  In this note we present explicit canonical forms for all the elements in the
two-qubit CNOT-Dihedral group, with minimal numbers of controlled-S (CS) and
controlled-X (CX) gates, using the generating set of quantum gates [X, T, CX,
CS]. We provide an algorithm to successively construct the n-qubit
CNOT-Dihedral group, asserting an optimal number of controlled-X (CX) gates.
These results are needed to estimate gate errors via non-Clifford randomized
benchmarking and may have further applications to circuit optimization over
fault-tolerant gate sets.
