Applying the quantum approximate optimization algorithm to the minimum vertex cover problem

Abstract

The minimum vertex cover problem belongs to a NP- complete problem, which is difficult to obtain the near-optimal solution in the polynomial time range using classical algorithms. In this paper, a quantum circuit solution scheme based on the quantum approximate optimization algorithm is presented for the minimum vertex cover problem. Firstly, the quantum Ising model and Hamiltonian of the problem are obtained based on the Ising model corresponding to the problem, which is quantized by the rotation operator and Pauli operator. Secondly, the parametric unitary transformation with the initial Hamiltonian and the problem Hamiltonian as the generator is obtained respectively. Through the alternating evolution of two parametric unitary transformations, the final quantum state and the problem Hamiltonian expectation are derived. In the process of evolution, the parameters in the parametric unitary transformations which are optimized by the classical processor can adjust the problem Hamiltonian expectation, so as to improve the probability of the problem solution. Then, the initial state of the algorithm and the quantum logic gate corresponding to the parametric unitary transformation are derived to generate the quantum circuit which can be implemented on the quantum computer. Simulation results show that the scheme can obtain the problem solution with high probability in polynomial time, realizes exponential acceleration, and has certain feasibility, effectiveness and innovation.