Speedup in adiabatic evolution based quantum algorithms

In this context, we study three different strategies to improve the time complexity of the widely used adiabatic evolution algorithms when solving a particular class of quantum search problems where both the initial and final Hamiltonians are one-dimensional projector Hamiltonians on the corresponding ground state. After some simple analysis, we find the time complexity improvement is always accompanied by the increase of some other “complexities” that should be considered. But this just gives the implication that more feasibilities can be achieved in adiabatic evolution based quantum algorithms over the circuit model, even though the equivalence between the two has been shown. In addition, we also give a rough comparison between these different models for the speedup of the problem.