# Nielsen Chang exercise 4.10

I' m trying to figure out how to decompose unitary operator U using only Rx and Ry rotations. I understand that I must use result of exercise 4.8 ($$U = e^{i\alpha}R_n(\theta)$$). But I don't understand how, my thoughts is use Bloch vector coordinates and place it instead n-vector coordinate and try to solve using this, but I think it is not correct way. Can anyone help me or give some insights how to solve this exercise?

• Please add the text of the exercise in full. Commented Apr 11, 2023 at 18:04

Problem 4.10 of Nielsen and Chuang's text asks the reader to produce an $$XY$$-decomposition of a single-qubit unitary $$U$$, in parallel with the $$ZY$$-decomposition given in Theorem 4.1 of the text. That is, the problem asks to show that any single-qubit unitary $$U$$ can be written as $$U = e^{i \alpha} R_x(\beta) R_y(\gamma) R_x(\delta)$$ for appropriate choices of $$\alpha, \beta, \gamma, \delta$$.

Certainly, we could proceed along similar lines as the text does for the $$ZY$$-decomposition of Theorem 4.1, producing the result essentially from scratch. Alternatively, we could simply use Theorem 4.1 and do a "change of reference", to interchange the $$x$$ and $$z$$ axes. Let's proceed the latter route and see how this works.

We know that the Hadamard $$H$$ (which we quickly note is unitary and self-inverse) implements the appropriate swap between $$x$$ and $$z$$, while sending $$y$$ to $$-y$$. So instead of working with $$U$$ directly, consider $$U' = H U H^{-1} = H U H$$ which is the frame-rotated $$U$$. Using Theorem 4.1 of the text, we can produce a $$ZY$$-decomposition of $$U'$$. $$U' = e^{i\alpha} R_z(\beta) R_y(\gamma) R_z(\delta)$$ What does this imply for $$U$$? We will see that the above implies an $$XY$$-decomposition for $$U$$. \begin{align} U = HU'H &= e^{i\alpha} (H R_z(\beta) H)(H R_y(\gamma)H)(HR_z(\delta) H) \\ &= e^{i\alpha} R_x(\beta) R_y(-\gamma) R_x(\delta). \end{align} Notice the change in sign for $$\gamma$$ that occurs since $$H Y H = -Y$$. Regardless, we have $$U$$ written in the required form.

As a final elaboration, this argument could be generalized. We could choose any two orthogonal axes $$\hat{n}, \hat{m}$$ for rotations in our decomposition, using the fact that we can always frame-rotate to the $$z$$ and $$x$$ axes.

• Thanks, understood! Commented Apr 12, 2023 at 7:09
• I tried to change $R_z$ rotation with $R_x$ rotations but I always got $HR_x(\beta)R_y(-\gamma)R_x(\delta)H$ And stopped at this moment without knowing how transform this in desired view.)) I didn't figure it out before this trick) Commented Apr 12, 2023 at 7:17