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arXiv:1407.6228 · International Journal of Quantum Information

Quantum stabilizer codes from Abelian and non-Abelian groups association schemes

Avaz Naghipour, M. A. Jafarizadeh, S. Shahmorad

Published Apr 1, 2015• 10 citations
arXivHTMLPDF

Abstract

A new method for the construction of the binary quantum stabilizer codes is provided, where the construction is based on Abelian and non-Abelian groups association schemes. The association schemes based on non-Abelian groups are constructed by bases for the regular representation from U6n, T4n, V8n and dihedral D2n groups. By using Abelian group association schemes followed by cyclic groups and non-Abelian group association schemes a list of binary stabilizer codes up to 40 qubits is given in tables 4, 5 and 10. Moreover, several binary stabilizer codes of minimum distances 5, 7 and 8 with good quantum parameters is presented. The preference of this method specially for Abelian group association schemes is that one can construct any binary quantum stabilizer code with any distance by using the commutative structure of association schemes.

DOI
10.1142/s0219749915500215

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A new method for the construction of the binary quantum stabilizer codes is provided, where the construction is based on Abelian and non-Abelian groups association schemes. The association schemes based on non-Abelian groups are constructed by bases for the regular representation from U6n, T4n, V8n and dihedral D2n groups. By using Abelian group association schemes followed by cyclic groups and non-Abelian group association schemes a list of binary stabilizer codes up to 40 qubits is given in tables 4, 5 and 10. Moreover, several binary stabilizer codes of minimum distances 5, 7 and 8 with good quantum parameters is presented. The preference of this method specially for Abelian group association schemes is that one can construct any binary quantum stabilizer code with any distance by using the commutative structure of association schemes.
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.
In this short letter, our main objective is to construct good quantum stabilizer codes over small fields, which have a more potential possibility to be employed in actual quantum processors. We firstly consider a suitable class of two-generators quasi-cyclic (QC) codes and determine their parameters. Then by virtue of their symplectic dual algebraic structure, we derive a sufficient condition to be symplectic dual-containing of these codes. Subsequently, an explicit construction of stabilizer codes from quasi-cyclic codes is presented. As for computational results, we provide many stabilizer codes with good parameters over binary, ternary and quaternary fields that all can not be deduced by the quantum Gilbert-Varshamov (GV) bound. In the binary case, our several codes strictly improve the lower bounds on the minimum distance in current records. In the ternary and quaternary occasions, numerous codes fill some gaps or have better parameters than the current results.
We propose a systematic scheme for the construction of graphs associated with binary stabilizer codes. The scheme is characterized by three main steps: first, the stabilizer code is realized as a codeword-stabilized (CWS) quantum code; second, the canonical form of the CWS code is uncovered; third, the input vertices are attached to the graphs. To check the effectiveness of the scheme, we discuss several graphical constructions of various useful stabilizer codes characterized by single and multi-qubit encoding operators. In particular, the error-correcting capabilities of such quantum codes are verified in graph-theoretic terms as originally advocated by Schlingemann and Werner. Finally, possible generalizations of our scheme for the graphical construction of both (stabilizer and nonadditive) nonbinary and continuous-variable quantum codes are briefly addressed.
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.
We propose a systematic scheme for the construction of graphs associated with binary stabilizer codes. The scheme is characterized by three main steps: first, the stabilizer code is realized as a codeword-stabilized (CWS) quantum code; second, the canonical form of the CWS code is uncovered; third, the input vertices are attached to the graphs. To check the effectiveness of the scheme, we discuss several graphical constructions of various useful stabilizer codes characterized by single and multi-qubit encoding operators. In particular, the error-correcting capabilities of such quantum codes are verified in graph-theoretic terms as originally advocated by Schlingemann and Werner. Finally, possible generalizations of our scheme for the graphical construction of both (stabilizer and nonadditive) nonbinary and continuous-variable quantum codes are briefly addressed.