基于连续时间量子行走的最小顶点覆盖问题迭代量子算法
Iterative quantum algorithms for the minimum vertex cover problem based on continuous-time quantum walks
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中文总结 AI 辅助
该研究提出一种保约束量子-经典混合贪心框架，结合连续时间量子行走与顶点排名，在随机图和有界度图上实现比经典贪心算法更优的最小顶点覆盖求解效果。
中文摘要 AI 辅助
我们提出了一种适用于最小顶点覆盖问题的保约束量子-经典混合贪心框架，该框架可通过按位补运算直接扩展至最大独立集问题。该框架使用投影的Pauli-X项，其总和可保持可行子空间，并在该子空间内精确充当可行覆盖的分层图的邻接矩阵。此图是连通的，因此每个可行覆盖都通过一系列允许的单顶点翻转与包含所有顶点的构型相连。从该构型出发，对应的连续时间量子行走将振幅传播到包含逐步更小覆盖的层中。我们使用顶点的边际覆盖概率，或在将每个候选顶点固定到覆盖后获得的预期覆盖大小对顶点进行排名，并利用这些排名指导递归贪心约简。在多个随机图族上，通过使用独立校准集合固定行走时间，与对应的经典贪心基线相比，量子启发算法实现了更低的平均近似比，并以更大的比例最优求解实例。条件能量策略在测试实例上表现最佳，且在低深度Trotter化下保持接近精确连续时间极限的算法性能。对于有界度图，每个Trotter层的电路深度与系统规模无关，且该框架既不需要惩罚项也不需要变分训练。
英文摘要
We introduce a constraint-preserving hybrid quantum-classical greedy framework for the minimum vertex cover problem, which extends directly to maximum independent set by bitwise complementation. The framework uses projected Pauli-X terms whose sum preserves the feasible subspace and acts within it exactly as the adjacency matrix of a layered graph of feasible covers. This graph is connected, so every feasible cover is linked to the configuration containing all vertices by a sequence of allowed single-vertex flips. Starting from this configuration, the corresponding continuous-time quantum walk propagates amplitude into layers containing progressively smaller covers. We rank vertices using either their marginal cover probabilities or the expected cover size obtained after fixing each candidate vertex in the cover, and use these rankings to guide recursive greedy reductions. Across several random-graph families, with walk times fixed using independent calibration ensembles, the quantum-informed algorithms achieve lower mean approximation ratios and solve a larger fraction of instances optimally than their corresponding classical greedy baselines. The conditioned-energy strategy performs best on the tested instances and retains algorithmic performance close to the exact continuous-time limit under low-depth Trotterisation. For bounded-degree graphs, each Trotter layer has circuit depth independent of system size, and the framework requires neither penalty terms nor variational training.
