  We study noise-averaged observables for a system of exchange-coupled quantum
spins (qubits), each subject to a stochastic drive, by establishing mappings
onto stochastic models in the strong-noise limit. Averaging over noise yields
Lindbladian equations of motion; when these are subjected to a strong-noise
perturbative treatment, classical master equations are found to emerge. The
dynamics of noise averages of operators displays diffusive behaviour or
exponential relaxation, depending on whether the drive conserves one of the
spin components or not. In the latter case, the second moment of operators --
from which the average subsystem purity and out-of-time-order correlation
functions can be extracted -- is described by the Fredrickson-Andersen model,
originally introduced as a model of cooperative relaxation near the glass
transition. It is known that fluctuations of a ballistically propagating front
in the model are asymptotically Gaussian in one dimension. We extend this by
conjecturing, with strong numerical evidence, that in two dimensions the
long-time fluctuations are in the Kardar-Parisi-Zhang universality class,
complementing a similar observation in random unitary circuits.
