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
Given the Euler angles: $R_z(a)R_y(b)R_z(c) = \left(R_z\left(a'\right)R_y\left(b'\right)R_z\left(c'\right)\right)^2$
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:
$R_z(a)R_y(b)R_z(c) = R_z(a') [R_z(A)R_y(B)R_z(C)] R_z(c')$, where $A,B,C$ can be related to $a,b,c$ via some nasty expressions found eg. on p4 of this paper.
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?