Quantum Algorithms for Finite-Horizon Markov Decision Processes

Abstract

In this work, we design quantum algorithms that are more efficient than classical algorithms to solve time-dependent and finite-horizon Markov Decision Processes (MDPs) in two distinct settings: (1) In the exact dynamics setting, where the agent has full knowledge of the environment’s dynamics (i.e., transition probabilities), we prove that our Quantum Value Iteration (QVI) algorithm QVI-1 achieves a quadratic speedup in the size of the action space $(A)$ compared with the classical value iteration algorithm for computing the optimal policy ($\pi^{\ast}$) and the optimal V-value function ($V_{0}^{\ast}$). Furthermore, our algorithm QVI-2 provides an additional speedup in the size of the state space $(S)$ when obtaining near-optimal policies and V-value functions. Both QVI-1 and QVI-2 achieve quantum query complexities that provably improve upon classical lower bounds, particularly in their dependences on $S$ and $A$. (2) In the generative model setting, where samples from the environment are accessible in quantum superposition, we prove that our algorithms QVI-3 and QVI-4 achieve improvements in sample complexity over the state-of-the-art (SOTA) classical algorithm in terms of $A$, estimation error $(\epsilon)$, and time horizon $(H)$. More importantly, we prove quantum lower bounds to show that QVI-3 and QVI-4 are asymptotically optimal, up to logarithmic factors, assuming a constant time horizon.