Quantum Quasi-Cyclic LDPC Codes

Abstract

In this paper, a construction of a pair of "regular" quasi-cyclic LDPC codes to construct a quantum errorcorrecting code is proposed. In other words, we find quantum regular LDPC codes with various weight distributions. Our construction method is based on algebraic combinatorics and achieves a lower bound of the code length, and has lots of variations for length, code rate. These codes are obtained by a descrete mathematical characterization for model matrices of quasi-cyclic LDPc codes.I. INTRODUCTION Quantum error-correcting codes provide detection or correction of the errors which occur in a communication through a noisy quantum channel. Hence the codes protect integrity of a message sent from a sender to a receiver. This paper presents a construction method of a pair of "regular" quasi-cyclic low-density parity-check codes (regular QC-LDPC codes) as ingredients of a CSS code. QC-LDPC codes are known as a practical class of classical error-correcting codes due to its compact representation and good performance, especially for short code lengths [1], [2]. Since CSS codes find their applications not only for quantum error-correction but also for privacy amplification of quantum cryptography, they have become important research objects for quantum information theory [3].Mackay proposed bicycle codes which are constructed by a combination of heuristic method and theoretical approach [4]. In his paper, we find the following requirement "we delete rows using the heuristic that the column weight of a matrix should be as uniform as possible". This implies that the design of weight distribution for quantum LDPC codes by deterministic method is a theoretically interesting problem. Recently, various construction methods of quantum LDPC codes have been proposed [5], [6], [7], [8], [9], [10]. Among these proposed constructions, it has been difficult to design the weight distribution of the related parity-check matrices. In this paper, we give a solution for the weight distribution design problem for regular weight case.Contributions of this paper are the following: 1. We find a characterization for model matrices of QC-LDPC codes with circulant permutation matrices to be ingredient codes of a CSS code. The characterization is easily treatable for not only theoretical use but also computer experiments. 2. We propose a construction for (λ 1 , ρ)-regular and (λ 2 , ρ)-regular parity-check matrices for any 1 ≤ λ 1 , λ 2 ≤ ρ/2 such that the associated QC-LDPC codes are ingredient codes of a CSS code. In other words, we find quantum regular LDPC codes. Our method for designing a pair of LDPC matrices is theoretical and deterministic.

Quantum serial turbo-codes

“…This shows that Clifford transformations are efficiently specifiable as claimed. It can readily be verified that the rows of , denoted , are equal to (18) (19) Since conjugation by a unitary matrix does not change the commutation relations, the above equations imply that the encoding matrix is a symplectic matrix, whose definition is recalled below.…”

“…The first attempts at obtaining quantum analogues of LDPC codes [9], [19], [28] have not yielded results as spectacular as their classical counterpart. This is due to several reasons.…”

Quantum LDPC Codes With Positive Rate and Minimum Distance Proportional to the Square Root of the Blocklength

“…In particular, the plain random constructions that work so well in the classical setting are pointless here. There have been a number of attempts at overcoming this difficulty and a variety of methods for constructing quantum LDPC codes have been proposed [33,27,31,10,11,30,19,21,24,15,35,1,2,23,38,25,12,3,4]. However, all of these constructions suffer from disappointingly small minimum distances, namely whenever they have non-vanishing rate and parity-check matrices with bounded row-weight, their minimum distance is either proved to be bounded, or unknown and with little hope for unboundedness.…”

A Construction of Quantum LDPC Codes From Cayley Graphs

“…Therefore, randomly choosing these matrices, the generic method which works very well in the classical case, is simply not an option in the Quantum case, because the probability of finding two sparse row-orthogonal matrices is extremely small. A number of constructions have been suggested by classical coding theorists nevertheless [15,1,2,8,13, 20] but they do not produce families of Quantum LDPC codes with a minimum distance growing with the blocklength. While this may be tolerable for practical constructions of fixed size, this is clearly an undesirable feature of any asymptotic construction and it raises the intriguing theoretical question of how large can the minimum distance of sparse (or LDPC) CSS codes be.…”