  Twirling operations, which average a quantum state with respect to a unitary
subgroup, have become a frequently-employed tool in quantum information
processing. We investigate the efficient implementation of twirling operations
with minimal resources, without necessitating the ability to perform all
possible unitary operations on the quantum system of interest. We present a
general algebraic method allowing us to choose a set of - typically very few -
unitary operators which, when applied randomly and repeatedly, produce the
given twirling operation exponentially quickly. The method is applied to
twirling operations for bipartite quantum systems with respect to the unitary
group $U(d)\otimes U(d)$, an essential ingredient in entanglement distillation
protocols. In particular, we provide a complete classification of sets of
unitary operators capable of performing twirling on two qubits. Moreover, we
construct a generic set containing at most three unitary operators achieving
the twirling operation for a general two-qudit system.
