Represent Hadamard gate in terms of rotations and reflections in Bloch sphere

Question

I read in a book that any single qubit operation can be decomposed as

$$ \bf{U} =e^{i\gamma}\begin{pmatrix}e^{-i\phi/2}&0\\ 0&e^{i\phi/2}\end{pmatrix}\begin{pmatrix}\cos{\theta/2}&-\sin{\theta/2}\\ \sin{\theta/2}&\cos{\theta/2}\end{pmatrix} $$

I figured out that $\theta$ and $\phi$ are angles on the bloch sphere corresponding to rotations about the y axis and z axis respectively, but how about $\gamma$?

Ultimately, I want to represent the Hadamard gate as a rotation about the y axis by $\pi/2$ followed by a reflection through the x-y plane.

Answer 1

but how about γ?

Gamma doesn't show up on the Bloch sphere. It's a global phase. It's unobservable without conditioning the operation on a second qubit, in which case it turns into phase kickback onto that qubit.

want to represent the Hadamard gate as a rotation about the y axis by π/2 followed by a reflection through the x-y plane.

Reflection in the Bloch sphere doesn't correspond to a physical operation.

Bloch sphere reflection would allow any qubit to be used to conjugate the wavefunction. Any other qubit could be used to detect that conjugation. This allows you to do things like construct the "impossible universal NOT gate" and communicate faster than light:

Answer 2

The Hadamard gate is a $\pi$ rotation about the diagonal axis in the XZ-plane.

It is not a $\pi/2$ rotation about the $y$ axis. This can be easily seen from the fact that the Hadamard squares to the identity; it must thus be a rotation by an angle of $\pi$, not $\pi/2$.