Stabilizer code

• “Most of the Quantum error correction codes that have been seriously considered are Stabilizer codes. The reason for this is similar to why Binary linear code play such a central role in classical error correction: namely, (1) it makes everything ​much​ easier to calculate and reason about, and (2) by insisting on it, we don’t seem to give up any of the error-correcting properties we want. As a result, the Stabilizer formalism is the lingua franca of Quantum error-correction; it’s completely indispensable there.”

• “A Stabilizer code is a quantum error-correcting code in which the encoding and decoding (at least if there are no errors!) can be done entirely by Stabilizer circuits. In particular, this means that all the code states are Stabilizer states.”

import Quantum_error_correction
import Stabilizer_circuit

def stabilizer_code := sorry

• Quantum codes satisfy certain properties;

Valid encoded operations in a stabilizer code;

Errors that anticommute with the stabilizer get detected;

Examples of Stabilizer codes:

• Surface code;

• Quantum code from two binary linear codes;

  • The five-qubit code;

  • Shor’s nine-qubit code;

  • Repetition code;

• Quantum code from two binary linear codes on expander graphs;

References

• Aaronson;

• http://codetables.de ; What is the largest distance for a given n and k?