Numerical Techniques for Finding the Distances of Quantum Codes

TL;DR

This paper reviews and advances methods for calculating the distances of quantum codes, introducing a new linked-cluster technique that improves efficiency for quantum LDPC codes.

Contribution

It presents a novel linked-cluster technique that reduces complexity for quantum LDPC codes and outperforms existing probabilistic methods at high code rates.

Findings

Reduces complexity exponent for classical and quantum LDPC code distance calculations

Surpasses probabilistic techniques for high code rates

Applicable to all known families of quantum LDPC codes

Abstract

We survey the existing techniques for calculating code distances of classical codes and apply these techniques to generic quantum codes. For classical and quantum LDPC codes, we also present a new linked-cluster technique. It reduces complexity exponent of all existing deterministic techniques designed for codes with small relative distances (which include all known families of quantum LDPC codes), and also surpasses the probabilistic technique for sufficiently high code rates.