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IEEE International Symposium on Information Theory (ISIT)

A class of quantum LDPC codes: construction and performances under iterative decoding

Thomas Camara, Harold Ollivier, Jean–Pierre Tillich

Published Jun 1, 2007• 49 citations
Publisher

Abstract

A generic method for constructing quantum LDPC codes is presented. We first explain how to overcome the difficulty of finding a set of low weight generators for the stabilizer group of the code. Our approach is based on a graph representation of the generators of the stabilizer group and on a simple local rule to ensure commutativity. We provide several specific examples of quantum LDPC codes obtained by our method, together with numerical simulations over the depolarizing channel and the erasure channel.

DOI
10.1109/isit.2007.4557324

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A generic method for constructing quantum LDPC codes is presented. We first explain how to overcome the difficulty of finding a set of low weight generators for the stabilizer group of the code. Our approach is based on a graph representation of the generators of the stabilizer group and on a simple local rule to ensure commutativity. We provide several specific examples of quantum LDPC codes obtained by our method, together with numerical simulations over the depolarizing channel and the erasure channel.
Two methods for constructing quantum LDPC codes are presented. We explain how to overcome the difficulty of finding a set of low weight generators for the stabilizer group of the code. Both approaches are based on some graph representation of the generators of the stabilizer group and on simple local rules that ensure commutativity. A message passing algorithm for generic quantum LDPC codes is also introduced. Finally, we provide two specific examples of quantum LDPC codes of rate 1/2 obtained by our methods, together with a numerical simulation of their performance over the depolarizing channel.
Nonbinary quantum codes are more suitable for error-correction in multi-level quantum systems.Based on stabilizer formalism over the finite field,a class of nonbinary quantum LDPC codes is presented in this paper.And the BP iterative decoding algorithm for these codes is described under the quantum depolarizing channel with the Monte Carlo simulation method.For example,a class of CSS structure 4-ary quantum LDPC codes is given with code rates 1/2 and 1/4.Compared with the present binary quantum LDPC codes of equivalent codelength,the channel threshold with 10-5FER is improved from 0.016 to 0.025.
We propose a method of constructing quantum LDPC codes from multiplicative groups of order p - 1, where p = 4n + 1 is a prime for some positive integer n. The proposed quantum LDPC codes of non-CSS structure are constructed from a pair of classical regular quasi-cyclic (QC) LDPC codes. We show that these classical regular QC-LDPC codes without cycles of length 4 are orthogonal with respect to the symplectic inner product. Moreover, the proposed construction method yields a large number of new quantum LDPC codes with various code lengths and rates. The performance of the proposed quantum LDPC codes over quantum depolarizing channels with an iterative belief-propagation decoding algorithm is evaluated by simulations.
Quantum low-density parity-check (LDPC) codes are stabilizer codes in which each stabilizer generator acts on O(1) qubits and each qubit participates in O(1) generators. Some famous classes of codes such as the toric code are quantum LDPC codes. There are several reasons that explain why this subclass of stabilizer codes is particularly interesting. For instance, last year it was shown by Gottesman that such codes would in principle allow fault-tolerant quantum computing with constant overhead.However, for this application we need quantum LDPC codes that are able to encode a linear fraction of qubitswith good minimum distance properties and that are equipped with an efficient decoding algorithm.The fact that their classical counterpart enjoys all these nice properties could lead to think that it will be easyto obtain the same result in the quantum setting. However it has turned out that decoding quantum LDPC codes is a much more involved issue than decoding classical LDPC codes.The latter class of codes is decoded successfully by low complexity iterative decoding algorithms.In principle, such algorithms could also be used to decode quantum LDPC codes.However, this approach fails for quantum LDPC codes due to the fact that suchcodes are inherently highly degenerate. I will survey and discuss in this talk alternative approaches that have been successful for some subclasses of quantum LDPC codes.
Low-density parity check (LDPC) codes are a significant class of classical codes with many applications. Several good LDPC codes have been constructed using random, algebraic, and finite geometries approaches, with containing cycles of length at least six in their Tanner graphs. However, it is impossible to design a self-orthogonal parity check matrix of an LDPC code without introducing cycles of length four. In this paper, a new class of quantum LDPC codes based on lines and points of finite geometries is constructed. The parity check matrices of these codes are adapted to be self- orthogonal with containing only one cycle of length four in each pair of two rows. Also, the column and row weights, and bounds on the minimum distance of these codes are given. As a consequence, these codes can be encoded using shift-register encoding algorithms and can be decoded using iterative decoding algorithms over various quantum depolarizing channels.