Quantum Adiabatic Evolution with a Special Kind of Interpolating Paths

We study a special kind of interpolating paths in quantum adiabatic search algorithms. With this special kind of adiabatic paths, notably we find that even when the parameter n within it tends to infinity, the adiabatic evolution would be a failure if the initial state is orthogonal to the final target state. But superficially, it seems that the minimum gap of the quantum system happening at the end of the computation would not affect the validity of the algorithm, since each ground state of the problem Hamiltonian encodes a solution to the problem. When the beginning state has a nonzero overlap with the final state, again if the parameter n within the special interpolating paths tends to infinity, it may give ones the counterintuitive impression that the adiabatic evolution could be considerably faster than the usual simple models of adiabatic evolution, even possible with constant time complexity. However, the fact is that as in the usual case, the quadratic speedup is the quantum algorithmic performance limit for which this kind of interpolating functions can provide for the adiabatic evolution. We also expose other easily made mistakes which may lead to draw the wrong conclusions about the validity of the adiabatic search algorithms.