t b a b a

5.5 Stabilizer Codes
The stabilizer formalism is a succinct manner to describe a quantum error correcting code
by a set of quantum operators [Got97]. We first review several concepts and properties of
stabilizer codes.
The 1-qubit Pauli group G1 consists of the Pauli matrices, σI, σx, σy, and σz together
with the multiplicative factors ±1 and ±i:
The generators of G1 are:
G1 ≡ {±σI, ±iσI, ±σx, ±iσx, ±σy, ±iσy, ±σz, ±iσz}.
〈σx, σz, iσI〉.
Indeed, every element of G1 can be expressed as a product of a finite number of generators.
For example:
−σx = iσIiσIσx, + iσx = iσIσx, − iσx = iσIiσIiσIσx.
The n-qubit Pauli group Gn consists of the 4 n tensor products of σI, σx, σy, and σz and an
overall phase of ±1 or ±i. Elements of the group can be used to describe the error operators
applied to an n-qubit register. The weight of such an operator in Gn is equal to the number
of tensor factors which are not equal to σI.
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