  Since Grover's seminal work which provides a way to speed up combinatorial
search, quantum search has been studied in great detail. We propose a new
method for designing quantum search algorithms for finding a marked element in
the state space of a graph. The algorithm is based on a local diabatic
evolution of the Hamiltonian associated with the graph. The main new idea is to
apply some techniques such as Krylov bspace projection methods, Lanczos
algorithm and spectral distribution methods. Indeed, using these techniques
together with the second-order perturbation theory, we give a systematic method
for calculating the approximate search time at which the marked state can be
reached. That is, for any undirected regular connected graph which is
considered as the state space of the database, the introduced algorithm
provides a systematic and programmable way for evaluation of the search time,
in terms of the corresponding graph polynomials.
