Quantum Algorithms for Stochastic Differential Equations: A Schr\"odingerisation Approach

Quantum computers are known for their potential to achieve up-to-exponential speedup compared to classical computers for certain problems. To exploit the advantages of quantum computers, we propose quantum algorithms for linear stochastic differential equations, utilizing the Schr\"odingerisation method for the corresponding approximate equation by treating the noise term as a (discrete-in-time) forcing term. Our algorithms are applicable to stochastic differential equations with both Gaussian noise and $\alpha$-stable L\'evy noise. The gate complexity of our algorithms exhibits an $\mathcal{O}(d\log(Nd))$ dependence on the dimensions $d$ and sample sizes $N$, where its corresponding classical counterpart requires nearly exponentially larger complexity in scenarios involving large sample sizes. In the Gaussian noise case, we show the strong convergence of first order in the mean square norm for the approximate equations. The algorithms are numerically verified for the Ornstein-Uhlenbeck processes, geometric Brownian motions, and one-dimensional L\'evy flights.