A blindness property of the Min-Sum decoding for the toric code

TL;DR

This paper analyzes the limitations of min-sum decoding for the toric code, revealing an intrinsic blindness in local information propagation and proposing a practical pre-processing method to improve error correction performance.

Contribution

It provides a theoretical analysis of min-sum decoding limitations for the toric code and introduces a stabiliser-blowup pre-processing method to enhance error correction.

Findings

MS decoding is locally blind when unsatisfied checks are distant by at least 5.

Decoding failures occur for errors of weight ≥4 beyond degeneracy issues.

Proposed stabiliser-blowup method corrects errors up to weight 3 with quadratic performance improvement.

Abstract

Kitaev's toric code is one of the most prominent models for fault-tolerant quantum computation, currently regarded as the leading solution for connectivity constrained quantum technologies. Significant effort has been recently devoted to improving the error correction performance of the toric code under message-passing decoding, a class of low-complexity, iterative decoding algorithms that play a central role in both theory and practice of classical low-density parity-check codes. Here, we provide a theoretical analysis of the toric code under min-sum (MS) decoding, a message-passing decoding algorithm known to solve the maximum-likelihood decoding problem in a localized manner, for codes defined by acyclic graphs. Our analysis reveals an intrinsic limitation of the toric code, which confines the propagation of local information during the message-passing process. We show that if the unsatisfied checks of an error syndrome are at distance greater or equal to 5 from each other, then the MS decoding is locally blind: the qubits in the direct neighborhood of an unsatisfied check are never aware of any other unsatisfied checks, except their direct neighbor. Moreover, we show that degeneracy is not the only cause of decoding failures for errors of weight at least 4, that is, the MS non-degenerate decoding radius is equal to 3, for any toric code of distance greater or equal to 9. Finally, complementing our theoretical analysis, we present a pre-processing method of practical relevance. The proposed method, referred to as stabiliser-blowup, has linear complexity and allows correcting all (degenerate) errors of weight up to 3, providing quadratic improvement in the logical error rate performance, as compared to MS only.