Provable Quantum Speedups for Reaction-Rate Estimation in High-Dimensional Fokker-Planck Dynamics

Abstract

The Fokker-Planck equation models rare events across sciences, but blue{direct solution of the PDE is intractable for classical computers due to } its high-dimensional nature. Classical stochastic methods circumvent this curse-of-dimensionality, and serve as the de facto standard for practicing computational scientists. Quantum algorithms for such non-unitary dynamics often suffer from exponential decay in success probability. We introduce a quantum algorithm that overcomes this bottleneck for estimating reaction rates {and dynamical correlation functions more generally}. Using a sum-of-squares representation, we develop a Gaussian linear combination of Hamiltonian simulations (Gaussian-LCHS) to represent the non-unitary propagator with $O\left(\sqrt{t\|H\|\log(1/\epsilon)}\right)$ queries to its block encoding. Crucially, we pair this with {a} novel technique to directly estimate matrix elements without exponential decay. For $\eta$ pairwise interacting particles discretized with $N$ plane waves per degree of freedom, we estimate reactive flux to error $\epsilon$ using $\widetilde{O}\left((\eta^{5/2}\sqrt{t\beta}\alpha_V + \eta^{3/2}\sqrt{t/\beta}N)/\epsilon\right)$ quantum gates, where $\alpha_V = \max_{r}|V'(r)/r|$. We further prove that under comparable worst-case analytical guarantees, the sharpest classical bounds for estimating reaction rates via simulation of the associated overdamped Langevin dynamics scale as $O(t\eta^2 e^{\Omega(\eta)}/\epsilon^4)$, yielding an exponential improvement in $\eta$, a quartic speedup in $\epsilon$, and quadratic speedup in the time horizon $t$. While classical algorithms may outperform these bounds in practice, this work demonstrates a rigorous route toward quantum advantage for high-dimensional dissipative dynamics.