  Unitary $2$-designs are random unitaries simulating up to the second order
statistical moments of the uniformly distributed random unitaries, often
referred to as Haar random unitaries. They are used in a wide variety of
theoretical and practical quantum information protocols, and also have been
used to model the dynamics in complex quantum many-body systems. Here, we show
that unitary $2$-designs can be approximately implemented by alternately
repeating random unitaries diagonal in the Pauli-$Z$ basis and that in the
Pauli-$X$ basis. We also provide a converse about the number of repetitions
needed to achieve unitary $2$-designs. These results imply that the process
after $\ell$ repetitions achieves a $\Theta(d^{-\ell})$-approximate unitary
$2$-design. Based on the construction, we further provide quantum circuits that
efficiently implement approximate unitary $2$-designs. Although a more
efficient implementation of unitary $2$-designs is known, our quantum circuit
has its own merit that it is divided into a constant number of commuting parts,
which enables us to apply all commuting gates simultaneously and leads to a
possible reduction of an actual execution time. We finally interpret the result
in terms of the dynamics generated by time-dependent Hamiltonians and provide
for the first time a random disordered time-dependent Hamiltonian that
generates a unitary $2$-design after switching interactions only a few times.
