What is the name of this quantum gate?
This is a phase shift gate.

Notice that

$$\begin{bmatrix}e^{i\sigma}&amp;0\\0&amp;e^{i\theta}\end{bmatrix}=
e^{i\sigma}\begin{bmatrix}1&amp;0\\0&amp;e^{i(\theta-\sigma)}\end{bmatrix}
$$

Multiplying the state by a constant (with unit norm) will not affect your measurements.