I am trying to figure out the action of logical gates on quantum codes. I am trying to check the logical operations that the logical gate performs on the code.

I can do it in simple cases, for example in the Steane code: $\bar{X}=XXX$ and $\bar{Z}=ZZZ$ are the logical operators. We know that $H$ maps $X$ to $Z$, meaning that if $U=HHH$ is to implement a logical Hadamard then we require $U$ maps $\bar{X}$ to $\bar{Z}$ i.e $U\bar{X}U^{\dagger}=\bar{Z}$.

However, I do not understand how to do this in more complex cases, say for example where we have $U$ is a tensor product of $T$ gates, which map pauli $X$ gates to non-Pauli $SX$ gates. Then $U\bar{X_{i}}U^{\dagger}$ results in tensor products of $SX$ and $I$ (or if $U$ consists of tensor products of $T$ and $T^{\dagger}$, then $U$ maps logical $X$ state $\bar{X_{i}}$ to a tensor product of $SX$ and $S^{\dagger}X$).

However, the tensor product of $SX$ (and maybe $S^{\dagger}X$) does not correspond to another logical operator of the code (as all the logical operators are obtained from different combinations of $\bar{X_{i}}, \bar{Z_{j}}$ (meaning that the logical states are all Pauli products).

How do we deal with figuring out the effect on logical operators in cases such as these, where they map logical states to states outside of the Pauli group?



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