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

TL;DR

This paper introduces a quantum algorithm that efficiently computes reaction rates in high-dimensional systems modeled by the Fokker-Planck equation, achieving exponential speedups over classical methods.

Contribution

The authors develop a Gaussian-LCHS technique for representing non-unitary propagators and a novel method for direct matrix element estimation, enabling sublinear-time quantum simulation of reaction rates.

Findings

Achieves a polynomial quantum speedup in error and time over classical bounds.

Provides a quantum algorithm with exponential advantage in particle number η.

Demonstrates a rigorous route toward quantum advantage in high-dimensional dissipative dynamics.

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 $O\left(\sqrt{t\|H\|\log(1/\epsilon)}\right)$ 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 $\epsilon$ using $\widetilde{O}\left((\eta^{5/2}\sqrt{t\beta}\alpha_V + \eta^{3/2}\sqrt{t/\beta}N)/\epsilon\right)$ quantum gates, where $\alpha_V = \max_{r}|V'(r)/r|$. For non-convex potentials, the {sharpest classical} worst-case analytical bounds to simulate the related overdamped Langevin {equation} scale as $O(te^{\Omega(\eta)}/\epsilon^4)$. This {implies} an exponential separation in particle number $\eta$, a quartic speedup in $\epsilon$, 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.