Quantum algorithm for minimum dominating set problem with circuit design

Abstract

Using quantum algorithms to solve various problems has attracted widespread attention with the development of quantum computing. Researchers are particularly interested in using the acceleration properties of quantum algorithms to solve NP-complete problems. This paper focuses on the well-known NP-complete problem of finding the minimum dominating set in undirected graphs. To expedite the search process, a quantum algorithm employing Grover’s search is proposed. However, a challenge arises from the unknown number of solutions for the minimum dominating set, rendering direct usage of original Grover’s search impossible. Thus, a swap test method is introduced to ascertain the number of iterations required. The oracle, diffusion operators, and swap test are designed with achievable quantum gates. The query complexity is O(1.414 n ) and the space complexity is O(n). To validate the proposed approach, qiskit software package is employed to simulate the quantum circuit, yielding the anticipated results.