A Construction of Quantum LDPC Codes From Cayley Graphs

• Published in: IEEE transactions on information theory Vol. 59; no. 9; pp. 6087 - 6098
• Main Authors: Couvreur, Alain, Delfosse, Nicolas, Zemor, Gilles
• Format: Journal Article
• Language: English
• Published: New York, NY IEEE 01.09.2013; Institute of Electrical and Electronics Engineers; The Institute of Electrical and Electronics Engineers, Inc. (IEEE)
• Subjects: Algebra; Applied sciences; Cascading style sheets; Cayley graphs; Codes; Coding theory; Coding, codes; Computer Science; Exact sciences and technology; Generators; graph covers; Graphs; Information Theory; Information, signal and communications theory; LDPC codes; Low density parity check codes; Mathematics; Parity check codes; quantum codes; Quantum computing; Quantum mechanics; Signal and communications theory; Sparse matrices; Telecommunications and information theory; Lower bound; Logarithmic function; Cayley graph; LDPC codes; quantum codes; graph covers; Sparse matrix; Error correcting code; Parity check codes; Minimal distance; Cayley graphs; sparse matrices; Parity check
• ISSN: 0018-9448; 1557-9654
• DOI: 10.1109/TIT.2013.2261116

We study a construction of quantum LDPC codes proposed by MacKay, Mitchison, and Shokrollahi. It is based on the Cayley graph of \BBF2 n together with a set of generators regarded as the columns of the parity-check matrix of a classical code. We give a general lower bound on the minimum distance of the quantum code in O(dn 2 ) where d is the minimum distance of the classical code. This bound is logarithmic in the blocklength 2 n of the quantum code. When the classical code is the [n,1,n] repetition code, we are able to compute the exact parameters of the associated quantum code which are [[2 n , 2 [(n+1)/2] , 2 [(n-1)/2] ]].