Relationship between the quantum approximate optimization algorithmand diabatic time evolution for ground-state preparation

The quantum approximate optimization algorithm (QAOA) has emerged as an accurate and efficient way to solve many optimization problems including ground-state preparation. In this work, we examine the transverse field Ising model and compare the QAOA procedure to prepare the ground state to a diabatic state preparation via time evolution. In the QAOA, the Ising piece is the problem Hamiltonian, while the magnetic field is the mixing Hamiltonian. We find that the ratio of the amplitudes used in a QAOA track closely to the local adiabatic magnetic field used in diabatic time evolution. But, the optimal QAOA profile has an additional oscillatory behavior for both the problem and the mixing amplitudes. By introducing an overall time-dependent amplitude to the Hamiltonian of a diabatic state preparation, we mimic this behavior as well. Using adiabatic perturbation theory we can understand why the amplitude of the oscillations must be large to optimize the fidelity of the final approximate ground state.