  We analyze in detail the discrete--time quantum walk on the line by
separating the quantum evolution equation into Markovian and interference
terms. As a result of this separation, it is possible to show analytically that
the quadratic increase in the variance of the quantum walker's position with
time is a direct consequence of the coherence of the quantum evolution. If the
evolution is decoherent, as in the classical case, the variance is shown to
increase linearly with time, as expected. Furthermore we show that this system
has an evolution operator analogous to that of a resonant quantum kicked rotor.
As this rotator may be described through a quantum computational algorithm, one
may employ this algorithm to describe the time evolution of the quantum walker.
