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For all stabilizer codes, we can represent its stabilizer generators using a standard form

$$ H = \left(\begin{array}{ccc|ccc} I & A_1 & A_2 & B & 0 & C \\ 0 & 0 & 0 & D & I & E \end{array}\right), $$ where the left half is for the $Z$ check and the right half is for the $X$ check.

However, the bottom-half block of $H$, which represents stabilizer generators containing only $I$ or $X$ (As it only contains one type of Paulis, I call it pure stabilizers ), is not necessary for a stabilizer code. For example, the $[[5, 1, 3]]$ code doesn't have pure stabilizers. In contrast, every stabilizer of CSS codes is pure stabilizer.

Is there any special properties about the stabilizer codes with no pure stabilizers? Does it necessarily perform better for error correction purposes? I would really appreciate if any can share any related literatures about this topic.

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  • $\begingroup$ As far as performance of the code as a QECC I don't see how pure stabilizers affect things. Now if you throw in "circuit noise" (which I'm still waiting to see a clear definition for) then things might change a little : pure stabilizers might have more favorable circuit implementations... $\endgroup$
    – unknown
    Mar 30 at 15:29

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