Suppose we have a $[[n,2,d]]$ stabilizer code whose one choice of basis of logical operators is $\{\bar{X}_1,\bar{Z}_1\}$ and $\{\bar{X}_2,\bar{Z}_2\}$. We can also choose different basis $\{\bar{X}_1\bar{X}_2,\bar{Z}_1\}$ and $\{\bar{X}_2,\bar{Z}_1\bar{Z}_2\}$. The question is, if the weight of the lowest-weight logical operators of the former case is $d_1$, and the latter case is $d_2$ ($d_1<d_2$), what is the distance of this code? One way of thinking this is that the distance should be determined by the lowest-weight logical operator among all possible basis, so $d_1$ is the distance of the code. Another way of thinking is that once we decide the basis of the logical operators, distance should be the lowest-weight logical operator of that basis, so we can use this code as $[[n,2,d_2]]$ code by choosing $\{\bar{X}_1\bar{X}_2,\bar{Z}_1\}$ and $\{\bar{X}_2,\bar{Z}_1\bar{Z}_2\}$ as the basis of the two logical qubits. I am confused which way of thinking is true.
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$\begingroup$ Also remember that if you have a stabilizer $g$, $\bar X_1 g$ also implements the first logical $X$, but could be lower weight. $\endgroup$– DaftWullieCommented Nov 25 at 7:13
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$\begingroup$ "once we decide the basis of the logical operators, distance should be the lowest-weight logical operator of that basis" - this is your problem: you actually need the lowest weight (nontrivial) operator in the "span" of that basis (i.e. the group generated by the basis), not just the lowest weight element of the basis. $\endgroup$– forky40Commented Nov 27 at 18:16
2 Answers
The distance of a (subspace) code is, by definition, the lowest weight operator that preserves the subspace while not acting identically on it. This definition is independent of whatever basis you choose for the subspace.
One way to think about this clearly: you can define many different logical Z operators. The distance of these operators has nothing to do with how well you protect against bit-flip errors. The protection against bit-flip errors is determined by the smallest operator that anticommutes with your logical Z operator, i.e. the shortest X logical operator. Similarly, the protection against phase flip errors is determined by the shortest Z logical operator.
The reason that increasing the length of the logical operator doesn't increase the distance of your code is that it has no effect on what actually matters: the shortest-distance operator that anticommutes with your logical operators.
