  In this paper for the first time, we construct quantum analogs starting from
classical stochastic processes, by replacing random which path decisions with
superpositions of all paths. This procedure typically leads to non-unitary
quantum evolution, where coherences are continuously generated and destroyed.
In spite of their transient nature, these coherences can change the scaling
behavior of classical observables. Using the zero temperature Glauber dynamics
in a linear Ising spin chain, we find quantum analogs with different domain
growth exponents. In some cases, this exponent is even smaller than for the
original classical process, which means that coherence can play an important
role to speed up the relaxation process.
