  The majorization theory has been applied to analyze the mathematical
structure of quantum algorithms. An empirical conclusion by numerical
simulations obtained in the previous literature indicates that step-by-step
majorization seems to appear universally in quantum adiabatic algorithms. In
this paper, a rigorous analysis of the majorization arrow in a special class of
quantum adiabatic algorithms is carried out. In particular, we prove that for
any adiabatic algorithm of this class, step-by-step majorization of the ground
state holds exactly. For the actual state, we show that step-by-step
majorization holds approximately, and furthermore that the longer the running
time of the algorithm, the better the approximation.
