# What is the general formula for unitary rotations in terms of Pauli spin operators?

Recently I have read a paper in which they have used a unitary transformation as follows:

$$U_{\frac{7\pi}{16}}=\cos\left(\frac{7\pi}{8}\right)\sigma_{z}+\sin\left(\frac{7\pi}{8}\right)\sigma_{x}$$

Here $$\sigma_{x}$$ and $$\sigma_{z}$$ are the Pauli operators. I didn't understand where this came from? Also are any other combinations of sigma operators with any angles is a Unitary rotation? Do anyone know of any general formula for Unitary rotation? Any references would be great. Please see Eq. (3) in the paper: Experimental test of local observer-independence.

Why I am concerned about the above unitary operator is: Please see the below definition for the rotation operators. Here there is an imaginary $$i$$ coming which is not in the paper I have mentioned.

• It may be helpful if you link the paper (and mention the section where that transformation is introduced) you're talking about Jun 22 '21 at 1:33
• @epelaaez Edited the question with the link Jun 22 '21 at 1:36
• $e^{-i\frac{\theta}{2}n\cdot\sigma}$ is rotation around $\vec{n}$ about $\theta$ angle. You can find this in Nielsen's book, chapter 4. Jun 22 '21 at 1:44
• @narip But then there would be a factor of i, I cant find that in the paper Jun 22 '21 at 1:45
• You only need to see that $UU^\dagger=I$, hence it's unitary. As for the reason why there is no imaginary part because it's a specific angle(e.g., when $\theta$ in your rotation is 2$\pi$, there will also miss the imaginary part). Jun 22 '21 at 2:05

Setting $$\theta=\pi/2$$ for a general rotation, you get $$\mathrm{e}^{-\mathrm{i}\pi/2 A} = \mathrm{i} A\,,$$ which is unitary and up to a global phase $$\mathrm{i}$$ the operator $$A$$ itself. Now define a new operator $$U=-\mathrm{i}A$$ and it follows that $$U$$ is unitary $$U^\dagger U = \mathrm{i}(-\mathrm{i}) A^\dagger A = I\,.$$
You can decompose $$A$$ into $$A = n_x \sigma_x + n_y \sigma_y + n_z \sigma_z$$ with $$n_x^2 + n_y^2 + n_z^2 = 1$$ and $$n_i \in[-1,1]$$, $$i=x,y,z$$. In your example $$n_x = \sin\left(\frac{7\pi}{8}\right) \qquad n_y = 0 \qquad n_z = \cos\left(\frac{7\pi}{8}\right)\,.$$ For more details see Nielsen and Chuang, Quantum Computation and Quantum Information.