Stabilizer codes

Definition of stabilizer codes

An $n$- qubit stabilizer code is specified by a list of $n$-qubit Pauli operations, $P_{1},...,P_{r}$. These operations are called stabilizer generators in this context, and they must satisfy the following three properties.

1. The stabilizer generators all commute with one another.

$$P_{j}P_{k} = P_{k}P_{j} \ (\text{for all} j,k \in \{1,...,r\})$$

1. The stabilizer generators form a minimal generating set.

$$P_{k} \notin <P_{1},...,P_{k-1},P_{k+1},...,P_{r}> (\text{for all} \k \in \{1,...,r\})$$

1. At leasr one quantum state vector is fiexed by all of the stabilizer generators.

$$-I^{\otimes n} \notin <P_{1},...,P_{r}>$$