  The adiabatic theorem has been recently used to design quantum algorithms of
a new kind, where the quantum computer evolves slowly enough so that it remains
near its instantaneous ground state which tends to the solution [Farhi et al.,
quant-ph/0001106]. We apply this time-dependent Hamiltonian approach to the
Grover's problem, i. e., searching a marked item in an unstructured database.
We find that, by adjusting the evolution rate of the Hamiltonian so as to keep
the evolution adiabatic on each infinitesimal time interval, the total running
time is of order $\sqrt{N}$, where $N$ is the number of items in the database.
We thus recover the advantage of Grover's standard algorithm as compared to a
classical search, scaling as $N$. This is in contrast with the constant-rate
adiabatic approach developed by Farhi et al., where the requirement of
adiabaticity is expressed only globally, resulting in a time of order $N$.
