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I was wondering what would be an equation for Hadamard operation for a single qubit, given the input as the current $\theta$ (0 to $+\pi/2$) and $\varphi$ ($-\pi$ to $+\pi$) and output expected in $\theta$ and $\varphi$ with same ranges. Most expressions of Hadamard that I saw use Cartesian transformation, but not $\theta/\varphi$ transform.

I could convert the input to Cartesian form, and convert the output back into $\theta/\varphi$ form well, but I'm looking for an equation that does it without using Cartesian conversion step? The goal is to understand direct relation between input and output. I tried interpreting Hadamard as subtracting $\pi/4$ from $\theta$ and adding $\pi$ to phase, but I see that it doesn't quite work for arbitrary input.

Note: here $\varphi$ refers to relative phase ($-\pi$ to $+\pi$) and $\theta$ refers to component amplitude factor (0 to $+\pi/2$).

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I think I got it. The two rotations with angles $-\pi/4$ and $\pi/2$ should do it. The first rotation is about Y-axis and the second one is about Z-axis. So it still not what I was looking for. I was looking for change in the angle that determines the amplitudes instead of Y-rotation (the two are not same). But the Y-rotation is enough for my intuition. How I got this answer? Well, I just observed the change through multiple trials and learned the pattern :)

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    $\begingroup$ Hi, welcome to QCSE. Although it's OK to answer your own question, could you provide a little more detail as to how you arrived at your answer? $\endgroup$ – Mark S Aug 20 '20 at 21:12
  • $\begingroup$ @Mark S - Thank you very much for adding the markup to my text!! $\endgroup$ – vrpbkp Aug 24 '20 at 12:19
  • $\begingroup$ I think @MartinVesely did... $\endgroup$ – Mark S Aug 24 '20 at 12:56

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