Repetition code

a|0⟩ + b|1⟩ → a|000⟩ + b|111⟩

• “A simple way to perform this encoding is to attach two ancilla qubits in the state |0⟩ to the original qubit and perform a CNOT gate from the first qubit to the second and from the first to the last.”

• “Encoded qubit protected against noise where either the first, second, or third qubit is subjected to a bit flip (and no other errors occur).

  • For example, if a bit flip occurs on the second qubit, the encoded state changes as follows:

    X₂ (a|000⟩ + b|111⟩) = a|010⟩ + b|101⟩
    

• The procedure for the receiver to recover from such an error is to perform collective measurements on all three qubits that learn only about the error; Nothing about the encoded quantum data.

  • The receiver can perform a measurement of the operators Z₁Z₂ and Z₂Z₃ to learn only about the error. Superposition is preserved.”

• Is this a Stabilizer code? The encoded state is stabilized by {ZZI, IZZ}.

References

• Wilde2011;