Quantum algorithm for solving nonlinear algebraic equations

Nonlinear equations are generally challenging to solve. As analytical solutions typically do not exist, numerical methods have been developed to find their solutions. In this article, building upon the classical Newton method and the recent quantum singular value transformation framework, we give a quantum algorithm for solving a system of nonlinear algebraic equations, in which each equation is a homogeneous polynomial of an even degree. A detailed analysis is then carried out to reveal that our method achieves polylogarithmic time relative to the number of variables. Furthermore, the number of required qubits is logarithmic in the number of variables. In particular, we also show that our method can be generalized to deal with polynomials of various types. Some potential applications stem from various contexts, such as the Gross-Pitaevski equation, the Lotka-Volterra equations, and the intersection of algebraic varieties. Our work thus marks a significant step forward toward quantum advantage in nonlinear science, highlighting a promising avenue for future exploration.