Quantum Algorithms for Finite-horizon Markov Decision Processes

TL;DR

This paper introduces quantum algorithms that significantly improve the efficiency of solving finite-horizon Markov Decision Processes, achieving quadratic speedups and optimal sample complexities in various settings.

Contribution

The paper presents novel quantum algorithms for finite-horizon MDPs that outperform classical methods in both exact dynamics and generative model settings, with proven speedups and optimal bounds.

Findings

Quadratic speedup in action space for classical value iteration

Additional speedup in state space for near-optimal policies

Quantum algorithms achieve asymptotic optimality in sample complexity

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 $\textbf{Quantum Value Iteration (QVI)}$ algorithm $\textbf{QVI-1}$ achieves a quadratic speedup in the size of the action space $(A)$ compared with the classical value iteration algorithm for computing the optimal policy ($\pi^{*}$) and the optimal V-value function ($V_{0}^{*}$). Furthermore, our algorithm $\textbf{QVI-2}$ provides an additional speedup in the size of the state space $(S)$ when obtaining near-optimal policies and V-value functions. Both $\textbf{QVI-1}$ and $\textbf{QVI-2}$ 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 $\textbf{QVI-3}$ and $\textbf{QVI-4}$ 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 $\textbf{QVI-3}$ and $\textbf{QVI-4}$ are asymptotically optimal, up to logarithmic factors, assuming a constant time horizon.