A Sublinear-Time Quantum Algorithm for High-Dimensional Reaction Rates

Abstract

The Fokker-Planck equation models rare events across sciences, but its high-dimensional nature challenges classical computers. Quantum algorithms for such non-unitary dynamics often suffer from exponential {decay in} success probability. We introduce a quantum algorithm that overcomes this for computing reaction rates. Using a sum-of-squares representation, we develop a Gaussian linear combination of Hamiltonian simulations (Gaussian-LCHS) to represent the non-unitary propagator with $O\left(\sqrt{t\|H\|\log(1/ε)}\right)$ queries to its block encoding. Crucially, we pair this with {a} novel technique to directly estimate matrix elements without exponential decay. For $η$ pairwise interacting particles discretized with $N$ plane waves per degree of freedom, we estimate reactive flux to error $ε$ using $\widetilde{O}\left((η^{5/2}\sqrt{tβ}α_V + η^{3/2}\sqrt{t/β}N)/ε\right)$ quantum gates, where $α_V = \max_{r}|V'(r)/r|$. For non-convex potentials, the {sharpest classical} worst-case analytical bounds to simulate the related overdamped Langevin {equation} scale as $O(te^{Ω(η)}/ε^4)$. This {implies} an exponential separation in particle number $η$, a quartic speedup in $ε$, and quadratic speedup in $t$. While specialized classical heuristics may outperform these bounds in practice, this demonstrates a rigorous route toward quantum advantage for high-dimensional dissipative dynamics.