Physics – Quantum Physics
Scientific paper
2010-04-07
Physics
Quantum Physics
10 pages
Scientific paper
A good quantum code corrects for a linear number of errors. It has been shown that random codes attains the QGVB with a relative distance $H^{-1}_{q^2}((1-R)/2)$ where the code is $q$-ary and $R$ is the rate of the code (number of encoded systems / block length). However, random codes have little structure. In this paper, we study a family of concatenated $q$-ary stabilizer codes. Each of the inner codes is a random rate 1 q-ary stabilizer code of block length $n$. The outer code is a quantum MDS code with block length $q^n$ and alphabet size $q^n$, an arbitrary rate $R \le 1$, and distance $N(1-R)/2 +1 $ that meets the Quantum Singleton bound. Fixing the outer code rate and letting $n$ grow, the concatenated stabilizer code has a distance that almost surely attains the QGVB. This partially generalizes Thommesen's result, where he showed that the distance of a concatenated code with a Reed-Solomon outer code and random inner linear codes of arbitrary rate attains the Gilbert-Varshamov bound almost surely.
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