  In this paper, we propose efficient quantum algorithms for solving nonlinear
stochastic differential equations (SDE) via the associated Fokker-Planck
equation (FPE). We discretize the FPE in space and time using two well-known
numerical schemes, namely Chang-Cooper and implicit finite difference. We then
compute the solution of the resulting system of linear equations using the
quantum linear systems algorithm. We present detailed error and complexity
analyses for both these schemes and demonstrate that our proposed algorithms,
under certain conditions, provably compute the solution to the FPE within
prescribed $\epsilon$ error bounds with polynomial dependence on state
dimension $d$. Classical numerical methods scale exponentially with dimension,
thus, our approach, under the aforementioned conditions, provides an
\emph{exponential speed-up} over traditional approaches.
