Stabilizer Codes over Frobenius Rings

Abstract

In quantum information processing, the information is stored in the state of quantum mechanical systems. Since the interaction with the environment is unavoidable, there is a need for quantum error correction to protect the stored information. Until now, the methods for quantum error correction were mainly based on quantum codes that rely on the arithmetic in finite fields. In contrast, this thesis aims to develop a basic framework for quantum error correcting codes over a class of rings known as the Frobenius rings. This thesis focuses on developing the theory of stabilizer codes over the Frobenius rings and provides a systematic construction of codes over these rings. A special class of Frobenius rings called finite chain rings will be the emphasis of this thesis. The theory needed for comparing the minimum distance of stabilizer codes over the finite chain rings to that over the fields is studied in detail. This thesis finally derives that the minimum distance of stabilizer codes over finite chain rings cannot exceed the minimum distance over the fields.