  We investigate decoupling, one of the most important primitives in quantum
Shannon theory, by replacing the uniformly distributed random unitaries
commonly used to achieve the protocol, with repeated applications of random
unitaries diagonal in the Pauli-$Z$ and -$X$ bases. This strategy was recently
shown to achieve an approximate unitary $2$-design after a number of
repetitions of the process, which implies that the strategy gradually achieves
decoupling. Here, we prove that even fewer repetitions of the process achieve
decoupling at the same rate as that with the uniform ones, showing that rather
imprecise approximations of unitary $2$-designs are sufficient for decoupling.
We also briefly discuss efficient implementations of them and implications of
our decoupling theorem to coherent state merging and relative thermalisation.
