  We study nonequilibrium dynamics of the quantum Ising chain at zero
temperature when the transverse field is varied stochastically. In the
equivalent fermion representation, the equation of motion of Majorana operators
is derived in the form of a one-dimensional, continuous-time quantum random
walk with stochastic, time-dependent transition amplitudes. This type of
external noise gives rise to decoherence in the associated quantum walk and the
semiclassical wave-packet generally has a diffusive behavior. As a consequence,
in the quantum Ising chain, the average entanglement entropy grows in time as
$t^{1/2}$ and the logarithmic average magnetization decays in the same form. In
the case of a dichotomous noise, when the transverse-field is changed in
discrete time-steps, $\tau$, there can be excitation modes, for which coherence
is maintained, provided their energy satisfies $\epsilon_k \tau\approx n\pi$
with a positive integer $n$. If the dispersion of $\epsilon_k$ is quadratic,
the long-time behavior of the entanglement entropy and the logarithmic
magnetization is dominated by these ballistically traveling coherent modes and
both will have a $t^{3/4}$ time-dependence.
