Stabilizer code

Remarks

The weight of a Pauli operator is the number of tensor factors different from $I$. Consider the Pauli operators that commute with every element of $S$ but do not lie in $S$, even up to a phase. The distance $d$ is the least weight of such an operator; one then writes $[\![n, k, d]\!]$. Such a code corrects arbitrary errors on any $\lfloor (d-1)/2 \rfloor$ qubits.

Example: $n = 3$ and $S = \set{I, Z_1 Z_2, Z_2 Z_3, Z_1 Z_3}$, generated by $Z_1 Z_2$ and $Z_2 Z_3$. The group is abelian since all elements are diagonal, and $-I \notin S$. A basis state $\lvert x_1 x_2 x_3 \rangle$ is fixed by $Z_1 Z_2$ and $Z_2 Z_3$ exactly when $x_1 = x_2 = x_3$, so $V_S = \operatorname{span}\set{\lvert 000 \rangle, \lvert 111 \rangle}$, of dimension $2^{3-2} = 2$. Its distance is $1$, because $Z_1$ commutes with $S$ but is not in $S$; it corrects bit flips only.

Non-example: $S = \set{\pm I, \pm Z_1}$ contains $-I$, and then $V_S = \set{0}$, since $-I \psi = \psi$ forces $\psi = 0$. Similarly, a group containing $X_1$ and $Z_1$ is not abelian, and these two operators have no common $+1$ eigenvector.

Standard examples are the five-qubit code $[\![5, 1, 3]\!]$, the Steane code $[\![7, 1, 3]\!]$ and the Shor code $[\![9, 1, 3]\!]$.