential speedup using a quantum computer. In this work, we introduce a class of nonlinear ODEs, called quantum solvable ODEs, that can be efficiently solved on quantum computers, where the efficiency is defined as solving the ODE with computational complexity of polylog(N) for a number N of variables in ODEs. Specifically, we employ Koopman-von-Neumann linearization to map nonliner ODEs to Hamiltonian dynamics and find conditions where the norm of the mapped Hamiltonian is preserved and the mapped Hamiltonian is sparse. This allows us to use the optimal Hamiltonian simulation technique for solving the quantum solvable ODEs with polylog(N) overhead. We further show that quantum solvable ODEs include a wide range of nonlinear ODEs, such as the extended short-range Kuramoto model. Since this is the first concrete example of solving nonlinear differential equations with exponential quantum speedup, these findings contribute significantly to the application of quantum computers in solving nonlinear problems.