So, in practise, the point I made about $iY$ not having a +1 eigenspace often doesn't come up. To start with, many of the best liked and interesting codes - from the 9-qubit code and the Steane $\! ]$ code, to the toric code - happen to be CSS codes, whose stabiliser groups are generated by separate X and Z stabilisers any stabiliser operators involving Y, will indeed have Y operators in pairs. The reason (I believe) why Gottesman didn't make the distinction there, that I am making now, is because his main interest in that article is in describing error correcting codes - in which the scalar factors preceding the Pauli operators are in principle less important. It must be said that is not just any old article which uses the Pauli group, nor will the point I made above have escaped its author. And because the operators $\mathbf 1$, $X$, $Y$, and $Z$ form an operator basis for the Hermitian operators on a single qubit, there are often very good theoretical reasons to include the operator $Y$ (in preference to $iY$) in one's analysis. This is only enough to simulate real stabiliser circuits, which is what you're noticing.įor that reason, I would use the first definition and anyone whose work involves taking the Pauli operators first (which consist of $\mathbf 1$, $X$, $Z$, and also $Y$) and then generating a group from them will inevitably use the first definition, for the simple reason that this is the group $\langle X, Y, Z\rangle$. That means that you can only describe states which are stabilised by operators with two or more factors of $\pm i Y$. ![]() ![]() Note that the second definition actually doesn't make more sense in the context of the stabiliser formalism, as neither of $\pm i Y$ have a +1 eigenspace.
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