We can rewrite the Hadamard gate as $R_x(-\pi/2)R_z(-\pi/2)R_x(-\pi/2)$. I am trying to visualize the effect of applying the Hadamard gate to $|+\rangle$ on the Bloch sphere.

The X gate takes $|+\rangle$ to $|+\rangle$. The Z gate takes $|+\rangle$ to $|-\rangle$. The last X gate takes $|-\rangle$ to $|-\rangle$ (ignoring the phase factors).

I know that the effect of Hadamard on $|+\rangle$ is $|0\rangle$.

I am trying to imagine a clockwise rotation of $\pi$/2 about the X "axis" on the Bloch sphere, followed by a clockwise rotation of $\pi$/2 about the Z "axis", followed by a $\pi$/2 about the X "axis".

Please advise my error. I have read up on wikipedia and StackExchange regarding the Bloch sphere and rotations. I am trying to use this understanding to find the Z-Y decomposition of the Hadamard gate, and use that decomposition to find $U=e^(i\alpha)AXBXC$ for the Hadamard gate.

The corollary of $U=e^(i\alpha)R_n(\beta)R_m(\gamma)R_n(\delta)$).

  • $\begingroup$ Did you try typing "hadamard rotation axis" into google? The 4th hit (for me) is vcpc.univie.ac.at/~ian/hotlist/qc/talks/…, which on slide 25 describes the Hadamard gate as a rotation. (Feel free to take that slide and make it into an answer!) $\endgroup$ – Norbert Schuch Jan 10 '16 at 0:10
  • $\begingroup$ Yes, I've looked through that link several times before, but it's not exactly addressing my issue. @NorbertSchuch $\endgroup$ – NewUser392 Jan 10 '16 at 0:31
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    $\begingroup$ So what is your issue? You are saying "I am trying to visualize the effect of applying the Hadamard gate to |+⟩ on the Bloch sphere." According to the link, it is a rotation about a diagonal axis by $\pi$. $\endgroup$ – Norbert Schuch Jan 10 '16 at 0:33
  • $\begingroup$ What you are saying is the Hadamard gate represented as a rotation about an n axis, which is very simple. I am looking to represent it as only X and Z rotations, a Z-Y decomposition, and the corollary above. @NorbertSchuch $\endgroup$ – NewUser392 Jan 10 '16 at 1:13
  • $\begingroup$ But you know how to represent it as an X-Z-X rotation -- you write it yourself! I don't understand. $\endgroup$ – Norbert Schuch Jan 10 '16 at 1:17

The Hadamard gate is a 180 degree rotation around the diagonal X+Z axis of the Bloch sphere.

Terrible picture (from a blog post):

enter image description here

In the diagram the |0> state is at the top (+Z) of the sphere and the |0>+|1> state is at the front (+X). Rotating around the back-bottom-to-front-top axis (X=Z,Y=0) moves the top point (|0>) to the front point (|0>+|1>) and cycle. And this same operation will move the bottom point (|1>) to the back point (|0>-|1>) and cycle. Just like the Hadamard gate.

Global Phase Matters

The second part of your question is about emulating this rotation using only rotations around the X and Z axis.

The construction you give is X-rot 90°, Z-rot 90°, X-rot 90°. But don't forget to include the phase correction term! The Bloch sphere doesn't track the global phase factor, which is usually fine since it only matters when you're performing a controlled operation, but your particular case does involve controls. That $e^{i\alpha}$ term, which ends up being an $R_z$ gate on the control wire of your construction, is important.

(If you wanted to get the phase right without that term, you could use a longer construction such as $H = X^{-0.5} \cdot Z^{-0.25} \cdot X \cdot Z^{0.25} \cdot X^{0.5}$, although then placing intermediate NOTs won't turn it into an identity operation.)

Verifying the Combined Rotation

A quick way to convince yourself that a combination of rotations matches some desired rotation is to track what it does to each of the six cardinal points. Any rotation that gets all of them right has to be correct for all the other points.

Based on what the Hadamard operation does: we want the top and front points to switch places, the bottom and back points to switch places, and the left and right points to switch places. Your sequence of moves does in fact do that.


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