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.

翻译：动态规划是图优化问题的基石方法。尽管有效，但其计算复杂度随问题规模增长而急剧上升。本研究提出QuantGraph，一种两阶段量子增强框架，将局部与全局图优化问题转化为离散轨迹空间上的量子搜索。该求解器设计为高效运行：首先在图中寻找一系列局部最优转移（局部阶段），而无需考虑完整轨迹。这些转移的累积成本作为阈值，对搜索空间进行剪枝（在特定示例中可实现高达60%的缩减）。随后的全局阶段基于此阈值对解进行优化。两个阶段均采用Grover自适应搜索算法的变体。为实现可扩展性与鲁棒性，我们借鉴控制理论原理，将QuantGraph的全局阶段嵌入滚动时域模型预测控制框架中。这一经典控制层能够稳定并引导量子搜索，提高精度并降低计算负担。实际应用中，由此形成的闭环系统展现出鲁棒行为与更低的整体复杂度。值得注意的是，在固定查询预算下，QuantGraph在控制离散化精度上实现了2倍提升，同时相较于经典方法仍能保持Grover搜索固有的二次加速优势。