Optimality of partial adiabatic search and its circuit model

In this paper, we first uncover a fact that a partial adiabatic quantum search with O(root N/M) time complexity is in fact optimal, in which N is the total number of elements in an unstructured database, and M(M >= 1) of them are the marked ones(one) (N >> M). We then discuss how to implement a partial adiabatic search algorithm on the quantum circuit model. From the implementing procedure on the circuit model, we can find out that the approximating steps needed are always in the same order of the time complexity of the adiabatic algorithm.