1004.1127 (Yingkai Ouyang)

Concatenated quantum codes can attain the quantum Gilbert-Varshamov
bound    [PDF]

Yingkai Ouyang

A good quantum code corrects a linear number of errors. The asymptotic quantum Gilbert-Varshamov (GV) bound states that there exist $q$-ary quantum codes of sufficiently long block length $N$ having fixed rate $R$ with distance at least $N H^{-1}_{q^2}((1-R)/2)$, where $H_{q^2}$ is the $q^2$-ary entropy function. For $q < 7$, only random quantum codes are known to asymptotically attain the quantum GV bound. However, random codes have little structure. In this paper, we generalize the classical result of Thommesen \cite{Tho83} to the quantum case, thereby demonstrating the existence of concatenated quantum codes that can asymptotically attain the quantum GV bound. The outer codes are quantum generalized Reed-Solomon codes, and the inner codes are random independently chosen stabilizer codes, where the rates of the inner and outer codes lie in a specified feasible region.

View original: http://arxiv.org/abs/1004.1127