Single-Shot Decoding of Linear Rate LDPC Quantum Codes with High Performance

Nikolas P. Breuckmann*, Vivien Londe

*Corresponding author for this work

Research output: Contribution to journalArticle (Academic Journal)peer-review

4 Citations (Scopus)
16 Downloads (Pure)


We construct and analyze a family of low-density parity check (LDPC) quantum codes with a linear encoding rate, polynomial scaling distance and efficient decoding schemes. The code family is based on tessellations of closed, four-dimensional, hyperbolic manifolds, as first suggested by Guth and Lubotzky. The main contribution of this work is the construction of suitable manifolds via finite presentations of Coxeter groups, their linear representations over Galois fields and topological coverings. We establish a lower bound on the encoding rate~k/n of~13/72 = 0.180... and we show that the bound is tight for the examples that we construct. Numerical simulations give evidence that parallelizable decoding schemes of low computational complexity suffice to obtain high performance. These decoding schemes can deal with syndrome noise, so that parity check measurements do not have to be repeated to decode. Our data is consistent with a threshold of around 4% in the phenomenological noise model with syndrome noise in the single-shot regime.
Original languageEnglish
Pages (from-to)272-286
Number of pages15
JournalIEEE Transactions on Information Theory
Issue number1
Early online date26 Oct 2021
Publication statusPublished - 10 Jan 2022

Bibliographical note

Publisher Copyright:
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  • belief-propagation
  • cellular automata
  • Coxeter groups
  • hyperbolic
  • Quantum codes
  • quantum error-correction
  • quantum fault-tolerance
  • single-shot decoding


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