Other, pp.1-97, 2017
Though classical computers have been developed day by day, a new machine which is based on quantum mechanics and is called quantum computer is expected more powerful than a classical one. For instance, RSA which is a powerful cryptographic algorithm in classical computers is used in recent security systems and this algorithm cannot be cracked by using a classical computer. However, it is expected that a quantum computer can easily break this algorithm. If these computers can be built in practice, then a quantum error correction process which is based on principles of quantum mechanics is needed. Hence, quantum error correcting codes have been developed. In this thesis, in Chapter 1 development of quantum coding theory from the beginning to today is mentioned and many studies made in this process are explained. In Chapter 2, main definitions and theorems of algebraic coding theory is gave to place. In Chapter 3, notations, matrices, operators which are used in quantum computation and their operations have been explained with properties. In Chapter 4, the differences of quantum error correction from classical error correction and quantum error correcting codes are explained. CSS code, stabilizer code and entanglement assisted quantum codes are analyzed in detail and they are illustrated with examples. In Chapter 5, quantum codes over Eisenstein-Jacobi integers have been constructed. Error detection and correction procedures of this new type of quantum codes have been explained and they are intensified with examples. Error matrices, error bases and a new distance of these codes are defined. Commutative condition of error operators is given and it is proved. Finally, it is showed that these new codes may give new and better parameters. Keywords: Quantum codes, error correcting codes, CSS code, stabilizer code, entanglement, Eisenstein- Jacobi integers.