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Journal of Physics Conference Series

Quantum Codes Constructed by Quantum Caps in PG (4, 5)

Chaofeng Guan, Ruihu Li, Husheng Li, Xuizhen Zhan, Yu Yao, Shiwei Lu, Xuan Chen

Published Jun 1, 2021• 0 citations
Publisher

Abstract

Abstract In this paper, 57 quantum caps in PG (4,5) are found using a computer-supported search method, and related quantum error-correcting codes are constructed via CSS construction. By comparison, most of these codes have good parameters, and some of them enhance the existing results.
DOI
10.1088/1742-6596/1948/1/012152

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•Jul 1, 2024•0 citations
In this paper, we present a computer-supported method of searching for quantum caps. By means of this method and relevant knowledge of combinatorics, many quantum caps inPG(3,9)PG(3,9)PG(3,9)andPG(4,9)PG(4,9)PG(4,9)are constructively proven to exist. Then, according to the theorem that each quantum cap corresponds to a quantum error-correcting code withd=4d=4d=4, we obtain 278 quantum error-correcting codes. Most of these results break the GV bound, and a number of them are optimal quantum codes or have improved parameters.
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Calderbank, Rains, Shor and Sloane (see \cite{Sloane}) showed that error-correction is possible in the context of quantum computations. Quantum stabilizer codes are a class of additive quaternary codes in binary projective spaces, which are self-orthogonal with respect to the symplectic form. A geometric description is given in \cite{Bierbra}, where also the notion of quantum cap is introduced. Quantum caps correspond to the special case of quantum stabilizer codes of distance d=4d=4d=4 when the code is linear over GF(4). In the present paper we review the translation from quantum error-correction to symplectic geometry and study quantum codes in PG(4,4) where we construct complete quantum caps with 20, 29, 30, 32, 33, 34, 36 and 38 points and incomplete quantum caps with 10, 12, 13, 20, 23, 24, 25 and 26 points and we prove the non existence of 11-quantum caps. In particular the quantum caps of sizes 36 and 38 yield positive answers to the existence questions of quantum codes [[36,26,4]][[36,26,4]][[36,26,4]] and [[38,28,4]][[38,28,4]][[38,28,4]] that remained open in the data base \cite{codetable}.
Abstract Calderbank, Rains, Shor, and Sloane (see [10]) showed that quantum stabilizer codes correspond to additive quaternary codes in binary projective spaces, which are self‐orthogonal with respect to the symplectic form. A geometric description is given in [8, 19]. In [8] the notion of a quantum cap is introduced. Quantum caps are equivalent to quantum stabilizer codes of minimum distance when the code is linear over . In this paper, we determine the valuesksuch that there exists a quantumk‐cap in , corresponding to pure linear quantum codes, proving, by exhaustive search, that no 11, 37, 39‐quantum caps exist. Moreover we give examples of quantum caps in not already known in the literature.
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.