t b a b a

2. The identity matrix multiplied by −1 is not in S:
−σ ⊗n
I
If E is an error operator, and E anti-commutes with some element M ∈ S, then E can
be detected, since for any | ψi〉, | ψj〉 ∈ Q:
/∈ S.
ME | ψi〉 = −EM | ψi〉 = −E | ψi〉
If our stabilizer code is a [n, k, d] code, the cardinality of the stabilizer S and its generator
M are:
| S |= 2 n−k , | M |= n − k
The error syndrome corresponding to the stabilizer Mj is a function of the error operator,
E, defined as:
fMj (E) : G ↦→ Z2 fMj
⎧
⎪⎨
(E) =
⎪⎩
0 if [Mj, E] = 0
Let f(E) be the (n − k)-bit integer given by the binary vector:
1 if {Mj, E} = 0.
f(E) = (fM1(E)fM2(E) . . . fMn−k (E)).
This (n − k)-bit integer is called the syndrome of error E.
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