Spatially Coupled Quasi-Cyclic Quantum LDPC Codes

TL;DR

This paper introduces spatially coupled LDPC codes tailored for quantum error correction, balancing decoding performance and minimum distance, and demonstrating capacity-achieving properties over depolarizing channels.

Contribution

The paper designs spatially coupled LDPC-CSS codes for quantum error correction, leveraging classical capacity-achieving code properties in a quantum context.

Findings

Spatially coupled LDPC-CSS codes achieve capacity over depolarizing channels.

The codes balance large row weights with good decoding performance.

Enhanced error-floor and water-fall performance in quantum LDPC codes.

Abstract

We face the following dilemma for designing low-density parity-check codes (LDPC) for quantum error correction. 1) The row weights of parity-check should be large: The minimum distances are bounded above by the minimum row weights of parity-check matrices of constituent classical codes. Small minimum distance tends to result in poor decoding performance at the error-floor region. 2) The row weights of parity-check matrices should not be large: The sum-product decoding performance at the water-fall region is degraded as the row weight increases. Recently, Kudekar et al. showed spatially-coupled (SC) LDPC codes exhibit capacity-achieving performance for classical channels. SC LDPC codes have both large row weight and capacity-achieving error-floor and water-fall performance. In this paper, we design SC LDPC-CSS (Calderbank, Shor and Steane) codes for quantum error correction over the depolarizing channels.