  Given a universal gate set on two qubits, it is well known that applying
random gates from the set to random pairs of qubits will eventually yield an
approximately Haar-distributed unitary. However, this requires exponential
time. We show that random circuits of only polynomial length will approximate
the first and second moments of the Haar distribution, thus forming approximate
1- and 2-designs. Previous constructions required longer circuits and worked
only for specific gate sets. As a corollary of our main result, we also improve
previous bounds on the convergence rate of random walks on the Clifford group.
