Kavli Affiliate: Birgitta Whaley
| Summary:
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 $Oleft(sqrtlog(1/ε)right)$ 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 $widetildeOleft((η^5/2sqrttβα_V + η^3/2sqrtt/βN)/εright)$ 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 $O(te^Ω(η)/ε^4)$. 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.
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