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

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