Stabilizer Code - Example of A Stabilizer Code

Example of A Stabilizer Code

An example of a stabilizer code is the five qubit stabilizer code. It encodes logical qubit into physical qubits and protects against an arbitrary single-qubit error. Its stabilizer consists of Pauli operators:

$\begin{array} {ccccccc} g_{1} & = & X & Z & Z & X & I\\ g_{2} & = & I & X & Z & Z & X\\ g_{3} & = & X & I & X & Z & Z\\ g_{4} & = & Z & X & I & X & Z \end{array}$

The above operators commute. Therefore the codespace is the simultaneous +1-eigenspace of the above operators. Suppose a single-qubit error occurs on the encoded quantum register. A single-qubit error is in the set $\left\{ X_{i},Y_{i},Z_{i}\right\}$ where denotes a Pauli error on qubit . It is straightforward to verify that any arbitrary single-qubit error has a unique syndrome. The receiver corrects any single-qubit error by identifying the syndrome and applying a corrective operation.

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