# Math Behind $X$ Gate With Arbitrary Phase is equivalent to $ZXZ$ Gate

An X gate where there is a phase shift $$\phi$$ to the applied sinusoidal wave $$U = e^{-i\frac{\theta}{2}(cos(\phi)\sigma_x+sin(\phi)\sigma_y)}$$ is equivalent to a series of gates $$Z_{-\phi}X_{\theta}Z_{\phi} = e^{i\phi\sigma_z/2}e^{-i\theta\sigma_x/2}e^{-i\phi\sigma_z/2}$$. What is the math used to get from $$U$$ to the series of gates?

One of the key steps is a relation that says $$Ue^{iH}U^\dagger=e^{iUHU^\dagger}$$ for any unitary $$U$$. So, if I consider $$Z_{-\phi}X_{\theta}Z_{\phi}$$ then really all I need to work out to see the equivalence is what goes on in the exponent: $$Z_{-\phi}XZ_{\phi}=XZ_{2\phi}=X(I\cos\phi-i\sin\phi Z)=X\cos\phi-Y\sin\phi$$ (I seem to have a different sign to what you were expecting. I could easily have messed up e.g. whichever convention you're taking for $$X_{\theta}$$.)
To prove my main claim, you could think about the Taylor expansion of the exponential: $$e^{-iUHU^\dagger}=\sum_{n=0}^{\infty}\frac{1}{n!}(-iUHU^\dagger)^n.$$ When you write out $$(UHU^\dagger)^n=UHU^\dagger UHU^\dagger\ldots$$ then all the $$U^\dagger U$$ pairs are identity, and you just get $$UH^nU^\dagger$$.