Toward a Union-Find decoder for quantum LDPC codes

TL;DR

This paper introduces a generalized Union-Find decoder for quantum LDPC codes, demonstrating its ability to correct errors effectively and outperform belief propagation in low error regimes through theoretical proofs and numerical simulations.

Contribution

It generalizes the Union-Find decoder for quantum LDPC codes and introduces a covering radius concept, with proofs of error correction capabilities across various code classes.

Findings

Corrects errors up to a certain weight for multiple quantum LDPC code classes.

Outperforms belief propagation decoder at low error rates.

Provides theoretical bounds and numerical validation.

Abstract

Quantum LDPC codes are a promising direction for low overhead quantum computing. In this paper, we propose a generalization of the Union-Find decoder as adecoder for quantum LDPC codes. We prove that this decoder corrects all errors with weight up to An^{\alpha} for some A, {\alpha} > 0 for different classes of quantum LDPC codes such as toric codes and hyperbolic codes in any dimension D \geq 3 and quantum expander codes. To prove this result, we introduce a notion of covering radius which measures the spread of an error from its syndrome. We believe this notion could find application beyond the decoding problem. We also perform numerical simulations, which show that our Union-Find decoder outperforms the belief propagation decoder in the low error rate regime in the case of a quantum LDPC code with length 3600.