Renormalization Group Decoder for a Four-Dimensional Toric Code

Kasper Duivenvoorden, N. P. Breuckmann, Barbara Terhal

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

19 Citations (Scopus)


We describe a computationally efficient heuristic algorithm based on a renormalization-group procedure which aims at solving the problem of finding a minimal surface given its boundary (curve) in any hypercubic lattice of dimension D>2. We use this algorithm to correct errors occurring in a four-dimensional variant of the toric code, having open as opposed to periodic boundaries. For a phenomenological error model which includes measurement errors we use a five-dimensional version of our algorithm, achieving a threshold of 4.35 ± 0.1%. For this error model, this is the highest known threshold of any topological code. Without measurement errors, a four-dimensional version of our algorithm can be used and we find a threshold of 7.3 ± 0.1%. For the gate-based depolarizing error model, we find a threshold of 0.31 ± 0.01% which is below the threshold found for the two-dimensional toric code.

Original languageEnglish
Article number8528891
Pages (from-to)2545-2562
Number of pages18
JournalIEEE Transactions on Information Theory
Issue number4
Publication statusPublished - Apr 2019

Bibliographical note

Funding Information:
Manuscript received September 19, 2017; accepted October 23, 2018. Date of publication November 9, 2018; date of current version March 15, 2019. This work was supported through the ERC Consolidator under Grant 682726. K. Duivenvoorden was supported by the Excellence Initiative of the DFG.

Publisher Copyright:
© 2018 IEEE.


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