  We investigate the ballistic spreading behavior of the one-dimensional
discrete time quantum walks whose time evolution is driven by any balanced
quantum coin. We obtain closed-form expressions for the long-time variance of
position of quantum walks starting from any initial qubit (spin-$1/2$ particle)
and position states following a delta-like (local), Gaussian and uniform
probability distributions. By averaging over all spin states, we find out that
the average variance of a quantum walk starting from a local state is
independent of the quantum coin, while from Gaussian and uniform states it
depends on the sum of relative phases between spin states given by the quantum
coin, being non-dispersive for a Fourier walk and large initial dispersion. We
also perform numerical simulations of the average probability distribution and
variance along the time to compare them with our analytical results.
