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

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 Oleft\(sqrt{t\|H\|log(1/ε\)}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\((η^5/2sqrt{tβ}α_V + η^3/2sqrt{t/β}N\)/εright) 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\(te^Ω(η\)/ε⁴). 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.