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

TL;DR

This paper introduces new quantum LDPC codes derived from finite projective planes, achieving increasing rates with block length and minimum weights proportional to the square root of the length, addressing previous limitations.

Contribution

It presents novel classes of quantum LDPC codes based on finite projective planes with scalable rates and minimum distances, overcoming prior constraints.

Findings

Codes have rates increasing with block length

Minimum weights scale as the square root of block length

Codes are constructed from finite projective planes

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 $n^{1/2}$.