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The paper A Fully Fault-Tolerant Representation of Quantum Circuits uses the $P$ gate which is given as: $$P = \begin{bmatrix}1 & 0\\0 & i\\\end{bmatrix}.$$

Nielsen and Chuang Quantum Computation and Information 10th edition uses the $S$ gate which is also given as: $$S = \begin{bmatrix}1 & 0\\0 & i\\\end{bmatrix}.$$

Why does the same gate has different names?

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Denoting phase gate by $P$ probably comes from its name Phase

Phase gate is usually denoted by $S$ as you can see from your Nielsen and Chuang book example. Moreoer, symbol $S$ is used also on IBM Q.

To sum up, it is a matter of a paper's author decission. As I remember, I have seen phase gate denoted by $S$ only.

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