Frobenius Automorphism and Quantum Cyclic Codes

Efficient quantum codes are a necessary building block in the eventual design of quantum computers. Despite some of the strange laws of quantum physics that seems difficult to handle --- the no cloning theorem for example which states one cannot make copies of quantum information --- the theory of quantum codes, in particular stabilize codes, have a remarkable similarity to that of classical linear codes. The talk aims at giving a general introduction to the theory of quantum codes followed by a discussion on some of our recent work in this area. Starting with classical linear codes, we quickly build up the necessary background on quantum stabilizer codes.

Having built the necessary background, I describe, in the second part of the talk, our recent work on constructing families of cyclic codes which we call the Frobenius codes. These codes include the well known Laflamme [5,1,3] code. We give efficient decoding algorithms that generalise the Berlekamp algorithm for classical cyclic codes. The theory we develop also proves that cyclic quantum codes for certain lengths do not exists. These negative results for quantum codes are purely field theoretic and not based on packing bounds (this is joint work with Sagarmoy Dutta).