Kavli Affiliate: Birgitta Whaley
| Summary:
The Fokker-Planck equation models rare events across sciences, but bluedirect 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 $Oleft(sqrtt\right)$ 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 $widetildeOleft((η^5/2sqrttβα_V + η^3/2sqrtt/βN)/εright)$ quantum gates, where $α_V = max_r|V'(r)/r|$. 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 $O(tη^2 e^Ω(η)/ε^4)$, yielding an exponential improvement in $η$, 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.
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