Diagonal Unitaries and Stabilizer Rank

In the notes on the Stabilizer Rank, we discussed the possibility og using the Stabilizer rank to study controlled diagonal rotations and their Stabilizer Rank. A related quesiton would be to study compiling diagonal unitaries into some combination of diagonal rotations.

In a 2014 paper on synthesising diagonal unitaries in the Clifford+T basis, Welch et al. note that a multi-controlled rotation can be decomposed into a Clifford+T circuit where the T-count relates diretly to the number of controls, and is directly proportional to it. (Note: Read Phys. Rev. A. 87, 032332 (2013) and Phys. Rev. A. 52, 3457, (1995)).

This suggests the T count of a single controlled Z is related singly to the T-count of the rotation. This is no good for exact stabilizer rank as small rotations typically cannot be exactly synthesised.