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DOI: 10.4230/LIPIcs.ITCS.2022.88
URN: urn:nbn:de:0030-drops-156846
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Jeronimo, Fernando Granha ; Mittal, Tushant ; O'Donnell, Ryan ; Paredes, Pedro ; Tulsiani, Madhur

Explicit Abelian Lifts and Quantum LDPC Codes

LIPIcs-ITCS-2022-88.pdf (0.8 MB)


For an abelian group H acting on the set [š“], an (H,š“)-lift of a graph Gā‚€ is a graph obtained by replacing each vertex by š“ copies, and each edge by a matching corresponding to the action of an element of H.
Expanding graphs obtained via abelian lifts, form a key ingredient in the recent breakthrough constructions of quantum LDPC codes, (implicitly) in the fiber bundle codes by Hastings, Haah and O'Donnell [STOC 2021] achieving distance Ī©Ģƒ(N^{3/5}), and in those by Panteleev and Kalachev [IEEE Trans. Inf. Theory 2021] of distance Ī©(N/log(N)). However, both these constructions are non-explicit. In particular, the latter relies on a randomized construction of expander graphs via abelian lifts by Agarwal et al. [SIAM J. Discrete Math 2019].
In this work, we show the following explicit constructions of expanders obtained via abelian lifts. For every (transitive) abelian group H ā©½ Sym(š“), constant degree d ā‰„ 3 and Īµ > 0, we construct explicit d-regular expander graphs G obtained from an (H,š“)-lift of a (suitable) base n-vertex expander Gā‚€ with the following parameters:
ii) Ī»(G) ā‰¤ 2āˆš{d-1} + Īµ, for any lift size š“ ā‰¤ 2^{n^{Ī“}} where Ī“ = Ī“(d,Īµ),
iii) Ī»(G) ā‰¤ Īµ ā‹… d, for any lift size š“ ā‰¤ 2^{n^{Ī“ā‚€}} for a fixed Ī“ā‚€ > 0, when d ā‰„ dā‚€(Īµ), or
iv) Ī»(G) ā‰¤ OĢƒ(āˆšd), for lift size "exactly" š“ = 2^{Ī˜(n)}. As corollaries, we obtain explicit quantum lifted product codes of Panteleev and Kalachev of almost linear distance (and also in a wide range of parameters) and explicit classical quasi-cyclic LDPC codes with wide range of circulant sizes.
Items (i) and (ii) above are obtained by extending the techniques of Mohanty, O'Donnell and Paredes [STOC 2020] for 2-lifts to much larger abelian lift sizes (as a byproduct simplifying their construction). This is done by providing a new encoding of special walks arising in the trace power method, carefully "compressing" depth-first search traversals. Result (iii) is via a simpler proof of Agarwal et al. [SIAM J. Discrete Math 2019] at the expense of polylog factors in the expansion.

BibTeX - Entry

  author =	{Jeronimo, Fernando Granha and Mittal, Tushant and O'Donnell, Ryan and Paredes, Pedro and Tulsiani, Madhur},
  title =	{{Explicit Abelian Lifts and Quantum LDPC Codes}},
  booktitle =	{13th Innovations in Theoretical Computer Science Conference (ITCS 2022)},
  pages =	{88:1--88:21},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-217-4},
  ISSN =	{1868-8969},
  year =	{2022},
  volume =	{215},
  editor =	{Braverman, Mark},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{},
  URN =		{urn:nbn:de:0030-drops-156846},
  doi =		{10.4230/LIPIcs.ITCS.2022.88},
  annote =	{Keywords: Graph lifts, expander graphs, quasi-cyclic LDPC codes, quantum LDPC codes}

Keywords: Graph lifts, expander graphs, quasi-cyclic LDPC codes, quantum LDPC codes
Collection: 13th Innovations in Theoretical Computer Science Conference (ITCS 2022)
Issue Date: 2022
Date of publication: 25.01.2022

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