On the use of P gate vs S gate

[Aerylia]

On page 50 you say that the S gate is sometimes called a P gate, where P stands for phase.

In the few cases where I've seen this distinct difference with their matrix, I noticed that often P equals S up-to-global-phase. In particular, it makes a difference in the use of Controlled-S and Controlled-P (e.g. Benoit's paper on complete re-write rules for the circuit model). I didn't keep a proper account of where this occurred when I was looking into this.

I'm not sure if this is a proper convention or accidental.

[jvdwetering]

Probably in Chapter 9 we should say somewhere clearly that while global phase is not important, when you add a control, it becomes a local phase and hence becomes important.