  We obtained analytical expressions considering a directed continuous-time
quantum walk on a directed infinite line using Bessel functions, expanding
previous results in the literature, for a general initial condition. We derive
the equation for the probability distribution, and show how to recover normal
and enhanced decay rates for the survival probability by adjusting the phase
factor of the direction of the graph. Our result shows that the mean and
standard deviation for a specific non-local initial condition does not depend
on the direction.
