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I have a little difficulty with understanding. How do I properly visualize the change of qubit's basis as a rotation?

Let's say that we have classical basis vectors, $|0\rangle$ and $|1\rangle$. Now, we can change this basis into the $\{|+\rangle, |-\rangle\}$ one by applying a Hadamard gate:

$$ H|0\rangle = \frac{1}{\sqrt{2}}\begin{bmatrix} 1 & 1 \\ 1 & -1 \end{bmatrix} \begin{bmatrix} 1 \\ 0 \end{bmatrix} = \frac{1}{\sqrt{2}} \begin{bmatrix} 1 \\ 1 \end{bmatrix} = \frac{|0\rangle + |1\rangle}{\sqrt{2}} = |+\rangle, $$

$$ H|1\rangle = \frac{1}{\sqrt{2}}\begin{bmatrix} 1 & 1 \\ 1 & -1 \end{bmatrix} \begin{bmatrix} 0 \\ 1 \end{bmatrix} = \frac{1}{\sqrt{2}} \begin{bmatrix} 1 \\ -1 \end{bmatrix} = \frac{|0\rangle - |1\rangle}{\sqrt{2}} = |-\rangle. $$

Is there some analogy between the Hadamard gate and classical rotation matrix, which can be defined as

$$ R(\theta) = \begin{bmatrix} \cos \theta & -sin \theta \\ \sin \theta & \cos \theta \end{bmatrix},$$

for a rotation in the counterclockwise direction by the angle of $\theta$?

I think that my problem lies in viewing the vectors $|0\rangle$ and $|1\rangle$ as two perpendicular vectors on a plane of real numbers. But, we are dealing with two vectors of **complex** numbers, so I guess we shouldn't think of them in this way, but rather than that, use the Bloch sphere?

Might be interesting: a nice blog post about VQE, where basis changes are obtained via rotations around the X and Y axis in the Bloch sphere https://www.mustythoughts.com/post/variational-quantum-eigensolver-explained.

– Davit Khachatryan – 2020-03-14T12:22:41.600