t b a b a

if:
Mj | ϕk〉 = (−1) | ϕk〉.
then | ϕk〉 = Ei | ψk〉, the state | ϕk〉 is a codeword | ψk〉 ∈ Q affected by error Ei. The
error operators affecting codewords in Q, E = {E1, E2 . . .} are also a subgroup of the n-qubit
Pauli group
E ⊂ Gn.
Each error operator Ei is a tensor product on n Pauli matrices. Its weight is equal to the
number of errors affecting a quantum word, thus with the number of Pauli matrices other
than σI.
The coding space
Q = {| ψ〉 ∈ Hn such that Mj | ψ〉 = (+1) | ψ〉, ∀Mj ∈ S }
is the space of all vectors | ψ〉 fixed by S.
It is easy to prove that S is a stabilizer of a non-trivial Hilbert subspace V2n ⊂ H2n if
and only if:
1. S = {S1, S2, . . .} is an Abelian group:
SiSj = SjSi, ∀Si, Sj ∈ S i �= j.
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