Distance verification for classical and quantum LDPC codes

TL;DR

This paper adapts distance verification methods for classical and quantum LDPC codes, deriving new complexity bounds and introducing an irreducible-cluster technique that leverages parity-check sparsity to improve verification efficiency.

Contribution

It presents novel complexity bounds for LDPC code distance verification and introduces an irreducible-cluster technique exploiting parity-check sparsity for classical and quantum codes.

Findings

Derived new complexity bounds based on average weight spectra.

Introduced an irreducible-cluster technique reducing verification complexity.

Applicable to all LDPC codes, improving existing deterministic methods.

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.