Stabilizer Code

A stabilizer code is a quantum error-correcting code that encodes logical quantum information into a larger set of physical qubits. It defines a protected codespace using a collection of commuting stabilizer operators, often constructed from Pauli operators.

A valid encoded state is typically a +1 eigenstate of every stabilizer operator Si:

Si |ψL⟩ = |ψL⟩

To detect errors, the stabilizer operators are measured. The collection of measurement results is called the error syndrome. If all results have the expected value, usually +1, no error is detected. An unexpected result, such as −1, indicates that an error has occurred and provides information that a classical decoder can use to infer the error and apply a correction.

Stabilizer measurements are designed to reveal information about errors without revealing the encoded logical state. For an error-free codeword, measuring a stabilizer leaves the state unchanged; after an error, the measurement may produce a different eigenvalue and thereby signal the error.