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.

翻译：本文提出通过关联的福克-普朗克方程（FPE）求解非线性随机微分方程（SDE）的高效量子算法。我们采用两种经典数值方案——Chang-Cooper格式和隐式有限差分法——对FPE进行空间和时间离散化，进而利用量子线性系统算法求解所得线性方程组。我们针对这两种方案给出了详细的误差与复杂度分析，并证明在特定条件下，所提出算法能在预设的$\epsilon$误差范围内以与状态维度$d$呈多项式依赖的复杂度精确求解FPE。经典数值方法随维度呈指数级增长，因此，在所述条件下，我们的方法相对于传统方法实现了\textit{指数级加速}。