  Periodic deterministic bang-bang dynamical decoupling and the quantum Zeno
effect are known to emerge from the same physical mechanism. Both concepts are
based on cycles of strong and frequent kicks provoking a subdivision of the
Hilbert space into independent subspaces. However, previous unification results
do not capture the case of random bang-bang dynamical decoupling, which can be
advantageous to the deterministic case but has an inherently acyclic structure.
Here, we establish a correspondence between random dynamical decoupling and the
quantum Zeno effect by investigating the average over random decoupling
evolutions. This protocol is a manifestation of the quantum Zeno dynamics and
leads to a unitary bath evolution. By providing a framework that we call
equitability of system and bath, we show that the system dynamics under random
dynamical decoupling converges to a unitary with a decoupling error that
characteristically depends on the convergence speed of the Zeno limit. This
reveals a unification of the random dynamical decoupling and the quantum Zeno
effect.
