I have made use of the following formulas, \begin{align} \theta&=2\cos^{-1}\bigg(\frac{e^{-i\alpha}Tr(X)}{2}\bigg)\\ n_i&=\frac{e^{-i\alpha}Tr(X\sigma_x)}{2\sin\theta/2}\\ e^{i\alpha}&=\sqrt{\det(X)} \end{align} Please check Can I find the axis of rotation for any single-qubit gate? for the reference. $$ X=\frac{1}{\sqrt{2}}\begin{bmatrix}-i&-1\\1&i\end{bmatrix}\\ e^{i\alpha}=\sqrt{\det(X)}=1\\ \theta=2\cos^{-1}(\frac{1.0}{2})=2\cos^{-1}(0)=2\frac{\pi}{2}=\pi\\ Tr(X\sigma_x)=0\implies n_1=0\\ X\sigma_y=X=\frac{1}{\sqrt{2}}\begin{bmatrix}-i&-1\\1&i\end{bmatrix}\begin{bmatrix}0&-i\\i&0\end{bmatrix}=\frac{1}{\sqrt{2}}\begin{bmatrix}-i&-1\\-1&-i\end{bmatrix}\\ \implies Tr(X\sigma_y)=\frac{1}{\sqrt{2}}.-2i=-\sqrt{2}i\implies n_2=\frac{i.1.-\sqrt{2}i}{2.1}=\frac{\sqrt{2}}{2}=\sqrt{2}\\ Tr(X\sigma_z)=0\implies n_3=0 $$
Why am I not getting a unit vector here ?
