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Transactions of the American Mathematical Society

A new construction of quantum error-correcting codes

Keqin Feng, Chaoping Xing

Published Oct 23, 2007• 22 citations
Publisher

Abstract

In this paper, we present a characterization of (binary and non-binary) quantum error-correcting codes. Based on this characterization, we introduce a method to construct p p -ary quantum codes using Boolean functions satisfying a system of certain quadratic relations. As a consequence of the construction, we are able to construct quantum codes of minimum distance 2 2 . In particular, we produce a class of binary quantum ( ( n , 2 n − 2 − 1 2 ( n − 1 ( n − 1 ) / 2 ) , 2 ) ) ((n,2^{n-2}-\frac 12{n-1\choose (n-1)/2},2)) -codes for odd length n ≥ 5 n\ge 5 . For n ≥ 11 n\ge 11 , this improves the result by Rains in Quantum codes of minimal distance two , 1999, showing the existence of binary quantum ( ( n , 3 ⋅ 2 n − 4 , 2 ) ) ((n,3\cdot 2^{n-4},2)) -codes for odd n ≥ 5 n\ge 5 . Moreover, our binary quantum

DOI
10.1090/s0002-9947-07-04242-0

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In this paper, we present a characterization of (binary and non-binary) quantum error-correcting codes. Based on this characterization, we introduce a method to construct p p -ary quantum codes using Boolean functions satisfying a system of certain quadratic relations. As a consequence of the construction, we are able to construct quantum codes of minimum distance 2 2 . In particular, we produce a class of binary quantum ( ( n , 2 n − 2 − 1 2 ( n − 1 ( n − 1 ) / 2 ) , 2 ) ) ((n,2^{n-2}-\frac 12{n-1\choose (n-1)/2},2)) -codes for odd length n ≥ 5 n\ge 5 . For n ≥ 11 n\ge 11 , this improves the result by Rains in Quantum codes of minimal distance two , 1999, showing the existence of binary quantum ( ( n , 3 ⋅ 2 n − 4 , 2 ) ) ((n,3\cdot 2^{n-4},2)) -codes for odd n ≥ 5 n\ge 5 . Moreover, our binary quantum
Stabilizer codes form a special class of quantum error correcting codes. Nonbinary quantum stabilizer codes are studied in this thesis. A lot of work on binary quantum stabilizer codes has been done. Nonbinary stabilizer codes have received much less attention. Various results on binary stabilizer codes such as various code families and general code constructions are generalized to the nonbinary case in this thesis. The lower bound on the minimum distance of a code is nothing but the minimum distance of the currently best known code. The focus of this research is to improve the lower bounds on this minimum distance. To achieve this goal, various existing quantum codes are studied that have good minimum distance. Some new families of nonbinary stabilizer codes such as quantum BCH codes are constructed. Different ways of constructing new codes from the existing ones are also found. All these constructions together help improve the lower bounds.
Abstract Many q-ary stabilizer quantum codes can be constructed from Hermitian self-orthogonal q2q^2q2 q 2 -ary linear codes. This result can be generalized to q2mq^{2 m}q2m q 2 m -ary linear codes, m>1m > 1m>1 m > 1 . We give a result for easily obtaining quantum codes from that generalization. As a consequence we provide several new binary stabilizer quantum codes which are records according to Grassl (Bounds on the minimum distance of linear codes, http://www.codetables.de , 2020) and new q-ary ones, with q≠2q \ne 2q=2 q ≠ 2 , improving others in the literature.
The most popular class of quantum error correcting codes is stabilizer codes. Binary quantum stabilizer codes have been well studied, and Calderbank, Rains, Shor and Sloane (July 1998) have constructed a table of upper bounds on the minimum distance of these codes using linear programming methods. However, not much is known in the case of nonbinary stabilizer codes. In this thesis, we establish a bridge between selforthogonal classical codes over the finite field containing q2 elements and quantum codes, extending and unifying previous work by Matsumoto and Uyematsu (2000), Ashikhmin and Knill (November 2001), Kim and Walker (2004). We construct a table of upper bounds on the minimum distance of the stabilizer codes using linear programming methods that are tighter than currently known bounds. Finally, we derive code construction techniques that will help us find new codes from existing ones. All these results help us to gain a better understanding of the theory of nonbinary stabilizer codes.
Quantum error-correcting codes protect quantum information against undesirable noise in quantum computation and quantum communication.It is one of important problems that constructing very good quantum codes.There are two methods that use binary codes to construct quantum codes,one is CSS construction,another is Steane construction.These two methods are all based on the construction of binary self-orthogonal codes.Combinational construction of binary self-orthogonal codes is investigated in this paper.From known codes chains of binary self-orthogonal codes and their dual codes chains with given dual distance,new binary self-orthogonal codes of dual distance three,four,five and six are constructed.Based on these results,for each satisfying,S-chains with parameters and are constructed.According to Steane's construction,many very good quantum codes of distance five and six are constructed by the obtained S-chains,some of these quantum codes are new and some of these quantum codes have improved parameters than previously known codes.