A quantum algorithm to solve nonlinear differential equations
  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. 
Sarah K. Leyton and Tobias J. Osborne