Synthesis of CNOT-Dihedral circuits with optimal number of two qubit gates

Abstract: 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.