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arXiv:quant-ph/0502086

Constructions and performance of classes of quantum LDPC codes

Thomas Camara, Ollivier, H., Jean–Pierre Tillich

Published Feb 14, 2005• 50 citations
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Abstract

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.

DOI
10.48550/arxiv.quant-ph/0502086

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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.
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.
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.
Due to their fast decoding algorithms, quantum generalizations of low-density parity check, or LDPC, codes have been invesitgated as a solution to the problem of decoherence in fragile quantum states [1, 2]. However, the additional twisted inner product requirements of quantum stabilizer codes force four-cycles and eliminate the possibility of randomly generated quantum LDPC codes. Moreover, the classes of quantum LDPC codes discovered thus far generally have unknown or small minimum distance, or a fixed rate (see [3, 4] and references therin). This paper presents several new classes of quantum LDPC codes constructed from finite projective planes. These codes have rates that increase with the block length n and minimum weights proportional to n1=2. For the sake of completeness, we include an introduction to classical error correction and LDPC codes, and provide a review of quantum communication, quantum stabilizer codes, and finite projective geometry.
We propose two constructions, namely Construction 1 and Construction 2, for the design of protograph quantum LDPC codes by exploiting conventional quasi cyclic protograph LDPC codes with lift operation, and tensor product operation, respectively. We show that by carefully designing the proto-matrix using quadratic residue sets of parameters p = 4n ± 1, the proposed Construction 1 yields protograph quantum LDPC codes of various rates. Furthermore, by performing tensor product between two non-binary parity-check matrices obtained from idempotent polynomials of a quadratic residue set, the proposed Construction 2 yields a class of protograph quantum LDPC codes with rate at least 0.9. The performance of the proposed codes over quantum depolarizing channel with iterative sum-product decoding algorithm is shown.