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arXiv:1906.04964 · IEEE International Symposium on Information Theory (ISIT)

Good Stabilizer Codes from Quasi-Cyclic Codes over F 4 and F 9

Martianus Frederic Ezerman, San Ling, Buket Özkaya, Patrick Solé

Published Jul 1, 2019• 29 citations
arXivHTMLPDF

Abstract

We apply quantum Construction X on quasi-cyclic codes with large Hermitian hulls over F4and F9to derive good qubit and qutrit stabilizer codes, respectively. In several occasions we obtain quantum codes with stricly improved parameters than the current record. In numerous other occasions we obtain quantum codes with best-known performance. For the qutrit ones we supply a systematic construction to fill some gaps in the literature.

DOI
10.1109/isit.2019.8849416

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We apply quantum Construction X on quasi-cyclic codes with large Hermitian hulls over F4and F9to derive good qubit and qutrit stabilizer codes, respectively. In several occasions we obtain quantum codes with stricly improved parameters than the current record. In numerous other occasions we obtain quantum codes with best-known performance. For the qutrit ones we supply a systematic construction to fill some gaps in the literature.
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.
Construction of quantum codes and entanglement-assisted quantum codes with good parameters via classical codes is an important task for quantum computing and quantum information. In this paper, by a family of one-generator quasi-cyclic codes, we provide quasi-cyclic extended constructions that preserve the self-orthogonality to obtain stabilizer quantum codes. As for the computational results, some binary and ternary stabilizer codes with good parameters are constructed. Moreover, we present methods to construct maximal-entanglement entanglement-assisted quantum codes by means of the class of quasi-cyclic codes and their extended codes. As an application, some good maximal-entanglement entanglement-assisted quantum codes are obtained and their parameters are compared.
In this paper, we introduce a class of dualcontaining 2 -generator quasi-cyclic codes over GF(4)G F(4)GF(4). By Hermitian construction, we show computation constructions of binary quantum codes with good parameters from quaternary quasi-cyclic codes. Furthermore, 3 new quantum codes [[42, 10, 9]], [[74,,36, 9]] and [[78, 40, 9]] are derived from Hermitian dual-containing quasi-cyclic codes by an exhaustive search, along with 24 newly derived codes that enhance the previously best-known lower bounds on minimum distance.
Building large-scale quantum computers is a very active area of research. A critical part of this effort is the ability to control quantum errors. Recently, many good quantum error correcting codes (QECC) over various finite fields Fq have been constructed from codes over extension rings or mixed alphabet rings via some version of a Gray map. We show that most of these codes can be obtained more directly from classical codes, including cyclic, constacyclic, quasicyclic (QC), quasi-twisted (QT), and polycyclic codes using the CSS construction, Hermitian construction, and Lisonek-Singh methods. We believe direct and more elementary methods should be preferred.