Quantum Circuit Based on Grover’s Algorithm to Solve Exact Cover Problem

Abstract

In this paper, we propose a quantum circuit based on Grover’s algorithm to solve the exact cover problem (ECP), which is NP-hard. A novel controlled counter is designed and used to implement the ECP oracle to check if every element is covered by a set exactly once. If so, a feasible solution is found. The phase of the qubit state representing a feasible solution is flipped by calling the oracle, and the state’s amplitude is amplified through the diffusion operator o($\sqrt{2^{n}}$) times, where n is the number of sets given in the ECP. Thanks to Grover’s algorithm, the proposed quantum circuit achieves quadratic speedup compared with the classical algorithm that needs to call the oracle O($2^{n}$) times. To validate our approach, we execute the proposed quantum circuit using the IBM Quantum Lab service.