  In leading fault-tolerant quantum computing schemes, accurate transformation
are obtained by a two-stage process. In a first stage, a discrete, universal
set of fault-tolerant operations is obtained by error-correcting noisy
transformations and distilling resource states. In a second stage, arbitrary
transformations are synthesized to desired accuracy by combining elements of
this set into a circuit. Here, we present a scheme which merges these two
stages into a single one, directly distilling complex transformations. We find
that our scheme can reduce the total overhead to realize certain gates by up to
a few orders of magnitude. In contrast to other schemes, this efficient gate
synthesis does not require computationally intensive compilation algorithms,
and a straightforward generalization of our scheme circumvents compilation and
synthesis altogether.
