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As I was watching a video about single qubit quantum gates, I came across the following gate: $$\begin{bmatrix}e^{i\sigma}&0\\0&e^{i\theta}\end{bmatrix}$$ The video didn't mention the name of the gate. It gave only the matrix and the following information:

  • $|0\rangle \rightarrow e^{i\sigma}|0\rangle$
  • $|1\rangle \rightarrow e^{i\theta}|1\rangle$
  • $\alpha |0\rangle + \beta|1\rangle \rightarrow \alpha e^{i\sigma}|0\rangle + \beta e^{i\theta}|1\rangle$

In the comments below the video, someone asked what $\sigma$ and $\theta$ could be, and the reply was all real numbers, but this was from someone other than the video maker, so I don't know if this is true.

What is the name of this gate?

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This is a phase shift gate.

Notice that

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

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

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