The $S$ gate has the following effect (when applied to a Pauli operator $P$ by conjugation $SPS^{\dagger}$):
$$S: X \rightarrow Y $$ $$S: Z \rightarrow Z $$
The logical operators for the $[[4,2,2]]$ code can be defined as follows: $$X_{L1}= XIXI$$ $$X_{L2}=XXII$$ $$Z_{L1}=ZZII$$ $$Z_{L2}=ZIZI$$
I have introduce $SSSS$ as a potential new logical gate for the $[[4,2,2]]$ code. I want to find its logical effect on the logical operators.
I know that $$SSSS: XIXI \rightarrow YIYI $$ $$SSSS: XXII \rightarrow YYII$$ $$SSSS: ZZII \rightarrow ZZII$$ $$SSSS: ZIZI \rightarrow ZIZI$$
In other words, if we denote $SSSS$ as $S_{L}$, then:
$$S_{L}: X_{L1} \rightarrow - X_{L1}Z_{L2} $$ $$S_{L}: X_{L2} \rightarrow - X_{L2}Z_{L1} $$ $$S_{L}: Z_{L1} \rightarrow Z_{L1} $$ $$S_{L}: Z_{L2} \rightarrow Z_{L2} $$
I have read on this post that $S_{L}$ implements a logical $CZ$ times a logical $ZZ$. However, I do not understand how this is the case.
Let me implement a logical $CZ$ followed by a logical $ZZ$ on the following, to check if the effect is the same:
$$X\otimes I \xrightarrow{CZ} X \otimes Z \xrightarrow{ZZ} ZX \otimes I$$ $$I \otimes X \xrightarrow{CZ} Z \otimes X \xrightarrow{ZZ} I \otimes ZX$$ $$Z \otimes I \xrightarrow{CZ} Z \otimes I \xrightarrow{ZZ} I \otimes Z$$ $$ I \otimes Z \xrightarrow{CZ} I \otimes Z \xrightarrow{ZZ} Z \otimes I$$
Which instead seems to have the effect:
$X_{L1} \rightarrow -X_{L1}Z_{L1}$
$X_{L2} \rightarrow -X_{L2}Z_{L2}$
$Z_{L1} \rightarrow Z_{L2} $
$Z_{L2} \rightarrow Z_{L1}$
Can anyone identify where I am going wrong?
