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IEEE International Symposium on Information Theory (ISIT)

On Construction of Two Classes of Efficient Quantum Error-Correction Codes

Peiyu Tan, Jing Li

Published Jun 1, 2007• 6 citations
Publisher

Abstract

Stabilizer codes are an important branch of quantum codes, but the existing stabilizer codes constructed from the classical binary codes almost exclusively belong to the special subclass of CSS codes. This paper develops simple and systematic constructions for two rich classes of non-CSS quantum stabilizer codes: quantum low density parity check (LDPC) codes based on classical quasi-cyclic LDPC codes and quantum convolutional codes based on classical LDPC-convolutional codes. Both classes enjoy a wide range of lengths and rates, and offer stronger error correction capability than the existing codes.

DOI
10.1109/isit.2007.4557531

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Stabilizer codes are an important branch of quantum codes, but the existing stabilizer codes constructed from the classical binary codes almost exclusively belong to the special subclass of CSS codes. This paper develops simple and systematic constructions for two rich classes of non-CSS quantum stabilizer codes: quantum low density parity check (LDPC) codes based on classical quasi-cyclic LDPC codes and quantum convolutional codes based on classical LDPC-convolutional codes. Both classes enjoy a wide range of lengths and rates, and offer stronger error correction capability than the existing codes.
Existing quantum stabilizer codes constructed from the classic binary codes exclusively belong to the special subclass of Calderbank-Shor-Steane (CSS) codes. This paper fills in the gap by proposing five systematic constructions for non-CSS stabilizer codes, the first four of which are based on classic binary quasi-cyclic low-density parity-check (LDPC) codes and last on classic binary LDPC-convolutional codes. These new constructions exploit structured sparse graphs, make essential use of simple and powerful coding techniques including concatenation, rotation and scrambling, and generate rich classes of codes with a wide range of lengths and rates. We derive the sufficient, and in some cases also the necessary, conditions for each construction to satisfy the general symplectic inner product (SIP) condition, and develop practical decoder algorithms for these codes. The resulting codes are the first classes of non-CSS quantum LDPC codes and non-CSS quantum convolutional codes rooted from classic binary codes (rather than codes in GF(4)), and some of them perform as well as or better than the existing quantum codes.
Quantum error correction codes are needed to protect quantum systems from interference. An important family of quantum error correction codes are stabilizer codes. Many stabilizer codes in the literature belong to a subclass of stabilizer codes, referred to as CSS codes since they are based on the so-called CSS formalism. CSS codes have received significant attention because the dual-containing condition for the CSS formalism is relatively straightforward to satisfy and also because it is easy to produce CSS codes from classical error correction codes. However, some stabilizer codes in the literature show superior error correction performance than existing CSS codes, and are based on binary low-density parity-check (LDPC) codes. Since it has been shown that CSS codes based on nonbinary LDPC codes have good performance over quantum qubit channels, in this paper we investigate stabilizer codes based on nonbinary LDPC codes. In particular, we propose constructions of stabilizer codes based on nonbinary quasi-cyclic low-density parity-check (QC-LDPC) codes for qubit channels. The simulation results show that our QC-LDPC stabilizer codes decoded by a nonbinary sum-product algorithm have better performance than their binary counterparts.
This paper proposes efficient construction of new classes of quantum low-density parity-check (LDPC) codes through quasi-cyclic matrices, symmetric matrices and scrambling matrices. Unlike many of the existing quantum LDPC codes, the proposed construction is based on classic binary codes rather than quaternary codes, targets the general stabilizer formalism rather than the special CSS formalism, employs systematic approaches rather than computer search, and generates a rich class of codes of different rates and lengths rather than a single code. Simulations demonstrate that these new codes outperform the existing non-CSS LDPC stabilizer codes reported in literature.
In this paper, a new but simple construction of stabilizer codes and related entanglement-assisted quantum error-correcting codes is proposed based on syndrome assignment by classical parity-check matrices. This method turns the construction of quantum stabilizer codes to the construction of classical parity-check matrices satisfying a specific commutative condition. The designed minimum distance 2t*+1 of the constructed quantum stabilizer codes can be achieved by a commutative classical parity-check matrix with classical minimum distance 4t*-m, where the parameterm, 0 ≤m≤ 2t*, depends on a property of the parity-check matrix. Asmdecreases, there is an increasing set of additional correctable error operators beyond the designed error correcting capability t*. The (asymptotic) coding efficiency is at least comparable to that of CSS codes. A class of quantum Reed-Muller codes is constructed and codes in this class have a larger set of correctable error operators than that of the quantum Reed-Muller codes previously developed in the literature. Quantum circulant codes are also constructed and many of them are optimal in terms of their coding parameters.
CSS code is a simple,efficient quantum error correction code construction,and has been found applications in construction of varieties of quantum codes,so far.In this paper,we propose a new construction method of quantum parity check matrix codes based on CSS by taking advantage of the method used in classical quasi-cyclic LDPC codes.Using quick encoding algorithm,we can obtain the quantum code based on this proposed method.At last,a quantum code(3,8)(16,6) is selected as the example to illustrate the process of construction,and its encoding circuit by Controlled-not gate and Hadard gate is also presented.
We investigate the construction of quantum low-density parity-check (LDPC) codes from classical quasi-cyclic (QC) LDPC codes with girth greater than or equal to 6. We have shown that the classical codes in the generalized Calderbank-Shor-Steane (CSS) construction do not need to satisfy the dual-containing property as long as pre-shared entanglement is available to both sender and receiver. We can use this to avoid the many 4-cycles which typically arise in dual-containing LDPC codes. The advantage of such quantum codes comes from the use of efficient decoding algorithms such as sum-product algorithm (SPA). It is well known that in the SPA, cycles of length 4 make successive decoding iterations highly correlated and hence limit the decoding performance. We show the principle of constructing quantum QC-LDPC codes which require only small amounts of initial shared entanglement.