Quantum LDPC Codes Constructed from Point-Line Subsets of the Finite Projective Plane

Abstract: Due to their fast decoding algorithms, quantum generalizations of low-density parity check, or LDPC, codes have been investigated as a solution to the problem of decoherence in fragile quantum states. 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. 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 $n^{1/2}$.