n=3 Clifford

page 131 of http://www.damtp.cam.ac.uk/user/tong/qhe/qhe.pdf

$\gamma^1=\sigma^1 \otimes \sigma^3 \otimes \sigma^3 = \left(\begin{array}{cccccccc}0 & 1 & 0 & 0 & 0 & 0 & 0 &0 \\\ 1 & 0 & 0 & 0 & 0 & 0 & 0 &0 \\\ 0 & 0 & 0 & -1 & 0 & 0 & 0 &0 \\\ 0 & 0 & -1 & 0 & 0 & 0 & 0 &0 \\\ 0 & 0 & 0 & 0 & 0 & -1 & 0 &0 \\\ 0 & 0 & 0 & 0 & -1 & 0 & 0 &0 \\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 &1 \\\ 0 & 0 & 0 & 0 & 0 & 0 & 1 &0\end{array}\right)$

$\gamma^2=\sigma^2 \otimes \sigma^3 \otimes \sigma^3 = \left(\begin{array}{cccccccc}0 & -i & 0 & 0 & 0 & 0 & 0 &0 \\\ i & 0 & 0 & 0 & 0 & 0 & 0 &0 \\\ 0 & 0 & 0 & i & 0 & 0 & 0 &0 \\\ 0 & 0 & -i & 0 & 0 & 0 & 0 &0 \\\ 0 & 0 & 0 & 0 & 0 & i & 0 &0 \\\ 0 & 0 & 0 & 0 & -i & 0 & 0 &0 \\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & -i \\\ 0 & 0 & 0 & 0 & 0 & 0 & i &0\end{array}\right)$

$|0\rangle \equiv \left(\begin{array}{c} 1 \\\ 0 \\\ 0 \\\ 0 \\\ 0 \\\ 0 \\\ 0 \\\ 0\end{array}\right) $, $\Psi_1^\dagger |0 \rangle =\frac{1}{2}\left(\gamma_1 - i \gamma_{2}\right) |0 \rangle = \left(\begin{array}{cccccccc}0 & 0 & 0 & 0 & 0 & 0 & 0 &0 \\\ 1 & 0 & 0 & 0 & 0 & 0 & 0 &0 \\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 &0 \\\ 0 & 0 & -1 & 0 & 0 & 0 & 0 &0 \\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 &0 \\\ 0 & 0 & 0 & 0 & -1 & 0 & 0 &0 \\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\\ 0 & 0 & 0 & 0 & 0 & 0 & 1 &0\end{array}\right) \left(\begin{array}{c} 1 \\\ 0 \\\ 0 \\\ 0 \\\ 0 \\\ 0 \\\ 0 \\\ 0\end{array}\right) = \left(\begin{array}{c} 0 \\\ 1 \\\ 0 \\\ 0 \\\ 0 \\\ 0 \\\ 0 \\\ 0\end{array}\right) \equiv |1 \rangle$

$\gamma^3={\bf 1} \otimes \sigma^1 \otimes \sigma^3 = \left(\begin{array}{cccccccc}0 & 0 & 1 & 0 & 0 & 0 & 0 &0 \\\ 0 & 0 & 0 & 1 & 0 & 0 & 0 &0 \\\ 1 & 0 & 0 & 0 & 0 & 0 & 0 &0 \\\ 0 & 1 & 0 & 0 & 0 & 0 & 0 &0 \\\ 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 \\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 \\\ 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 \\\ 0 & 0 & 0 & 0 & 0 & -1 & 0 &0\end{array}\right)$

$\gamma^4={\bf 1} \otimes \sigma^2 \otimes \sigma^3 = \left(\begin{array}{cccccccc}0 & 0 & -i & 0 & 0 & 0 & 0 &0 \\\ 0 & 0 & 0 & -i & 0 & 0 & 0 &0 \\\ i & 0 & 0 & 0 & 0 & 0 & 0 &0 \\\ 0 & i & 0 & 0 & 0 & 0 & 0 &0 \\\ 0 & 0 & 0 & 0 & 0 & 0 & i & 0 \\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & i \\\ 0 & 0 & 0 & 0 & -i & 0 & 0 & 0 \\\ 0 & 0 & 0 & 0 & 0 & -i & 0 &0\end{array}\right)$

$\Psi_2^\dagger |0 \rangle =\frac{1}{2}\left(\gamma_3 - i \gamma_4\right) |0 \rangle = \left(\begin{array}{cccccccc}0 & 0 & 0 & 0 & 0 & 0 & 0 &0 \\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 &0 \\\ 1 & 0 & 0 & 0 & 0 & 0 & 0 &0 \\\ 0 & 1 & 0 & 0 & 0 & 0 & 0 &0 \\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 &0 \\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 &0 \\\ 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 \\\ 0 & 0 & 0 & 0 & 0 & 1 & 0 &0\end{array}\right) \left(\begin{array}{c} 1 \\\ 0 \\\ 0 \\\ 0 \\\ 0 \\\ 0 \\\ 0 \\\ 0\end{array}\right) = \left(\begin{array}{c} 0 \\\ 0 \\\ 1 \\\ 0 \\\ 0 \\\ 0 \\\ 0 \\\ 0\end{array}\right) \equiv |2 \rangle$

$\gamma^5={\bf 1} \otimes {\bf 1} \otimes \sigma^1 = \left(\begin{array}{cccccccc}0 & 0 & 0 & 0 & 1 & 0 & 0 &0 \\\ 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 \\\ 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 \\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 \\\ 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\\ 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 \\\ 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 \\\ 0 & 0 & 0 & 1 & 0 & 0 & 0 &0\end{array}\right)$

$\gamma^6={\bf 1} \otimes {\bf 1} \otimes \sigma^2 = \left(\begin{array}{cccccccc}0 & 0 & 0 & 0 & -i & 0 & 0 &0 \\\ 0 & 0 & 0 & 0 & 0 & -i & 0 & 0 \\\ 0 & 0 & 0 & 0 & 0 & 0 & -i & 0 \\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & -i \\\ i & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\\ 0 & i & 0 & 0 & 0 & 0 & 0 & 0 \\\ 0 & 0 & i & 0 & 0 & 0 & 0 & 0 \\\ 0 & 0 & 0 & i & 0 & 0 & 0 &0\end{array}\right)$

$\Psi_3^\dagger |0 \rangle =\frac{1}{2}\left(\gamma_5 - i \gamma_6\right) |0 \rangle = \left(\begin{array}{cccccccc}0 & 0 & 0 & 0 & 0 & 0 & 0 &0 \\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 &0 \\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 &0 \\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 &0 \\\ 1 & 0 & 0 & 0 & 0 & 0 & 0 &0 \\\ 0 & 1 & 0 & 0 & 0 & 0 & 0 &0 \\\ 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 \\\ 0 & 0 & 0 & 1 & 0 & 0 & 0 &0\end{array}\right) \left(\begin{array}{c} 1 \\\ 0 \\\ 0 \\\ 0 \\\ 0 \\\ 0 \\\ 0 \\\ 0\end{array}\right) = \left(\begin{array}{c} 0 \\\ 0 \\\ 0 \\\ 0 \\\ 1 \\\ 0 \\\ 0 \\\ 0\end{array}\right) \equiv |4 \rangle$

$\Psi_2^\dagger \Psi_1^\dagger |0 \rangle =\Psi_2^\dagger |1 \rangle =\frac{1}{2}\left(\gamma_3 - i \gamma_4\right) |1 \rangle = \left(\begin{array}{cccccccc}0 & 0 & 0 & 0 & 0 & 0 & 0 &0 \\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 &0 \\\ 1 & 0 & 0 & 0 & 0 & 0 & 0 &0 \\\ 0 & 1 & 0 & 0 & 0 & 0 & 0 &0 \\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 &0 \\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 &0 \\\ 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 \\\ 0 & 0 & 0 & 0 & 0 & 1 & 0 &0\end{array}\right) \left(\begin{array}{c} 0 \\\ 1 \\\ 0 \\\ 0 \\\ 0 \\\ 0 \\\ 0 \\\ 0\end{array}\right) = \left(\begin{array}{c} 0 \\\ 0 \\\ 0 \\\ 1 \\\ 0 \\\ 0 \\\ 0 \\\ 0\end{array}\right) \equiv |3 \rangle$

$\Psi_3^\dagger \Psi_1^\dagger |0 \rangle =\Psi_3^\dagger |1 \rangle =\frac{1}{2}\left(\gamma_5 - i \gamma_6\right) |1 \rangle = \left(\begin{array}{cccccccc}0 & 0 & 0 & 0 & 0 & 0 & 0 &0 \\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 &0 \\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 &0 \\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 &0 \\\ 1 & 0 & 0 & 0 & 0 & 0 & 0 &0 \\\ 0 & 1 & 0 & 0 & 0 & 0 & 0 &0 \\\ 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 \\\ 0 & 0 & 0 & 1 & 0 & 0 & 0 &0\end{array}\right) \left(\begin{array}{c} 0 \\\ 1 \\\ 0 \\\ 0 \\\ 0 \\\ 0 \\\ 0 \\\ 0\end{array}\right) = \left(\begin{array}{c} 0 \\\ 0 \\\ 0 \\\ 0 \\\ 0 \\\ 1 \\\ 0 \\\ 0\end{array}\right) \equiv |5 \rangle$

$\Psi_3^\dagger \Psi_2^\dagger |0 \rangle =\Psi_3^\dagger |2 \rangle =\frac{1}{2}\left(\gamma_5 - i \gamma_6\right) |2 \rangle = \left(\begin{array}{cccccccc}0 & 0 & 0 & 0 & 0 & 0 & 0 &0 \\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 &0 \\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 &0 \\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 &0 \\\ 1 & 0 & 0 & 0 & 0 & 0 & 0 &0 \\\ 0 & 1 & 0 & 0 & 0 & 0 & 0 &0 \\\ 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 \\\ 0 & 0 & 0 & 1 & 0 & 0 & 0 &0\end{array}\right) \left(\begin{array}{c} 0 \\\ 0 \\\ 1 \\\ 0 \\\ 0 \\\ 0 \\\ 0 \\\ 0\end{array}\right) = \left(\begin{array}{c} 0 \\\ 0 \\\ 0 \\\ 0 \\\ 0 \\\ 0 \\\ 1 \\\ 0\end{array}\right) \equiv |6 \rangle$

$\Psi_3^\dagger \Psi_2^\dagger \Psi_1^\dagger |0 \rangle =\Psi_3^\dagger |3 \rangle =\frac{1}{2}\left(\gamma_5 - i \gamma_6\right) |3 \rangle = \left(\begin{array}{cccccccc}0 & 0 & 0 & 0 & 0 & 0 & 0 &0 \\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 &0 \\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 &0 \\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 &0 \\\ 1 & 0 & 0 & 0 & 0 & 0 & 0 &0 \\\ 0 & 1 & 0 & 0 & 0 & 0 & 0 &0 \\\ 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 \\\ 0 & 0 & 0 & 1 & 0 & 0 & 0 &0\end{array}\right) \left(\begin{array}{c} 0 \\\ 0 \\\ 0 \\\ 1 \\\ 0 \\\ 0 \\\ 0 \\\ 0\end{array}\right) = \left(\begin{array}{c} 0 \\\ 0 \\\ 0 \\\ 0 \\\ 0 \\\ 0 \\\ 0 \\\ 1\end{array}\right) \equiv |7 \rangle$

$\gamma^7=\sigma^3 \otimes \sigma^3 \otimes \sigma^3 = \left(\begin{array}{cccccccc} 1 & 0 & 0 & 0 & 0 & 0 & 0 &0 \\\ 0 & -1 & 0 & 0 & 0 & 0 & 0 &0 \\\ 0 & 0 & -1 & 0 & 0 & 0 & 0 &0 \\\ 0 & 0 & 0 & 1 & 0 & 0 & 0 &0 \\\ 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 \\\ 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 \\\ 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 \\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1\end{array}\right)$


$\gamma^7 |0> = 1 |0>$, 　$\gamma^7 |3> = 1 |3>$, 　$\gamma^7 |5> = 1 |5>$, 　$\gamma^7 |6> = 1 |6>$,

$\gamma^7 |1> = -1 |1>$, 　$\gamma^7 |2> = -1 |2>$, 　$\gamma^7 |4> = -1 |4>$, 　$\gamma^7 |7> = -1 |7>$, 

$R_1=\frac{1}{\sqrt{2}}\left(1+\gamma_2 \gamma_1\right)=\frac{1}{\sqrt{2}}\left(1+\left( \begin{array}{cccccccc}0 & -i & 0 & 0 & 0 & 0 & 0 &0 \\\ i & 0 & 0 & 0 & 0 & 0 & 0 &0 \\\ 0 & 0 & 0 & i & 0 & 0 & 0 &0 \\\ 0 & 0 & -i & 0 & 0 & 0 & 0 &0 \\\ 0 & 0 & 0 & 0 & 0 & i & 0 &0 \\\ 0 & 0 & 0 & 0 & -i & 0 & 0 &0 \\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & -i \\\ 0 & 0 & 0 & 0 & 0 & 0 & i &0\end{array} \right)\left(\begin{array}{cccccccc}0 & 1 & 0 & 0 & 0 & 0 & 0 &0 \\\ 1 & 0 & 0 & 0 & 0 & 0 & 0 &0 \\\ 0 & 0 & 0 & -1 & 0 & 0 & 0 &0 \\\ 0 & 0 & -1 & 0 & 0 & 0 & 0 &0 \\\ 0 & 0 & 0 & 0 & 0 & -1 & 0 &0 \\\ 0 & 0 & 0 & 0 & -1 & 0 & 0 &0 \\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 &1 \\\ 0 & 0 & 0 & 0 & 0 & 0 & 1 &0\end{array}\right) \right)$
$=\frac{1}{\sqrt{2}}\left( \begin{array}{cccccccc}1-i & 0 & 0 & 0 & 0 & 0 & 0 &0 \\\ 0 & 1+i & 0 & 0 & 0 & 0 & 0 &0 \\\ 0 & 0 & 1-i & 0 & 0 & 0 & 0 &0 \\\ 0 & 0 & 0 & 1+i & 0 & 0 & 0 &0 \\\ 0 & 0 & 0 & 0 & 1-i & 0 & 0 &0 \\\ 0 & 0 & 0 & 0 & 0 & 1+i & 0 &0 \\\ 0 & 0 & 0 & 0 & 0 & 0 & 1-i & 0 \\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1+i\end{array} \right)$
$=\left( \begin{array}{cccccccc} e^{-i\pi/4} & 0 & 0 & 0 & 0 & 0 & 0 &0 \\\ 0 &e^{i\pi/4} & 0 & 0 & 0 & 0 & 0 &0 \\\ 0 & 0 & e^{-i\pi/4} & 0 & 0 & 0 & 0 &0 \\\ 0 & 0 & 0 & e^{i\pi/4} & 0 & 0 & 0 &0 \\\ 0 & 0 & 0 & 0 & e^{-i\pi/4} & 0 & 0 &0 \\\ 0 & 0 & 0 & 0 & 0 & e^{i\pi/4} & 0 &0 \\\ 0 & 0 & 0 & 0 & 0 & 0 & e^{-i\pi/4} & 0 \\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & e^{i\pi/4}\end{array} \right)$

We have

$|0\rangle$, 　$|1 \rangle = \Psi_1^\dagger |0 \rangle$, 　$ |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 $