Concatenated quantum codes can attain the quantum Gilbert-Varshamov bound

TL;DR

This paper demonstrates that concatenated quantum codes, combining quantum Reed-Solomon outer codes and random stabilizer inner codes, can asymptotically reach the quantum Gilbert-Varshamov bound, advancing quantum coding theory.

Contribution

It generalizes classical results to quantum codes, showing concatenated codes can attain the quantum GV bound with structured code components.

Findings

Existence of concatenated quantum codes reaching the quantum GV bound.

Outer codes are quantum generalized Reed-Solomon codes.

Inner codes are random stabilizer codes within a feasible rate region.

Abstract

A family of quantum codes of increasing block length with positive rate is asymptotically good if the ratio of its distance to its block length approaches a positive constant. 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 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.