Visualizing a Hadamard gate

If we want to understand the visualization of pure states in Quantum Computing, the most used tool is the Bloch sphere.

Let´s draw the rotations using python and the QuTip library.

The first step is to get a Bloch sphere:

from qutip import *

b = qutip.Bloch()
b.make_sphere()
b.show()

Empty Bloch sphere

We can identify any point in the sphere following this formula:

Being Theta and Phi the following angles:

Over the Bloch sphere, when you apply a Hadamard gate is similar to a rotation of 90º over the Y-axe, followed by a rotation of 180º over the X-axe (it rotates over itself).

Let’s use a function than rotate the state |0> (also with the 180º over X than doesn’t change anything)

Let’s apply a Hadamard gate to the |0> state.

states =  rotate(pi/2,0)
animate_bloch(states, duration=0.1, save_all=False,file_name='output/bloch_anim0.gif')

And also over |1>

As you can see if you apply a Hadamard gate to the state |0> you get a state |+> and if you do it over |1> you get |->

If we want to see it in a non-fundamental state, we need to apply the rotation equation for X-axis.

You can multiply for those matrices to rotate a vector:

If we do it, for example to a vector with these angles:

origin_theta = pi\8
origin_phi = 0

And apply a Hadamard gate with the two rotations:

Hadamard gate applied

So you need two rotations to apply an H gate.

References

Tanay Roy: How to animate a Bloch Sphere

Wikipedia: Esfera de Bloch

Wikipedia: Bloch sphere