  Given an arbitrary single-qubit operation, an important task is to
efficiently decompose this operation into an (exact or approximate) sequence of
fault-tolerant quantum operations. We derive a depth-optimal canonical form for
single-qubit quantum circuits, and the corresponding rules for exactly reducing
an arbitrary single-qubit circuit to this canonical form. We focus on the
single-qubit universal H,T basis due to its role in fault-tolerant quantum
computing, and show how our formalism might be extended to other universal
bases. We then extend our canonical representation to the family of
Solovay-Kitaev decomposition algorithms, in order to find an
\epsilon-approximation to the single-qubit circuit in polylogarithmic time. For
a given single-qubit operation, we find significantly lower-depth
\epsilon-approximation circuits than previous state-of-the-art implementations.
In addition, the implementation of our algorithm requires significantly fewer
resources, in terms of computation memory, than previous approaches.
