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It is easy to see that Pauli's $X$ matrix represents the bit flip operation, i.e. $X \lvert 0 \rangle = \lvert 1 \rangle$ and $X \lvert 1 \rangle = \lvert 0 \rangle$.

Similarly, Pauli's $Z$ matrix represents the phase flip operation, i.e. $Z\cdot z = \overline{z}$ for any $z \in \mathbb{C}$.

But what about Pauli's $Y$ matrix; does it also have a simple interpretation?

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$Y = iXZ = -iZX$, so it can be thought of as both a bit flip and a phase flip, plus an overall $i$ phase.

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    $\begingroup$ My favorite mnemonic is $XYZ=i$ which implies that $Y=iXZ$... $\endgroup$ May 19, 2023 at 4:09
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    $\begingroup$ @AdamZalcman Thanks, fixed. I visualized the circle of $X,Y,Z$ in my head but somehow I went right-to-left instead of left-to-right.Your method seems less error-prone. $\endgroup$ May 19, 2023 at 4:51

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