  For a continuous-time quantum walk on a line the variance of the position
observable grows quadratically in time, whereas, for its classical counterpart
on the same graph, it exhibits a linear, diffusive, behaviour. A quantum walk,
thus, propagates at a rate which is linear in time, as compared to the square
root rate for a classical random walk. Indeed, it has been suggested that there
are graphs that can be traversed by a quantum walker exponentially faster than
by the classical random analogue. In this note we adopt the approach of
exploring the conditions to impose on a Markov process in order to emulate its
quantum counterpart: the central issue that emerges is the problem of taking
into account, in the numerical generation of each sample path, the causative
effect of the ensemble of trajectories to which it belongs. How to deal
numerically with this problem is shown in a paradigmatic example.
