I am trying to find a nice way to represent the square root of an arbitrary single qubit unitary to implement Lemma 6.1 from this paper: https://arxiv.org/pdf/quant-ph/9503016.pdf
Given the Euler angles: Rz(a)Ry(b)Rz(c) = (Rz(a')Ry(b')Rz(c'))^2
Is there a closed form expression that relates the angles a', b', c' to the angles a,b,c?
I have tried some simultaneous equations after I do the Euler angle exchange to have:
Rz(a)Ry(b)Rz(c) = Rz(a') [Rz(A)Ry(B)Rz(C)] Rz(c'), where A,B,C can be related to a,b,c via some nasty expressions found eg. on p4 of https://arxiv.org/pdf/1812.09114.pdf.
But from there I cannot find a closed form expression for all of a',b',c'.
This seems like the sort of problem that must have been solved before - or at least it must be known whether a nice relation exists - but I cannot find work on it. Is it possible to go via quaternions or something?
In a nutshell...
1) Convert $R_z(a)R_y(b)R_z(c)$ to axis-angle representation (link). You will then have a representation of $\{\theta, x, y, z\}$ where $\theta$ is the angle and $(x, y, z)$ is the axis.
2) Divide the angle by 2. I.e. $\theta' = \theta/2$. You will now have $\{\theta', x, y, z\}$
3) Convert back to Euler angles (link)
And you are done.