Let $\vec v \in \mathbb{C}^2 $ be the following quantum state: $$ \vec v = \frac{1}{\sqrt{2}}\begin{bmatrix} v_{1} \\ v_{2} \\ \end{bmatrix},\space \lvert v_1 \rvert = 1, \space \lvert v_2 \rvert = 1 $$

Let $\mathbb{A}$ be the following quantum gate:

$$ \mathbb{A} = cos (\phi)\cdot\sigma_1+sin(\phi)\cdot\sigma_2, $$ where, $\sigma_1,\sigma_2$ are Pauli matrices and the angle $\phi$ is the solution to the equation: $$ \frac{v_2}{v_1} = e^{i\phi} $$

With these definitions the matrix $\mathbb{A}$ has the property: $$\mathbb{A}\vec v = \vec v $$

I am a mathematical physicist not well versed in quantum computing terminology, does this gate have a specific name in the field? I would call it a state control gate. I would believe it must be an important notion in the field, I would welcome any further explanation of the importance of this gate in the field of quantum computing.

Note: $\mathbb{A}$ and $\vec{v}$ can be generalized to $n$-dimensions. The general form of the matrix is the following: $$ \mathbb{A} = \begin{bmatrix} 0 & \frac{v_2}{v_1} & \cdots & \frac{v_n}{v_1}\\ \frac{v_1}{v_2} & 0 & \cdots & \frac{v_n}{v_2}\\ \vdots & \vdots & \ddots & \vdots\\ \frac{v_1}{v_n} & \frac{v_2}{v_n} & \cdots & 0\\ \end{bmatrix}, $$ where, $$ \mathbb{A}\vec{v} = (n-1)\vec{v} $$

In dimension 3 the matrix is a combination of the Gell-mann matrices.



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