# Are these two 'divided by two' terms related?

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.

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.

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

• "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. Apr 4, 2021 at 19:56
• @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." Apr 4, 2021 at 21:21
• @forky40 Thanks for the comment:) How are the two groups $SO(3)$ and $SU(2)$ related in this sense?
– ZR-
Apr 4, 2021 at 23:14
• @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 Apr 5, 2021 at 1:48
• @forky40 Thanks!
– ZR-
Apr 5, 2021 at 2:46