Divergence-free algorithms for solving nonlinear differential equations on quantum computers

Abstract: From weather to neural networks, modeling is not only useful for understanding various phenomena, but also has a wide range of potential applications. Although nonlinear differential equations are extremely useful tools in modeling, their solutions are difficult to obtain. Based on the expectation of quantum transcendence, quantum algorithms for efficiently solving nonlinear differential equations continue to be developed. However, even the latest promising algorithms have been pointed out to have an evolution time limit. This limit is the theoretically predestined divergence of solutions. We propose algorithms of divergence-free simulation for nonlinear differential equations in quantum computers. For Hamiltonian simulations, a pivot state $\bf{s}$ in the neighborhood of state $\bf{x}$ is introduced. Divergence of the solutions is prevented by moving $\bf{s}$ to a neighborhood of $\bf{x}$ whenever $\bf{x}$ leaves the neighborhood of $\bf{s}$. Since updating $\bf{s}$ is directly related to computational cost, to minimize the number of updates, the nonlinear differential equations are approximated by nonlinear polynomials around $\bf{s}$, which are then Carleman linearized. Hamiltonian simulations of nonlinear differential equations based on several representative models are performed to show that the proposed method breaks through the theoretical evolution time limit. The solution of nonlinear differential equations free from evolution time constraints opens the door to practical applications of quantum computers.