I have a question about the two equations:
- Any matrix in $SU(2)$ could be parametrized as $$ R_{\hat{n}}(\theta) = \cos\left(\frac{\theta}{2}\right)I-i\sin\left(\frac{\theta}{2}\right)(\hat{n}\cdot\vec\sigma) $$
- And the $U_3$ gate in qiskit is defined as $$ U_3(\gamma,\beta,\delta) = \begin{bmatrix} \cos\left(\frac{\gamma}{2}\right) & -e^{i\delta} \sin\left(\frac{\gamma}{2}\right) \\ e^{i\beta} \sin\left(\frac{\gamma}{2}\right) & e^{i(\delta + \beta)} \cos\left(\frac{\gamma}{2}\right) \end{bmatrix} $$
For the first equation, I think the $\theta/2$ term is kind of relevant to the Dirac belt trick: the electron spin state will negate under a $2\pi$ rotation, so $4\pi$ will return to the original state. For the second equation, on the other hand, I think $\gamma/2$ is because on the Bloch sphere it looks like $|0\rangle$ and $|1\rangle$ are on the same line pointing toward the opposite directions, but they're orthogonal states. I'm wondering if there're any connections between the two cases when we divide $\theta$ or $\gamma$ by 2. Thanks!!
PS: In my understanding, $\theta$ refers to the angle of rotation along the axis $\hat n$ following the right-hand rule, and $\gamma$ here is the included angle of $\hat n$ and $z$-axis in the spherical coordinate.