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.


Normalizer exception N&C


1 Answer 1


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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