Local Decoders for the 2D and 4D Toric Code

TL;DR

This paper evaluates local decoders for 2D and 4D toric codes, demonstrating their thresholds under noise and faulty syndromes, and compares different decoding strategies for these topological quantum error-correcting codes.

Contribution

It introduces and analyzes local decoders for 2D and 4D toric codes, providing threshold estimates and performance comparisons under realistic noise models.

Findings

Threshold of 0.133% for 2D Harrington decoder

Threshold of 1.59% for 4D decoder with faulty syndromes

Comparison of different local decoding strategies

Abstract

We analyze the performance of decoders for the 2D and 4D toric code which are local by construction. The 2D decoder is a cellular automaton decoder formulated by Harrington which explicitly has a finite speed of communication and computation. For a model of independent $X$ and $Z$ errors and faulty syndrome measurements with identical probability we report a threshold of $0.133\%$ for this Harrington decoder. We implement a decoder for the 4D toric code which is based on a decoder by Hastings arXiv:1312.2546 . Incorporating a method for handling faulty syndromes we estimate a threshold of $1.59\%$ for the same noise model as in the 2D case. We compare the performance of this decoder with a decoder based on a 4D version of Toom's cellular automaton rule as well as the decoding method suggested by Dennis et al. arXiv:quant-ph/0110143 .