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While reviewing the stabilizer formalism in Nielsen and Chuang, I could not understand why the Toffoli and the $\frac{\pi}{8}$ cannot be described as normalizers, even though they map Pauli group elements to other Pauli group elements. I've attached the screenshot from Nielsen and Chuang as reference below.

Thanks

Normalizer exception N&C

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The key thing is that while each of those gates maps some of the Pauli operators to other Pauli operators (e.g. the left-hand side of 10.92), other Pauli operators (e.g. right-hand side of 10.92) do not get mapped to single Pauli operators but, instead, a linear combination of different Pauli operators: $(X+Y)/\sqrt{2}$ is not in the Pauli group. You need it to be true for all Pauli operators.

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