QuantFPFlow: Quantum Amplitude Estimation for Fokker--Planck Policy Optimisation in Continuous Reinforcement Learning

We introduce QuantFPFlow, a reinforcement learning framework that integrates quantum amplitude estimation into the Fokker--Planck~(FP) formulation of stochastic policy optimisation. Classical continuous-space RL agents must estimate the FP partition function Z = int e-V(x)/D,dx at cost calO(1/varepsilon2); QuantFPFlow replaces this with a Grover-amplified amplitude estimator achieving calO(1/varepsilon) -- a provable quadratic speedup. While the full quantum acceleration requires fault-tolerant hardware, the quantum-inspired classical simulation demonstrated here already exhibits the calO(1/varepsilon) algorithmic structure. The estimated stationary distribution rhostar drives a theoretically grounded exploration bonus Raug = Renv + αlog(1/rhostar(s)). This bonus steers the agent toward globally optimal regions of multimodal reward landscapes while simultaneously constraining policy variance through FP diffusion matching. On a continuous-control task specifically designed to expose local-optima failure, QuantFPFlow achieves mean reward 1,295.7 ± 423.2 versus 1,284.0 ± 474.0 for Soft Actor-Critic~(SAC), while discovering the global optimum 10.4,% more frequently (33.9,% vs. 30.7,%). Policy entropy remains near H(π)≈ 6.5,nats throughout training, whereas SAC collapses to 1.5,nats, confirming that FP diffusion matching actively prevents premature convergence. Dimensionality experiments further show computational scaling of calO(d0.35) for QuantFPFlow versus calO(d0.76) for classical FP estimation.