Distance verification for classical and quantum LDPC codes

Abstract

The techniques of distance verification known for general linear codes are re-applied to quantum stabilizer codes. Then distance verification is addressed for classical and quantum LDPC codes. New complexity bounds for distance verification with provable performance are derived using the average weight spectra of the ensembles of LDPC codes. These bounds are expressed in terms of the erasure-correcting capacity of the corresponding ensemble. We also present a new irreducible-cluster technique that can be applied to any LDPC code and takes advantage of parity-checks' sparsity for both classical and quantum LDPC codes. This technique reduces complexity exponents of all existing deterministic techniques designed for generic stabilizer codes with small relative distances, which also include all known families of quantum LDPC codes.