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