In this paper, we propose an efficient quantum algorithm for solving nonlinear stochastic differential equations (SDE) via the associated Fokker-Planck equation (FPE). We discretize FPE in space and time using the Chang-Cooper scheme, and compute the solution of the resulting system of linear equations using the quantum linear systems algorithm. The Chang-Cooper scheme is second order accurate and satisfies conservativeness and positivity of the solution. We present detailed error and complexity analyses that demonstrate that our proposed quantum scheme, which we call the Quantum Linear Systems Chang-Cooper Algorithm (QLSCCA), computes 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 provides an \emph{exponential speed-up} over traditional approaches.

翻译：本文提出了一种高效量子算法，用于通过关联的福克-普朗克方程求解非线性随机微分方程。我们采用Chang-Cooper格式在空间和时间上对福克-普朗克方程进行离散化，并利用量子线性系统算法计算所得线性方程组的解。该Chang-Cooper格式具有二阶精度，同时满足解的守恒性和正性。我们给出了详细的误差与复杂度分析，证明所提出的量子方案——称为量子线性系统Chang-Cooper算法——能够在规定的$\epsilon$误差范围内计算福克-普朗克方程的解，且其计算复杂度与状态维度$d$呈多项式关系。经典数值方法的复杂度随维度呈指数增长，因此，我们的方法相比传统方法实现了指数级加速。