  We present an extension of Adiabatic Quantum Computing (AQC) algorithm for
the unstructured search to the case when the number of marked items is unknown.
The algorithm maintains the optimal Grover speedup and includes a small
counting subroutine.
  Our other results include a lower bound on the amount of time needed to
perform a general Hamiltonian-based quantum search, a lower bound on the
evolution time needed to perform a search that is valid in the presence of
control error and a generic upper bound on the minimum eigenvalue gap for
evolutions.
  In particular, we demonstrate that quantum speedup for the unstructured
search using AQC type algorithms may only be achieved under very rigid control
precision requirements.
