Quantum algorithm for solving McKean-Vlasov stochastic differential equations

TL;DR

This paper introduces the first quantum algorithm using Quantum Monte Carlo integration to efficiently solve McKean-Vlasov stochastic differential equations, achieving a quadratic speed-up over classical methods.

Contribution

It develops a novel quantum algorithm combining QMCI with high-order discretization for MVSDEs, enabling faster expectation computations in complex stochastic models.

Findings

Quantum algorithm estimates expectations with accuracy ε using fewer queries.

Demonstrates quadratic speed-up over classical particle methods.

Numerical tests confirm expected accuracy and complexity behavior.

Abstract

Quantum Monte Carlo integration, 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 $\epsilon$, 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 a numerical demonstration of our quantum algorithm applied to an example of MVSDEs, with some parts emulated classically, and observe that the accuracy and complexity behave as expected.