eng
Schloss Dagstuhl ā Leibniz-Zentrum fĆ¼r Informatik
Leibniz International Proceedings in Informatics
1868-8969
2022-01-25
88:1
88:21
10.4230/LIPIcs.ITCS.2022.88
article
Explicit Abelian Lifts and Quantum LDPC Codes
Jeronimo, Fernando Granha
1
Mittal, Tushant
2
https://orcid.org/0000-0002-4017-2662
O'Donnell, Ryan
3
Paredes, Pedro
3
Tulsiani, Madhur
4
Institute for Advanced Study, Princeton, NJ, USA
Department of Computer Science, University of Chicago, IL, USA
Computer Science Department, Carnegie Mellon University, Pittsburgh, PA, USA
Toyota Technological Institute at Chicago, IL, USA
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©Ģ(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¤ 2ā{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¤ OĢ(ād), 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.
https://drops.dagstuhl.de/storage/00lipics/lipics-vol215-itcs2022/LIPIcs.ITCS.2022.88/LIPIcs.ITCS.2022.88.pdf
Graph lifts
expander graphs
quasi-cyclic LDPC codes
quantum LDPC codes