However, it will not be able to correct the product error Next, let's introduce a unitary $U$ such thatĭefining a new code $C'$ as $UCU^\dagger$ (please allow the abuse of notation I hope it's clear what I mean), it must be able to correct errors $X_1$ and $Z_1$. So, the code will not be able to correct this error (or, at least, there surely exist codes that do not correct this error). On the other hand, the product of the two errors is $Y_1Z_2Z_3Z_4$, which contains 4 $Z$ errors. This code can correct an error $X_1Z_2Z_3$ or an error $Z_1Z_4$ because CSS codes are independently distance 5 on the two X/Z types. In that context, the following construction may be of assistance:Ĭonsider a standard distance 5 CSS code, $C$. I've understood from the comments that the OP is willing to consider a more general error set, rather than focussing specifically on the more standard case of distance being a measure of the number of single-qubit errors that can be tolerated.
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