A Construction of Quantum LDPC Codes From Cayley Graphs

We study a construction of quantum LDPC codes proposed by MacKay, Mitchison, and Shokrollahi. It is based on the Cayley graph of BBF2ⁿ 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(dn²) where d is the minimum distance of the classical code. This bound is logarithmic in the blocklength 2ⁿ 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 [[2ⁿ, 2^[(n+1)/2], 2^[(n-1)/2]]].