Reducing the quantum-computing overhead with complex gate distillation

In leading fault-tolerant quantum-computing schemes, accurate transformations 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 that 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.