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I have a question about the two equations:

  1. Any matrix in $SU(2)$ could be parametrized as $$ R_{\hat{n}}(\theta) = \cos\left(\frac{\theta}{2}\right)I-i\sin\left(\frac{\theta}{2}\right)(\hat{n}\cdot\vec\sigma) $$
  2. 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.

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Yes, the referenced terms divided by two are directly related. First note that $R_{\hat n}(\theta) \in SU(2)$ is isomorphic to $U_3 \in U(2)$. You can see this by rewriting $U_3$ as an arbitrary element of $SU(2)$ multiplied by the global phase $e^{i(\delta+\beta)/2} \in U(1)$.

In other words, $$SU(2) \ni U_3^\prime =U_3\times e^{-i(\delta+\beta)/2} = \begin{bmatrix} e^{-i(\delta+\beta)/2}\cos\left(\frac{\gamma}{2}\right) & -e^{i(\delta-\beta)/2} \sin\left(\frac{\gamma}{2}\right) \\ e^{-i(\delta-\beta)/2} \sin\left(\frac{\gamma}{2}\right) & e^{i(\delta + \beta)/2} \cos\left(\frac{\gamma}{2}\right) \end{bmatrix},$$ such that $U_3^\prime$ is simply an alternative parameterization of an arbitrary element of $SU(2)$, and $U_3^\prime \cong U_3$ up to global phase.

In reference to your comments on the Bloch sphere, the Bloch sphere is a Riemann sphere (i.e. the complex projective line), not a Euclidean 2-sphere. The former is a part of elliptic geometry, the latter of spherical geometry. Elliptic geometry is essentially spherical geometry with antipodal points identified. This is why orthogonal states appear as antipodal points in a Bloch sphere representation. As you suggested this is all related to spinors ("the Dirac trick"), but that rabbit hole goes very deep.

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With $$ \vec\sigma = \begin{bmatrix}\begin{bmatrix}0 & -i\\ i &0 \end{bmatrix}\begin{bmatrix} 0 & -1 \\1 & 0\end{bmatrix}\begin{bmatrix}-1 & 0 \\ 0 & -1 \end{bmatrix} \end{bmatrix} $$ For the first equation I like to think that defining the formula with $\frac{\theta}{2}$ is a nice way to have a direct understanding of the effect on the Bloch sphere.
if $\hat{n}=[1\space 0\space 0]$ then $R_{\hat{n}}(\theta) =R_X(\theta)$ and $R_X(\theta)$ is a $\theta$ rotation around the X axis of the Bloch Sphere.
if $\hat{n}=[0\space 1\space 0]$ then $R_{\hat{n}}(\theta) =R_Y(\theta)$ and $R_Y(\theta)$ is a $\theta$ rotation around the Y axis of the Bloch Sphere.
if $\hat{n}=[0\space 0\space 1]$ then $R_{\hat{n}}(\theta) =R_Z(\theta)$ and $R_Z(\theta)$ is a $\theta$ rotation around the Z axis of the Bloch Sphere.

Essentially $R_{\hat{n}}(\theta)$ is a generalization of the rotation matrix hence the need of $\frac{\theta}{2}$. Another nice consequence is that for the Pauli gates $X=R_X(\pi)$, $Y=R_Y(\pi)$, $Z=R_Z(\pi)$

For the second equation the reasoning is essentially the same.

$U_3$ is a generalization of the $R_\phi$, $T$, $S$, $Z$ gates and defining it with $\frac{\gamma}{2}$ allows to have this nice identities that links to the axis rotation matrix.

$U_3(\gamma,0,\pi/2) = R_X(\gamma)$
$U_3(\gamma,0,0) = R_Y(\gamma)$
$U_3(0,0,\delta) = R_\phi(\delta)$

Remark : There is no relation between $U_3$ and $Z$

https://qiskit.org/textbook/ch-states/single-qubit-gates.html

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    $\begingroup$ "Essentially $R_n(θ)$ is a generalization of the rotation matrix hence the need of $\theta/2$" - this doesn't really explain the factor of two; when one represents ordinary rotations just working in $\mathbb{R}^3$ no such factor appears. Its more due to the unique relationship between the rotation group $SO(3)$ that we use to visualize Bloch sphere rotations and the special unitary group $SU(2)$ that we use to manipulate two level systems. $\endgroup$ – forky40 Apr 4 at 19:56
  • $\begingroup$ @forky40 thanks for the comment. I think your remark is clearly aligned with what I wrote "For the first equation [...] θ/2 is a nice way to have a direct understanding of the effect on the Bloch sphere." $\endgroup$ – PilouPili Apr 4 at 21:21
  • $\begingroup$ @forky40 Thanks for the comment:) How are the two groups $SO(3)$ and $SU(2)$ related in this sense? $\endgroup$ – ZR- Apr 4 at 23:14
  • $\begingroup$ @ZR- I can't do the explanation justice in a comment but if you search for terms like "$SU(2)$ $SO(3)$ double cover" or "dirac belt trick" you should find tons of material about this relationship from either a group theory or physics (but still group theory) perspective $\endgroup$ – forky40 Apr 5 at 1:48
  • $\begingroup$ @forky40 Thanks! $\endgroup$ – ZR- Apr 5 at 2:46

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