  Standard error correction techniques only provide a quantum memory and need
extra gadgets to perform computation. Central to quantum algorithms are small
angle rotations, which can be fault-tolerantly implemented given a supply of an
unconventional species of magic state. We present a low-cost distillation
routine for preparing these small angle magic states. Our protocol builds on
the work of Duclos-Cianci and Poulin [Phys. Rev. A, 91, 042315 (2015)] by
compressing their circuit. Additionally, we present a method of diluting magic
states that reduces costs associated with very small angle rotations. We
quantify performance by the expected number of noisy magic states consumed per
rotation, and compare with other protocols. For modest size angles, our
protocols offer a factor 24 improvement over the best known gate synthesis
protocols and a factor 2 over the Duclos-Cianci and Poulin protocol. For very
small angle rotations, the dilution protocol dramatically reduces costs, giving
several orders magnitude improvement over competitors. There also exists an
intermediary regime of small, but not very small, angles where our approach
gives a marginal improvement over gate synthesis. We discuss how different
performance metrics may alter these conclusions.
