How to obtain Y rotation with only X and Z rotations gates?

Question

Let's say you have a system with which you can perform arbitrary rotations around the X and Z axis. How would you then be able to use these rotations to obtain an arbitrary rotation around the Y axis?

I have seen somewhere that rotation around an arbitrary axis can be achieved by doing three rotations around two fixed axis, that is, $$\hat{R}_\vec{n}(\theta)=R_Z(\gamma)R_X(\beta)R_Z(\alpha)$$ for some angles $\gamma, \alpha, \beta$. But how do you actually use this? What if I want to rotate around the Y axis with an angle of $\theta$ i.e. $\hat{R}_Y(\theta)$? Then how do I figure out what $\gamma,\alpha,\beta$ to use?

Edit: I've found a nice answer on Physics SE.

Answer 1

Single-qubit unitaries are just 3D rotations, multiplied by a phase. So in order to find the actual angles, you can resort to the theory of rotation matrices, in particular to Euler's rotation theorem, which states that any rotation is a composition of 3 rotations (the theorem proof is constructive, so you get the actual angles).

Answer 2

Try selecting $\gamma$ and $\alpha$ so that you get the rotation $$ \sqrt{Z}R_X(\beta)\sqrt{Z}^\dagger. $$ There's two little tricks here that make this work. Firstly, the $R_X(\beta)$ has an $\mathbb{I}$ component and an $X$ component (I always get factors of $1/2$ wrong here, so I won't write out the cos and sin functions explicitly unless you define your $R_x$ function). Now, $$ \sqrt{Z}\mathbb{I}\sqrt{Z}^\dagger=\mathbb{I} $$ while there's some funky anti-commutation that goes on with $X$: $$ \sqrt{Z}X\sqrt{Z}^\dagger=ZX=iY, $$ so you've managed to effectively change the $X$ into a $Y$, so you're getting $Y$ rotations.

Of course, this doesn't answer your more general question about how you get a more general rotation....