In surface codes and color codes, when the code distance is $d$, you can correct up to $[(d-1)/2]$ Pauli errors. I would like to know what this $[(d-1)/2]$ Pauli errors means for $X$, $Y$, and $Z$. Here we consider the case where $d=3$. I understand that one $X$ error can be corrected, two $X$ errors cannot be corrected, one $Z$ error can be corrected, and two $Z$ errors cannot be corrected. However, if one $X$ error and one $Z$ error occur on different qubits, or if one $X$ error and one $Z$ error occur on the same qubit (i.e., a $Y$ error), is it definitely correctable? I would appreciate it if you could also explain the reason for this.

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    $\begingroup$ A single $X$, $Y$ or $Z$ is a Pauli error. Provided there are no more than $\lfloor(d-1)/2\rfloor$ of these, they can be corrected. (Two errors on the same qubit = 1 error). $\endgroup$
    – DaftWullie
    Jun 6, 2023 at 14:22
  • $\begingroup$ @DaftWullie Surface codes and color codes have defined X stabilizers and Z stabilizers, allowing them to correct X errors and Z errors separately. If an X error can be corrected, then a Z error can also be corrected. I think that two errors, one X error and one Z error, occurring on separate qubits can be corrected. Am I misunderstanding something? $\endgroup$
    – david
    Jun 6, 2023 at 14:45
  • $\begingroup$ One X and one Z error occurring on separate qubits = 2 Pauli errors (just as it does for two X errors on different qubits). Provided $2\leq\lfloor(d-1)/2\rfloor$, they can both be corrected. $\endgroup$
    – DaftWullie
    Jun 7, 2023 at 6:33


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