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 $O\left(\sqrt{t\Vert H\Vert \log (1/\hat{I}\mu )}\right)$ queries to its block encoding. Crucially, we pair this with {a} novel technique to directly estimate matrix elements without exponential decay. For $\hat{I}\cdot$ pairwise interacting particles discretized with $N$ plane waves per degree of freedom, we estimate reactive flux to error $\hat{I}\mu$ using $\tilde{O}\left((\hat{I}{\cdot }^{5/2}\sqrt{t{\hat{I}}^2}\hat{I}{\pm }_V+\hat{I}{\cdot }^{3/2}\sqrt{t/{\hat{I}}^2}N)/\hat{I}\mu \right)$ quantum gates, where $\hat{I}{\pm }_V={\max }_r|V^{\prime }(r)/r|$. 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 $O(t\hat{I}{\cdot }^2{e}^{\hat{I}\text{c◯}(\hat{I}\cdot )}/\hat{I}{\mu }^4)$, yielding an exponential improvement in $\hat{I}\cdot$, a quartic speedup in $\hat{I}\mu$, 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.

Updated version, includes detailed resource analysis, planewave convergence results, and a construction for non-polynomial potentials. 79 pages, 14 figures