Quantgraph: A receding-horizon quantum graph solver

Dynamic programming is a cornerstone of graph-based optimization. While effective, it scales unfavorably with problem size. In this work, we present QuantGraph, a two-stage quantum-enhanced framework that casts local and global graph-optimization problems as quantum searches over discrete trajectory spaces. The solver is designed to operate efficiently by first finding a sequence of locally optimal transitions in the graph (local stage), without considering full trajectories. The accumulated cost of these transitions acts as a threshold that prunes the search space (up to 60% reduction for certain examples). The subsequent global stage, based on this threshold, refines the solution. Both stages utilize variants of the Grover-adaptive-search algorithm. To achieve scalability and robustness, we draw on principles from control theory and embed QuantGraph's global stage within a receding-horizon model-predictive-control scheme. This classical layer stabilizes and guides the quantum search, improving precision and reducing computational burden. In practice, the resulting closed-loop system exhibits robust behavior and lower overall complexity. Notably, for a fixed query budget, QuantGraph attains a 2x increase in control-discretization precision while still benefiting from Grover-search's inherent quadratic speedup compared to classical methods.