R1～R5 のユニタリ変換 Tong's QHE p.130あたり

page 130 of http://www.damtp.cam.ac.uk/user/tong/qhe/qhe.pdf

$R_3 = \frac{1}{\sqrt{2}} \left ( 1 -i +2i \Psi_2^\dagger \Psi_2 \right )$
$R_5 = \frac{1}{\sqrt{2}} \left ( 1 -i +2i \Psi_3^\dagger \Psi_3 \right )$

$R_1|0\rangle = \frac{1}{\sqrt{2}} \left ( 1- i \right ) |0\rangle = e^{-i\pi/4}|0\rangle$,　 $R_1|1\rangle = \frac{1}{\sqrt{2}} \left ( 1+ i \right ) |1\rangle = e^{i\pi/4}|0\rangle$,
$R_1|2\rangle = \frac{1}{\sqrt{2}} \left ( 1- i \right ) |2\rangle = e^{-i\pi/4}|0\rangle$,　 $R_1|3\rangle = \frac{1}{\sqrt{2}} \left ( 1+ i \right ) |3\rangle = e^{i\pi/4}|0\rangle$,
$R_1|4\rangle = \frac{1}{\sqrt{2}} \left ( 1- i \right ) |4\rangle = e^{-i\pi/4}|0\rangle $, 　$R_1|5\rangle = \frac{1}{\sqrt{2}} \left ( 1+ i \right ) |5\rangle = e^{i\pi/4}|0\rangle$,
$R_1|6\rangle = \frac{1}{\sqrt{2}} \left ( 1- i \right ) |6\rangle = e^{-i\pi/4}|0\rangle$, 　 $R_1|7\rangle = \frac{1}{\sqrt{2}} \left ( 1+ i \right ) |7\rangle = e^{i\pi/4}|0\rangle$, 　

$R_1 = \frac{1}{\sqrt{2}} \left ( 1 -i +2i \Psi_1^\dagger \Psi_1 \right )$

　$ |2 \rangle=\Psi_2^\dagger |0 \rangle$, 　$|3 \rangle = \Psi_2^\dagger |1 \rangle = \Psi_2^\dagger \Psi_1^\dagger |0 \rangle$, 　$|4 \rangle = \Psi_3^\dagger |0 \rangle$, 　$|5 \rangle=\Psi_3^\dagger \Psi_1^\dagger |0 \rangle =\Psi_3^\dagger |1 \rangle$, 　$|6 \rangle = \Psi_3^\dagger |2 \rangle = \Psi_3^\dagger \Psi_2^\dagger |0 \rangle$, 　$|7 \rangle = \Psi_3^\dagger |3 \rangle = \Psi_3^\dagger \Psi_2^\dagger |1 \rangle = \Psi_3^\dagger \Psi_2^\dagger \Psi_1^\dagger |0 \rangle $

$UR_1U^\dagger = \left(\begin{array}{cccccccc}
1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 \\
0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 \\
0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 \\
0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 \end{array}\right)
\left(\begin{array}{cccccccc}
e^{-i\pi/4} & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\
0 & e^{i\pi/4} & 0 & 0 & 0 & 0 & 0 & 0 \\
0 & 0 & e^{-i\pi/4} & 0 & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & e^{i\pi/4} & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & e^{-i\pi/4} & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & 0 & e^{i\pi/4} & 0 & 0 \\
0 & 0 & 0 & 0 & 0 & 0 & e^{-i\pi/4} & 0 \\
0 & 0 & 0 & 0 & 0 & 0 & 0 & e^{i\pi/4} \end{array}\right)
\left(\begin{array}{cccccccc}
1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 \\
0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 \\
0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & 0 & 0 & 0 & 1\end{array}\right)$
$=\left(\begin{array}{cccccccc}
e^{-i\pi/4} & 0 & 0 & 0 & 0 & 0 & 0 &0 \\
0 & 0 & 0 & e^{i\pi/4} & 0 & 0 & 0 &0 \\
0 & 0 & 0 & 0 & 0 & e^{i\pi/4} & 0 &0 \\
0 & 0 & 0 & 0 & 0 & 0 & e^{-i\pi/4} &0 \\
0 & e^{i\pi/4} & 0 & 0 & 0 & 0 & 0 &0 \\
0 & 0 & e^{-i\pi/4} & 0 & 0 & 0 & 0 &0 \\
0 & 0 & 0 & 0 & e^{-i\pi/4} & 0 & 0 &0 \\
0 & 0 & 0 & 0 & 0 & 0 & 0 &e^{i\pi/4}\end{array}\right)
\left(\begin{array}{cccccccc}
1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 \\
0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 \\
0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & 0 & 0 & 0 & 1\end{array}\right)$
$=\left(\begin{array}{cccccccc}
e^{-i\pi/4} & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\
0 & e^{i\pi/4} & 0 & 0 & 0 & 0 & 0 & 0 \\
0 & 0 & e^{i\pi/4} & 0 & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & e^{-i\pi/4} & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & e^{i\pi/4} & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & 0 & e^{-i\pi/4} & 0 & 0 \\
0 & 0 & 0 & 0 & 0 & 0 & e^{-i\pi/4} & 0 \\
0 & 0 & 0 & 0 & 0 & 0 & 0 & e^{i\pi/4}\end{array}\right)$

$R_1^{(+)} = \left(\begin{array}{cccc}
e^{-i\pi/4} & 0 & 0 & 0 \\
0 & e^{i\pi/4} & 0 & 0 \\
0 & 0 & e^{i\pi/4} & 0 \\
0 & 0 & 0 & e^{-i\pi/4} \\
\end{array}\right)$, 　$R_1^{(-)} = \left(\begin{array}{cccc}
e^{i\pi/4} & 0 & 0 & 0 \\
0 & e^{-i\pi/4} & 0 & 0 \\
0 & 0 & e^{-i\pi/4} & 0 \\
0 & 0 & 0 & e^{i\pi/4}\end{array}\right)$

$R_3|0\rangle = \frac{1}{\sqrt{2}} \left ( 1- i \right ) |0\rangle = e^{-i\pi/4}|0\rangle$,　 $R_3|1\rangle = \frac{1}{\sqrt{2}} \left ( 1- i \right ) |1\rangle = e^{-i\pi/4}|1\rangle$,
$R_3|2\rangle = \frac{1}{\sqrt{2}} \left ( 1+ i \right ) |2\rangle = e^{i\pi/4}|2\rangle$,　 $R_3|3\rangle = \frac{1}{\sqrt{2}} \left ( 1+ i \right ) |3\rangle = e^{i\pi/4}|3\rangle$,
$R_3|4\rangle = \frac{1}{\sqrt{2}} \left ( 1- i \right ) |4\rangle = e^{-i\pi/4}|4\rangle $, 　$R_3|5\rangle = \frac{1}{\sqrt{2}} \left ( 1- i \right ) |5\rangle = e^{-i\pi/4}|5\rangle$,
$R_3|6\rangle = \frac{1}{\sqrt{2}} \left ( 1+ i \right ) |6\rangle = e^{i\pi/4}|6\rangle$, 　 $R_3|7\rangle = \frac{1}{\sqrt{2}} \left ( 1+ i \right ) |7\rangle = e^{i\pi/4}|7\rangle$, 　

$R_3 = \frac{1}{\sqrt{2}} \left ( 1 -i +2i \Psi_2^\dagger \Psi_2 \right )$

　$ |2 \rangle=\Psi_2^\dagger |0 \rangle$, 　$|3 \rangle = \Psi_2^\dagger |1 \rangle = \Psi_2^\dagger \Psi_1^\dagger |0 \rangle$, 　$|4 \rangle = \Psi_3^\dagger |0 \rangle$, 　$|5 \rangle=\Psi_3^\dagger \Psi_1^\dagger |0 \rangle =\Psi_3^\dagger |1 \rangle$, 　$|6 \rangle = \Psi_3^\dagger |2 \rangle = \Psi_3^\dagger \Psi_2^\dagger |0 \rangle$, 　$|7 \rangle = \Psi_3^\dagger |3 \rangle = \Psi_3^\dagger \Psi_2^\dagger |1 \rangle = \Psi_3^\dagger \Psi_2^\dagger \Psi_1^\dagger |0 \rangle $

$UR_3U^\dagger = \left(\begin{array}{cccccccc}
1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 \\
0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 \\
0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 \\
0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 \end{array}\right)
\left(\begin{array}{cccccccc}
e^{-i\pi/4} & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\
0 & e^{-i\pi/4} & 0 & 0 & 0 & 0 & 0 & 0 \\
0 & 0 & e^{i\pi/4} & 0 & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & e^{i\pi/4} & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & e^{-i\pi/4} & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & 0 & e^{-i\pi/4} & 0 & 0 \\
0 & 0 & 0 & 0 & 0 & 0 & e^{i\pi/4} & 0 \\
0 & 0 & 0 & 0 & 0 & 0 & 0 & e^{i\pi/4} \end{array}\right)
\left(\begin{array}{cccccccc}
1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 \\
0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 \\
0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & 0 & 0 & 0 & 1\end{array}\right)$
$=\left(\begin{array}{cccccccc}
e^{-i\pi/4} & 0 & 0 & 0 & 0 & 0 & 0 &0 \\
0 & 0 & 0 & e^{i\pi/4} & 0 & 0 & 0 &0 \\
0 & 0 & 0 & 0 & 0 & e^{-i\pi/4} & 0 &0 \\
0 & 0 & 0 & 0 & 0 & 0 & e^{i\pi/4} &0 \\
0 & e^{-i\pi/4} & 0 & 0 & 0 & 0 & 0 &0 \\
0 & 0 & e^{i\pi/4} & 0 & 0 & 0 & 0 &0 \\
0 & 0 & 0 & 0 & e^{-i\pi/4} & 0 & 0 &0 \\
0 & 0 & 0 & 0 & 0 & 0 & 0 &e^{i\pi/4}\end{array}\right)
\left(\begin{array}{cccccccc}
1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 \\
0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 \\
0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & 0 & 0 & 0 & 1\end{array}\right)$
$=\left(\begin{array}{cccccccc}
e^{-i\pi/4} & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\
0 & e^{i\pi/4} & 0 & 0 & 0 & 0 & 0 & 0 \\
0 & 0 & e^{-i\pi/4} & 0 & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & e^{i\pi/4} & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & e^{-i\pi/4} & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & 0 & e^{i\pi/4} & 0 & 0 \\
0 & 0 & 0 & 0 & 0 & 0 & e^{-i\pi/4} & 0 \\
0 & 0 & 0 & 0 & 0 & 0 & 0 & e^{i\pi/4}\end{array}\right)$

$R_3^{(+)} = \left(\begin{array}{cccc}
e^{-i\pi/4} & 0 & 0 & 0 \\
0 & e^{i\pi/4} & 0 & 0 \\
0 & 0 & e^{-i\pi/4} & 0 \\
0 & 0 & 0 & e^{i\pi/4} \\
\end{array}\right)$, 　$R_3^{(-)} = \left(\begin{array}{cccc}
e^{-i\pi/4} & 0 & 0 & 0 \\
0 & e^{i\pi/4} & 0 & 0 \\
0 & 0 & e^{-i\pi/4} & 0 \\
0 & 0 & 0 & e^{i\pi/4}\end{array}\right)$

$R_5|0\rangle = \frac{1}{\sqrt{2}} \left ( 1- i \right ) |0\rangle = e^{-i\pi/4}|0\rangle$,　 $R_5|1\rangle = \frac{1}{\sqrt{2}} \left ( 1- i \right ) |1\rangle = e^{-i\pi/4}|1\rangle$,
$R_5|2\rangle = \frac{1}{\sqrt{2}} \left ( 1- i \right ) |2\rangle = e^{-i\pi/4}|2\rangle$,　 $R_5|3\rangle = \frac{1}{\sqrt{2}} \left ( 1- i \right ) |3\rangle = e^{-i\pi/4}|3\rangle$,
$R_5|4\rangle = \frac{1}{\sqrt{2}} \left ( 1+ i \right ) |4\rangle = e^{i\pi/4}|4\rangle $, 　$R_5|5\rangle = \frac{1}{\sqrt{2}} \left ( 1+ i \right ) |5\rangle = e^{i\pi/4}|5\rangle$,
$R_5|6\rangle = \frac{1}{\sqrt{2}} \left ( 1+ i \right ) |6\rangle = e^{i\pi/4}|6\rangle$, 　 $R_5|7\rangle = \frac{1}{\sqrt{2}} \left ( 1+ i \right ) |7\rangle = e^{i\pi/4}|7\rangle$, 　

$R_5 = \frac{1}{\sqrt{2}} \left ( 1 -i +2i \Psi_3^\dagger \Psi_3 \right )$

　$ |2 \rangle=\Psi_2^\dagger |0 \rangle$, 　$|3 \rangle = \Psi_2^\dagger |1 \rangle = \Psi_2^\dagger \Psi_1^\dagger |0 \rangle$, 　$|4 \rangle = \Psi_3^\dagger |0 \rangle$, 　$|5 \rangle=\Psi_3^\dagger \Psi_1^\dagger |0 \rangle =\Psi_3^\dagger |1 \rangle$, 　$|6 \rangle = \Psi_3^\dagger |2 \rangle = \Psi_3^\dagger \Psi_2^\dagger |0 \rangle$, 　$|7 \rangle = \Psi_3^\dagger |3 \rangle = \Psi_3^\dagger \Psi_2^\dagger |1 \rangle = \Psi_3^\dagger \Psi_2^\dagger \Psi_1^\dagger |0 \rangle $

$UR_5U^\dagger = \left(\begin{array}{cccccccc}
1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 \\
0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 \\
0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 \\
0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 \end{array}\right)
\left(\begin{array}{cccccccc}
e^{-i\pi/4} & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\
0 & e^{-i\pi/4} & 0 & 0 & 0 & 0 & 0 & 0 \\
0 & 0 & e^{-i\pi/4} & 0 & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & e^{-i\pi/4} & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & e^{i\pi/4} & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & 0 & e^{i\pi/4} & 0 & 0 \\
0 & 0 & 0 & 0 & 0 & 0 & e^{i\pi/4} & 0 \\
0 & 0 & 0 & 0 & 0 & 0 & 0 & e^{i\pi/4} \end{array}\right)
\left(\begin{array}{cccccccc}
1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 \\
0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 \\
0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & 0 & 0 & 0 & 1\end{array}\right)$
$=\left(\begin{array}{cccccccc}
e^{-i\pi/4} & 0 & 0 & 0 & 0 & 0 & 0 &0 \\
0 & 0 & 0 & e^{-i\pi/4} & 0 & 0 & 0 &0 \\
0 & 0 & 0 & 0 & 0 & e^{i\pi/4} & 0 &0 \\
0 & 0 & 0 & 0 & 0 & 0 & e^{i\pi/4} &0 \\
0 & e^{-i\pi/4} & 0 & 0 & 0 & 0 & 0 &0 \\
0 & 0 & e^{-i\pi/4} & 0 & 0 & 0 & 0 &0 \\
0 & 0 & 0 & 0 & e^{i\pi/4} & 0 & 0 &0 \\
0 & 0 & 0 & 0 & 0 & 0 & 0 &e^{i\pi/4}\end{array}\right)
\left(\begin{array}{cccccccc}
1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 \\
0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 \\
0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & 0 & 0 & 0 & 1\end{array}\right)$
$=\left(\begin{array}{cccccccc}
e^{-i\pi/4} & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\
0 & e^{-i\pi/4} & 0 & 0 & 0 & 0 & 0 & 0 \\
0 & 0 & e^{i\pi/4} & 0 & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & e^{i\pi/4} & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & e^{-i\pi/4} & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & 0 & e^{-i\pi/4} & 0 & 0 \\
0 & 0 & 0 & 0 & 0 & 0 & e^{i\pi/4} & 0 \\
0 & 0 & 0 & 0 & 0 & 0 & 0 & e^{i\pi/4}\end{array}\right)$

$R_5^{(+)} = \left(\begin{array}{cccc}
e^{-i\pi/4} & 0 & 0 & 0 \\
0 & e^{-i\pi/4} & 0 & 0 \\
0 & 0 & e^{i\pi/4} & 0 \\
0 & 0 & 0 & e^{i\pi/4} \\
\end{array}\right)$, 　$R_5^{(-)} = \left(\begin{array}{cccc}
e^{-i\pi/4} & 0 & 0 & 0 \\
0 & e^{-i\pi/4} & 0 & 0 \\
0 & 0 & e^{i\pi/4} & 0 \\
0 & 0 & 0 & e^{i\pi/4}\end{array}\right)$

One can find the matrix forms for $R_2$ and $R_4$ here ( https://qiita.com/KogaSense/items/9f349a18ff6ed6088802 ).

$UR_2U^\dagger = \left(\begin{array}{cccccccc}
1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 \\
0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 \\
0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 \\
0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 \end{array}\right)
\frac{1}{\sqrt{2}}\left( \begin{array}{cccccccc}
1 & 0 & 0 & i & 0 & 0 & 0 &0 \\
0 & 1 & -i & 0 & 0 & 0 & 0 &0 \\
0 & -i & 1 & 0 & 0 & 0 & 0 &0 \\
i & 0 & 0 & 1 & 0 & 0 & 0 &0 \\
0 & 0 & 0 & 0 & 1 & 0 & 0 &i \\
0 & 0 & 0 & 0 & 0 & 1 & -i &0 \\
0 & 0 & 0 & 0 & 0 & -i & 1 & 0 \\
0 & 0 & 0 & 0 & i & 0 & 0 &1\end{array} \right)
\left(\begin{array}{cccccccc}
1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 \\
0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 \\
0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & 0 & 0 & 0 & 1\end{array}\right)$
$=\frac{1}{\sqrt{2}}\left(\begin{array}{cccccccc}
1 & 0 & 0 & i & 0 & 0 & 0 &0 \\
i & 0 & 0 & 1 & 0 & 0 & 0 &0 \\
0 & 0 & 0 & 0 & 0 & 1 & -i &0 \\
0 & 0 & 0 & 0 & 0 & -i & 1 &0 \\
0 & 1 & -i & 0 & 0 & 0 & 0 &0 \\
0 & -i & 1 & 0 & 0 & 0 & 0 &0 \\
0 & 0 & 0 & 0 & 1 & 0 & 0 &i \\
0 & 0 & 0 & 0 & i & 0 & 0 &1 \end{array}\right)
\left(\begin{array}{cccccccc}
1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 \\
0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 \\
0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & 0 & 0 & 0 & 1\end{array}\right)$
$=\frac{1}{\sqrt{2}}\left(\begin{array}{cccccccc}
1 & i & 0 & 0 & 0 & 0 & 0 & 0 \\
i & 1 & 0 & 0 & 0 & 0 & 0 & 0 \\
0 & 0 & 1 & -i & 0 & 0 & 0 & 0 \\
0 & 0 & -i & 1 & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & 1 & -i & 0 & 0 \\
0 & 0 & 0 & 0 & -i & 1 & 0 & 0 \\
0 & 0 & 0 & 0 & 0 & 0 & 1 & i \\
0 & 0 & 0 & 0 & 0 & 0 & i & 1 \end{array}\right)$

$R_2^{(+)} = \frac{1}{\sqrt{2}}\left(\begin{array}{cccc}
1 & i & 0 & 0 \\
i & 1 & 0 & 0 \\
0 & 0 & 1 & -i \\
0 & 0 & -i & 1 \\
\end{array}\right)$, 　$R_2^{(-)} = \left(\begin{array}{cccc}
1 & -i & 0 & 0 \\
-i & 1 & 0 & 0 \\
0 & 0 & 1 & i \\
0 & 0 & i & 1\end{array}\right)$

$UR_4U^\dagger = \left(\begin{array}{cccccccc}
1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 \\
0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 \\
0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 \\
0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 \end{array}\right)
\frac{1}{\sqrt{2}}\left( \begin{array}{cccccccc}
1 & 0 & 0 & 0 & 0 & 0 & i &0 \\
0 & 1 & 0 & 0 & 0 & 0 & 0 &i \\
0 & 0 & 1 & 0 & -i & 0 & 0 &0 \\
0 & 0 & 0 & 1 & 0 & -i & 0 &0 \\
0 & 0 & -i & 0 & 1 & 0 & 0 &0 \\
0 & 0 & 0 & -i & 0 & 1 & 0 &0 \\
i & 0 & 0 & 0 & 0 & 0 & 1 & 0 \\
0 & i & 0 & 0 & 0 & 0 & 0 &1\end{array} \right)
\left(\begin{array}{cccccccc}
1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 \\
0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 \\
0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & 0 & 0 & 0 & 1\end{array}\right)$
$=\frac{1}{\sqrt{2}}\left(\begin{array}{cccccccc}
1 & 0 & 0 & 0 & 0 & 0 & i &0 \\
0 & 0 & 0 & 1 & 0 & -i & 0 &0 \\
0 & 0 & 0 & -i & 0 & 1 & 0 &0 \\
i & 0 & 0 & 0 & 0 & 0 & 1 &0 \\
0 & 1 & 0 & 0 & 0 & 0 & 0 &i \\
0 & 0 & 1 & 0 & -i & 0 & 0 &0 \\
0 & 0 & -i & 0 & 1 & 0 & 0 & 0 \\
0 & i & 0 & 0 & 0 & 0 & 0 &1 \end{array}\right)
\left(\begin{array}{cccccccc}
1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 \\
0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 \\
0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & 0 & 0 & 0 & 1\end{array}\right)$
$=\frac{1}{\sqrt{2}}\left(\begin{array}{cccccccc}
1 & 0 & 0 & i & 0 & 0 & 0 & 0 \\
0 & 1 & -i & 0 & 0 & 0 & 0 & 0 \\
0 & -i & 1 & 0 & 0 & 0 & 0 & 0 \\
i & 0 & 0 & 1 & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & 1 & 0 & 0 & i \\
0 & 0 & 0 & 0 & 0 & 1 & -i & 0 \\
0 & 0 & 0 & 0 & 0 & -i & 1 & 0 \\
0 & 0 & 0 & 0 & i & 0 & 0 & 1 \end{array}\right)$

$R_4^{(+)} = \frac{1}{\sqrt{2}}\left(\begin{array}{cccc}
1 & 0 & 0 & i \\
0 & 1 & -i & 0 \\
0 & -i & 1 & 0 \\
i & 0 & 0 & 1 \\
\end{array}\right)$, 　$R_4^{(-)} = \left(\begin{array}{cccc}
1 & 0 & 0 & i \\
0 & 1 & -i & 0 \\
0 & -i & 1 & 0 \\
i & 0 & 0 & 1\end{array}\right)$