Quantum algorithm for solving McKean-Vlasov stochastic differential equations

Quantum Monte Carlo integration (QMCI), a quantum algorithm for calculating expectations that provides a quadratic speed-up compared to its classical counterpart, is now attracting increasing interest in the context of its industrial and scientific applications. In this paper, we propose the first application of QMCI to solving McKean-Vlasov stochastic differential equations (MVSDEs), a nonlinear class of SDEs whose drift and diffusion coefficients depend on the law $\mu _{t}$ of the solution $X_{t}$—appearing in fields such as finance and fluid mechanics. We focus on the problem setting where the coefficients depend on $\mu _{t}$ through expectations of some functions $\mathbb{E}[\varphi _{k}(X_{t})]$, and the goal is to compute the expectation of a function $\mathbb{E}[\phi (X_{T})]$ at a terminal time T. We devise a quantum algorithm that leverages QMCI to compute these expectations, combined with a high-order time discretization method for SDEs and extrapolation of the expectations in time. The proposed algorithm estimates $\mathbb{E}[\phi (X_{T})]$ with accuracy ϵ, making $O(1/\epsilon ^{1+2/p})$ queries to the quantum circuit for time evolution over one step, where $p\in (1,2]$ is the weak order of the SDE discretization method. This demonstrates the speed-up over the well-known classical algorithm called the particle method with complexity of $O(1/\epsilon ^{3})$. We conduct numerical demonstrations of our quantum algorithm applied to examples of MVSDEs, with some parts emulated classically, and observe that the accuracy and complexity behave as expected.