Quantum Error Correction Methods

This study surveys the mathematical structure of a quantum error correcting codes and the way they are developed through certain stages of error correction. In particular, the families of Calderbank-Shor-Steane codes (CSS) and the stabilizer codes are discussed and through elaborative examples it will be shown that the CSS codes are in the family of the stabilizer codes. Since the study of the CSS codes depends on a firm knowledge of classical coding theory, a rigorous mathematical review of the linear codes is done separately. Analysing the structure of the stabilizer formalism is highly depended on the effective use of some group theoretic notions. This structure is discussed in more detail and examples will be given. As the ultimate application of the quantum error correction the rules of the fault-tolerant quantum computing is explored and finding the threshold condition of an example will be done. Keywords: QEC, Coding theory, Stabilizer formalism, CSS codes, Fault-tolerant quantum computing, Threshold condition