Stabilizer Code - Stabilizer Error-correction Conditions

Stabilizer Error-correction Conditions

One of the fundamental notions in quantum error correction theory is that it suffices to correct a discrete error set with support in the Pauli group . Suppose that the errors affecting an encoded quantum state are a subset of the Pauli group :

An error that affects an encoded quantum state either commutes or anticommutes with any particular element in . The error is correctable if it anticommutes with an element in . An anticommuting error is detectable by measuring each element in and computing a syndrome identifying . The syndrome is a binary vector with length whose elements identify whether the error commutes or anticommutes with each . An error that commutes with every element in is correctable if and only if it is in . It corrupts the encoded state if it commutes with every element of but does not lie in . So we compactly summarize the stabilizer error-correcting conditions: a stabilizer code can correct any errors in if

or

where is the centralizer of .