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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: d 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.

glS
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Jasmine
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  • It may be helpful if you link the paper (and mention the section where that transformation is introduced) you're talking about – epelaez Jun 22 '21 at 01:33
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    @epelaaez Edited the question with the link – Jasmine Jun 22 '21 at 01:36
  • $e^{-i\frac{\theta}{2}n\cdot\sigma}$ is rotation around $\vec{n}$ about $\theta$ angle. You can find this in Nielsen's book, chapter 4. – narip Jun 22 '21 at 01:44
  • @narip But then there would be a factor of i, I cant find that in the paper – Jasmine Jun 22 '21 at 01:45
  • 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). – narip Jun 22 '21 at 02:05
  • @narip How many combinations of Pauli matrices will be there then such that the rotations are unitary – Jasmine Jun 22 '21 at 02:20
  • see also https://quantumcomputing.stackexchange.com/a/16552/55: the $2\times2$ (special) unitary matrices are all and only the matrices of the form $a_0 I + i\sum_{i=1}^3 a_i \sigma_i$ with $(a_0,a_1,a_2,a_3)\in S^3$. – glS Jun 23 '21 at 08:51

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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.

Auric Goldfinger
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