A Blindness Property of the Min-Sum Decoding for the Toric Code

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 c...

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Vydané v:IEEE journal on selected areas in information theory Ročník 6; s. 138 - 149
Hlavní autori: Crest, Julien Du, Mhalla, Mehdi, Savin, Valentin
Médium: Journal Article
Jazyk:English
Vydavateľské údaje: Piscataway IEEE 2025
The Institute of Electrical and Electronics Engineers, Inc. (IEEE)
Predmet:
Algorithms
Blindness
Codes
Complexity
Computer Science
Electronic mail
Error analysis
Error correcting codes
Error correction
Fault tolerance
Fault tolerant systems
Iterative decoding
Maximum likelihood decoding
Message passing
message-passing decoding
minimum-sum algorithm
Noise
Quantum computing
Quantum error correction
Qubit
Qubits (quantum computing)
toric code
ISSN:2641-8770, 2641-8770
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Shrnutí: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 <inline-formula> <tex-math notation="LaTeX">\ge 5 </tex-math></inline-formula> from each other, then 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 <inline-formula> <tex-math notation="LaTeX">\ge 9 </tex-math></inline-formula>. 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 alone.
Bibliografia:ObjectType-Article-1
SourceType-Scholarly Journals-1
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ISSN:2641-8770
2641-8770
DOI:10.1109/JSAIT.2025.3571313