发表机构
University of North Texas（北德克萨斯大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
针对最小顶点覆盖问题，提出并评估了三种基于Grover算法的编码与预言机策略，在量子比特数、电路深度和迭代次数间取得权衡，为硬件适配提供选择指导。
AI 中文摘要
最小顶点覆盖（MVC）问题是一个基础的NP难组合优化问题，在网络分析和资源分配中有着广泛应用。Grover算法为无结构搜索提供了查询复杂度的二次方缩减，但现有的基于Grover的MVC公式可能因昂贵的顶点计数电路和复杂的预言机结构而产生大量的量子资源开销。我们开发并评估了几种编码和预言机设计策略，以减少基于Grover的MVC搜索的量子比特数、电路深度和门复杂度。首先，我们构建了一个Dicke-Parallel公式，将搜索限制在固定基数的子集上，消除了显式的顶点计数，并配以并行边验证预言机以减少可行性检查开销。然后，我们开发了一个Edge-Counting公式，用对数大小的计数寄存器替代每条边的辅助存储，大幅减少了辅助量子比特需求。最后，我们提出了一种Edge-Centric编码，直接表示端点选择，并通过关联边的布尔运算推导出顶点选择状态，从而实现更深度高效的预言机构建。我们的资源分析揭示了量子比特宽度、电路深度、门数量和Grover迭代次数之间的互补权衡。在量子比特约束严格的情况下，Edge-Counting特别有吸引力，而Edge-Centric在图的边数适中且表示多重性足够时，可以减少电路深度和Grover迭代次数。当这种多重性有限或图密度导致基于边的搜索空间较大时，Dicke-Parallel提供了更稳健的选择。这些结果为根据硬件约束和图结构选择基于Grover的MVC公式提供了实用指导。
英文摘要
The Minimum Vertex Cover (MVC) problem is a fundamental NP-hard combinatorial optimization problem with applications in network analysis and resource allocation. Grover's algorithm provides a quadratic reduction in query complexity for unstructured search, but existing Grover-based MVC formulations can incur substantial quantum resource overhead due to costly vertex-counting circuits and complex oracle constructions. We develop and evaluate several encoding and oracle-design strategies for reducing the qubit count, circuit depth, and gate complexity of Grover-based MVC search. First, we construct a Dicke-Parallel formulation that restricts the search to fixed-cardinality subsets, eliminating explicit vertex counting, together with a parallel edge-verification oracle that reduces feasibility-checking overhead. We then develop an Edge-Counting formulation that replaces per-edge auxiliary storage with a logarithmic-size counting register, substantially reducing ancillary-qubit requirements. Finally, we propose an Edge-Centric encoding that represents endpoint selections directly and derives vertex-selection states through incident-edge Boolean operations, enabling more depth-efficient oracle construction. Our resource analysis reveals complementary trade-offs among qubit width, circuit depth, gate count, and Grover iteration count. Edge-Counting is particularly attractive under tight qubit constraints, while Edge-Centric can reduce both circuit depth and Grover iteration count when the graph has a moderate edge count and sufficient representation multiplicity. Dicke-Parallel provides a more robust choice when such multiplicity is limited or graph density makes the edge-based search space large. These results provide practical guidance for selecting Grover-based MVC formulations according to hardware constraints and graph structure.
Comments13 pages, 12 figures
