Problem
Research questions and friction points this paper is trying to address.
Minimum Vertex Cover
Grover's algorithm
quantum resource overhead
combinatorial optimization
oracle construction
Resource-Aware Grover Search for Minimum Vertex Cover
🤖 AI Summary
This study addresses the substantial quantum resource overhead incurred when solving the Minimum Vertex Cover (MVC) problem via Grover’s algorithm. To mitigate this, we propose three encoding and oracle design strategies: Dicke-Parallel, Edge-Counting, and Edge-Centric. By eliminating explicit vertex counting, introducing logarithmic-scale counting registers, and adopting an edge-centric encoding scheme, our approach significantly reduces ancillary qubit requirements while optimizing circuit depth and gate complexity. Furthermore, this work elucidates the resource trade-offs under diverse hardware constraints, offering practical guidance for selecting optimal quantum MVC formulations tailored to specific graph structures.
📝 Abstract
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.
Innovation
Methods, ideas, or system contributions that make the work stand out.
Grover search
Minimum Vertex Cover
quantum resource optimization
oracle design
qubit encoding
🔎 Similar Papers
No similar papers found.
