  In this paper we describe a quantum algorithm to solve sparse systems of
nonlinear differential equations whose nonlinear terms are polynomials. The
algorithm is nondeterministic and its expected resource requirements are
polylogarithmic in the number of variables and exponential in the integration
time. The best classical algorithm runs in a time scaling linearly with the
number of variables, so this provides an exponential improvement. The algorithm
is built on two subroutines: (i) a quantum algorithm to implement a nonlinear
transformation of the probability amplitudes of an unknown quantum state; and
(ii) a quantum implementation of Euler's method.
