$r$. This is done by extending each of the four diagonal components to a diagonal matrix in ${\mathbb C}^{(r/2) \times (r/2)}$ while conserving the structure. For instance, the matrix $$T= \sqrt{3}\left(
\begin{array}{ccc|ccc} \alpha &&&\beta\\ &\alpha &&&-{\overline \beta}
\\&&\alpha &&& \beta \\ \hline &&&&\\ [-0.35cm] {\overline \beta} &&&
-\alpha \\ & \beta &&& \alpha \\ && {\overline \beta} &&& -\alpha
\end{array}\right)$$ is extended to the matrix $$T_8=\sqrt{3}\left( \begin{array}{cccc|cccc}
\alpha &&&&\beta\\ &\alpha &&&&-{\overline \beta} \\&&\alpha &&&&
\beta \\ &&& \alpha &&&& -{\overline \beta} 
\\ \hline &&&&\\ [-0.35cm] {\overline \beta} &&&& -\alpha \\ & \beta
&&&& \alpha \\ && {\overline \beta} &&&& -\alpha\\ &&& \beta &&&& \alpha
\end{array}\right).$$ Furthermore, in the factorization $T=Q^\dagger {\tilde M} (I_2 \otimes
F_m^\dagger)$ the matrix $Q$ is replaced by a permutation matrix $Q_r
\in {\mathbb C}^{r \times r}$ that fixes the first row and maps the $(r/2+2)$nd row to the second row. In qubit notation, this permutation matrix can be described by ${|0 \ldots 0\rangle} \mapsto {|0 \ldots 0\rangle}$ and ${|10 \ldots 01\rangle} \mapsto {|00 \ldots 01\rangle}$. This permutation can be implemented by an XOR-gate on the first qubit controlled by the last qubit. Other implementations that satisfy the two constraints are also possible. The Fourier transform $F_m$ 