  We study quantum dynamics of Grover's adiabatic search algorithm with the
equivalent two-level system. Its adiabatic and non-adiabatic evolutions are
visualized as trajectories of Bloch vectors on a Bloch sphere. We find the
change in the non-adiabatic transition probability from exponential decay for
short running time to inverse-square decay for long running time. The size
dependence of the critical running time is expressed in terms of Lambert $W$
function. The transitionless driving Hamiltonian is obtained to make a quantum
state follow the adiabatic path. We demonstrate that a constant Hamiltonian,
approximate to the exact time-dependent driving Hamiltonian, can alter the
non-adiabatic transition probability from the inverse square decay to the
inverse fourth power decay with running time. This may open up a new way of
reducing errors in adiabatic quantum computation.
