Exercise 4.3#
The rotation operator about the \(\hat{z}\) axis is given by
(1)#\[\begin{split}
R_z(\theta) = \cos\frac{\theta}{2}\ I - i\sin\frac{\theta}{2}Z = \begin{pmatrix}
e^{-i\theta/2} & 0 \\ 0 & e^{i\theta/2}
\end{pmatrix}
\end{split}\]
When \(\theta = \pi/4\), we have
(2)#\[\begin{split}
R_z(\pi/4) = \begin{pmatrix}
e^{-i\pi/8} & 0 \\ 0 & e^{i\pi/8}
\end{pmatrix}
\end{split}\]
Note that the \(T\) gate is defined by
(3)#\[\begin{split}
T = e^{i\pi/8}\begin{pmatrix}
e^{-i\pi/8} & 0 \\ 0 & e^{i\pi/8}
\end{pmatrix}
\end{split}\]
Compare eq. (2) and eq. (3), we conclude that, up to a global phase \(e^{i\pi/8}\), the \(\pi/8\) gate \(T\) satisfies \(T = R_z(\pi/4)\).