  We generalize the quantum random walk protocol for a particle in a
one-dimensional chain, by using several types of biased quantum coins, arranged
in aperiodic sequences, in a manner that leads to a rich variety of possible
wave function evolutions. Quasiperiodic sequences, following the Fibonacci
prescription, are of particular interest, leading to a sub-ballistic
wavefunction spreading. In contrast, random sequences leads to diffusive
spreading, similar to the classical random walk behaviour. We also describe how
to experimentally implement these aperiodic sequences.
