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Quantum Quasi-Cyclic LDPC Codes

Manabu Hagiwara, Hideki Imai

Published Jan 1, 2007• 66 citations
OpenAlex

Abstract

Abstract — In this paper, a construction of a pair of quasicyclic LDPC codes to construct a quantum error-correcting code is proposed. Our construction method is based on algebraic combinatorics and have lots of variations for length, code rate. I.

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Apologizing Comment on `Quantum Quasi-Cyclic Low-Density Parity-Check codes \,"
In our recent paper entitled "Quantum Quasi-Cyclic Low-Density Parity-Check codes" [ICIC 2009. LNCS 5754], it was claimed that some new quantum codes can be constructed via the CSS encoding/decoding approach with various lengths and rates. However, the further investigation shows that the proposed construction may steal some ideas from the paper entitled "Quantum Quasi-Cyclic LDPC codes" [quant-ph/0701020v2]. We feel that the apologizing point of the original protocol is that some results are almost similar to that of construction methods with algebraic combinatorics although we suggest the different approach for improving them. Also, there is a weak point of the original coding approach while considering the application of codes in imperfect channels.
In this paper, we propose a construction method for two-level quantum error-correcting codes via non-binary LDPC codes over an extended field of order 2p, p an integer p >; 1. The proposed quantum error-correcting codes are based on binary quasi-cyclic LDPC codes which have almost achieved a “Bounded Distance Decoding (BDD)” limit but have not surpassed the limit yet. Quantum codes constructed from the proposed method surpass the BDD limit. Furthermore the codes outperform the efficiently-decodable state-of-the-art quantum codes.
In the late 1950's and early 1960's, finite fields were successfully used to construct linear block codes, especially cyclic codes, with large minimum distances for correcting random errors with algebraic decoding, such as Bose-Chaudhuri-Hocqenghem (BCH) and Reed-Solomon (RS) codes. Recently it has been shown that finite fields can also be used successfully to construct binary quasi-cyclic (QC)-LDPC codes that perform very well not only over the AWGN channel but also over the binary erasure channel with iterative decoding, besides being efficiently encodable. This paper is concerned with constructions of nonbinary QC-LDPC codes based on finite fields
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.