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arXiv:0901.0042 · IEEE Transactions on Information Theory

A Family of Asymptotically Good Quantum Codes Based on Code Concatenation

Zhuo Li, Lijuan Xing, Xinmei Wang

Published Jul 18, 2009• 21 citations
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Abstract

We explicitly construct an infinite family of asymptotically good concatenated quantum stabilizer codes where the outer codes use quantum Reed-Solomon codes and the inner codes use a set of special quantum codes. In the field of quantum error-correcting codes, this is the first time that a family of asymptotically good quantum codes is derived from bad codes. Its occurrence supplies a gap in quantum coding theory.

DOI
10.1109/tit.2009.2023715

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We explicitly construct an infinite family of asymptotically good concatenated quantum stabilizer codes where the outer codes use quantum Reed-Solomon codes and the inner codes use a set of special quantum codes. In the field of quantum error-correcting codes, this is the first time that a family of asymptotically good quantum codes is derived from bad codes. Its occurrence supplies a gap in quantum coding theory.
Using the idea of code concatenation in classical coding theory, we make the suggestion of constructing general concatenated quantum codes by choosing certain quantum codes as outer code and inner code. Then by concatenating quantum RS outer codes with a set of special quantum inner codes, one can construct a family of concatenated quantum codes, which is asymptotically good. In the field of quantum error-correcting codes, this is the first time that a family of asymptotically good quantum code is constructed using bad classical codes.
AbstractA good quantum code corrects for a linear number of errors. It has been shownthat random codes attains the QGVB with a relative distance H −1q 2 ((1−R)/2) wherethe code is q-ary and R is the rate of the code (number of encoded systems / blocklength). However,randomcodeshavelittlestructure. Inthispaper,westudyafamilyofconcatenatedq-arystabilizercodes. Eachoftheinnercodesisarandomrate1q-arystabilizer code of block length n. The outer code is a quantum MDS code with blocklengthq n andalphabetsizeq n ,anarbitraryrateR≤1,anddistanceN(1−R)/2+1thatmeets 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 heshowed thatthedistanceof aconcatenated code with a Reed-Solomon outer code and random inner linear codes ofarbitraryrateattainstheGilbert-Varshamovboundalmostsurely. 1 Introduction A family of q-ary quantum codes of increasing block length is defined to be good if the ratioof its distance to its block length approaches a non-zero constant. Designing good quantumcodes is highly nontrivial, just as it is in the classical case. The quantum Gilbert-Varshamovbound (QGVB) is a lower bound on an achievable relative distance of a quantum code ofa fixed rate. Explicit constructions of good quantum codes for q ≤ 7 have been studied[1, 2, 3], but they all fail to satisfy the QGVB. The QGVB bound is attainable for the familyof all random quantum codes [4], the family of random stabilizer codes [5], and the familyof random nondegenerate stabilizer codes [6], and the family of random degenerate stabilizercodes [6]. We show that concatenated quantum codes, with a quantum outer code havinga known structure and being efficiently decodable, and randomly chosen independent innerquantum codes also attains the QGVB. Thus our family of random quantum codes has morestructure than previously studied examples.In this paper, we generalize a special case of Thommesen’s result [7] to the quantumcase. He showed that a code from the family of binary concatenated codes made with a1
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 NH-1((1-R)/2), where Hq2is the q2-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.