# On the spreading of quantum walks starting from local and delocalized states. (arXiv:1706.06257v2 [quant-ph] UPDATED)

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.