  Nonlinear equations are challenging to solve due to their inherently
nonlinear nature. As analytical solutions typically do not exist, numerical
methods have been developed to tackle their solutions. In this article, we give
a quantum algorithm for solving a system of nonlinear algebraic equations, in
which each equation is a multivariate polynomial of known coefficients.
Building upon the classical Newton method and some recent works on quantum
algorithm plus block encoding from the quantum singular value transformation,
we show how to invert the Jacobian matrix to execute Newton's iterative method
for solving nonlinear equations, where each contributing equation is a
homogeneous polynomial of an even degree. A detailed analysis are then carried
out to reveal that our method achieves polylogarithmic time in 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
modified with little effort to deal with polynomial of various types, thus
implying the generality of our approach. Some examples coming from physics and
algebraic geometry, such as Gross-Pitaevski equation, Lotka-Volterra equations,
and intersection of algebraic varieties, involving nonlinear partial
differential equations are provided to motivate the potential application, with
a description on how to extend our algorithm with even less effort in such a
scenario. Our work thus marks a further important step towards quantum
advantage in nonlinear science, enabled by the framework of quantum singular
value transformation.
