Additive Quantum Error Correcting Code, The Nature Of Research

Because there exist quantum errors and quantum decoherences, the quantum information will be changed as the information transmitting through a quantum channel. By using quantum error correction, it can decrease or even eliminate the influence on the quantum information when the error occurs independently in different qubits. Therefore, quantum error correction is a promise for reliable quantum computation and quantum communication. An ((n, K)) quantum code is a K dimension linear subspace of a 2~n dimension Hilbert space. In 1998, Calderbank et al. create one of the most important class of quantum error correcting codes-additive quantum error correcting codes. Since the problem of finding a certain additive quantum error correcting code can be translated into a problem of finding a corresponding self-orthogonal binary code with respect to symplectic inner product, it enable us to study the quantum error correcting codes directly from a point of mathematic problem rather than from a point of quantum mechanical problem. Hence it greatly simplifies the study on quantum error correcting codes. In this thesis we study several topics of additive quantum error correcting codes.In Chapter 1, we give an introduction on the background of quantum error correcting codes, the basic defination and terminology of classical error correcting codes and quantum error correcting codes, and the relationship between additive quantum error correcting codes with self-orthogonal binary codes and self-orthogonal additive codes. We survey what is presently known about the additive quantum error correcting codes at the end of this chapter.By using Markov inequality and Chebyshev inequality, In Chapter 2 we give the asymptotic probability distribution of the relative minimal distance of the additive quantum error correcting codes. And as an application, this distribution property is uesd to reprove the Gilbert-Varshamov bound.By using Krawtchouk polynomials, In Chapter 3 we give a linear program-...