Explicit Abelian Lifts and Quantum LDPC Codes

Authors Fernando Granha Jeronimo, Tushant Mittal , Ryan O'Donnell, Pedro Paredes, Madhur Tulsiani

Thumbnail PDF


  • Filesize: 0.83 MB
  • 21 pages

Document Identifiers

Author Details

Fernando Granha Jeronimo
  • Institute for Advanced Study, Princeton, NJ, USA
Tushant Mittal
  • Department of Computer Science, University of Chicago, IL, USA
Ryan O'Donnell
  • Computer Science Department, Carnegie Mellon University, Pittsburgh, PA, USA
Pedro Paredes
  • Computer Science Department, Carnegie Mellon University, Pittsburgh, PA, USA
Madhur Tulsiani
  • Toyota Technological Institute at Chicago, IL, USA


We would like to thank the anonymous reviewers for their comments and helpful feedback.

Cite AsGet BibTex

Fernando Granha Jeronimo, Tushant Mittal, Ryan O'Donnell, Pedro Paredes, and Madhur Tulsiani. Explicit Abelian Lifts and Quantum LDPC Codes. In 13th Innovations in Theoretical Computer Science Conference (ITCS 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 215, pp. 88:1-88:21, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)


For an abelian group H acting on the set [𝓁], an (H,𝓁)-lift of a graph G₀ is a graph obtained by replacing each vertex by 𝓁 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 Ω̃(N^{3/5}), and in those by Panteleev and Kalachev [IEEE Trans. Inf. Theory 2021] of distance Ω(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 ⩽ Sym(𝓁), constant degree d ≥ 3 and ε > 0, we construct explicit d-regular expander graphs G obtained from an (H,𝓁)-lift of a (suitable) base n-vertex expander G₀ with the following parameters: ii) λ(G) ≤ 2√{d-1} + ε, for any lift size 𝓁 ≤ 2^{n^{δ}} where δ = δ(d,ε), iii) λ(G) ≤ ε ⋅ d, for any lift size 𝓁 ≤ 2^{n^{δ₀}} for a fixed δ₀ > 0, when d ≥ d₀(ε), or iv) λ(G) ≤ Õ(√d), for lift size "exactly" 𝓁 = 2^{Θ(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.

Subject Classification

ACM Subject Classification
  • Theory of computation → Pseudorandomness and derandomization
  • Theory of computation → Expander graphs and randomness extractors
  • Graph lifts
  • expander graphs
  • quasi-cyclic LDPC codes
  • quantum LDPC codes


  • Access Statistics
  • Total Accesses (updated on a weekly basis)
    PDF Downloads


  1. Naman Agarwal, Karthekeyan Chandrasekaran, Alexandra Kolla, and Vivek Madan. On the Expansion of Group-Based Lifts. SIAM J. Discret. Math., 33(3):1338-1373, 2019. URL: https://doi.org/10.1137/17M1141047.
  2. Noga Alon. Explicit expanders of every degree and size. Combinatorica, February 2021. URL: https://doi.org/10.1007/s00493-020-4429-x.
  3. Noga Alon, Shlomo Hoory, and Nathan Linial. The Moore bound for irregular graphs. Graphs and Combinatorics, 18:53-57, 2002. URL: https://doi.org/10.1007/s003730200002.
  4. Noga Alon and Yuval Roichman. Random cayley graphs and expanders. Random Struct. Algorithms, 5(2):271-285, 1994. URL: https://doi.org/10.1002/rsa.3240050203.
  5. László Babai, Amir Shpilka, and Daniel Stefankovic. Locally testable cyclic codes. IEEE Trans. Inf. Theory, 51(8):2849-2858, 2005. URL: https://doi.org/10.1109/TIT.2005.851735.
  6. László Babai. On the minimum order of graphs with given group. Canadian Mathematical Bulletin, 17(4):467-470, 1974. URL: https://doi.org/10.4153/CMB-1974-082-9.
  7. L.M.J. Bazzi and S.K. Mitter. Some randomized code constructions from group actions. IEEE Transactions on Information Theory, 52(7):3210-3219, 2006. URL: https://doi.org/10.1109/TIT.2006.876244.
  8. Avraham Ben-Aroya and Amnon Ta-Shma. A combinatorial construction of almost-ramanujan graphs using the zig-zag product. In Proceedings of the 40th ACM Symposium on Theory of Computing, pages 325-334, 2008. URL: https://doi.org/10.1145/1374376.1374424.
  9. Yonatan Bilu and Nathan Linial. Lifts, discrepancy and nearly optimal spectral gap. Combinatorica, 26(5):495-519, October 2006. URL: https://doi.org/10.1007/s00493-006-0029-7.
  10. Nikolas P. Breuckmann and Jens N. Eberhardt. Balanced Product Quantum Codes. IEEE Transactions on Information Theory, 67(10):6653-6674, 2021. URL: https://doi.org/10.1109/TIT.2021.3097347.
  11. A. R. Calderbank and Peter W. Shor. Good quantum error-correcting codes exist. Phys. Rev. A, 54:1098-1105, August 1996. URL: https://doi.org/10.1103/PhysRevA.54.1098.
  12. C.L. Chen, W.W. Peterson, and E.J. Weldon. Some results on quasi-cyclic codes. Information and Control, 15(5):407-423, November 1969. URL: https://doi.org/10.1016/s0019-9958(69)90497-5.
  13. Michael B. Cohen. Ramanujan graphs in polynomial time. In Proceedings of the 57th IEEE Symposium on Foundations of Computer Science, 2016. URL: https://doi.org/10.1109/FOCS.2016.37.
  14. Shai Evra, Tali Kaufman, and Gilles Zémor. Decodable quantum LDPC codes beyond the square root distance barrier using high dimensional expanders. In Proceedings of the 61st IEEE Symposium on Foundations of Computer Science, pages 218-227. IEEE, 2020. http://arxiv.org/abs/2004.07935, URL: https://doi.org/10.1109/FOCS46700.2020.00029.
  15. R. Gallager. Low-density parity-check codes. IRE Transactions on Information Theory, 8(1):21-28, 1962. URL: https://doi.org/10.1109/TIT.1962.1057683.
  16. Oded Goldreich and Avi Wigderson. Robustly self-ordered graphs: Constructions and applications to property testing. In Valentine Kabanets, editor, Proceedings of the 36th IEEE Conference on Computational Complexity, volume 200 of LIPIcs, pages 12:1-12:74. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2021. URL: https://doi.org/10.4230/LIPIcs.CCC.2021.12.
  17. Matthew B. Hastings, Jeongwan Haah, and Ryan O'Donnell. Fiber bundle codes: breaking the n^1/2 polylog (n) barrier for quantum LDPC codes. In Proceedings of the 53rd ACM Symposium on Theory of Computing, pages 1276-1288. ACM, 2021. URL: https://doi.org/10.1145/3406325.3451005.
  18. Shlomo Hoory, Nathan Linial, and Avi Wigderson. Expander graphs and their applications. Bull. Amer. Math. Soc., 43(04):439-562, August 2006. URL: https://doi.org/10.1090/S0273-0979-06-01126-8.
  19. Akhil Jalan and Dana Moshkovitz. Near-optimal cayley expanders for abelian groups. CoRR, abs/2105.01149, 2021. URL: http://arxiv.org/abs/2105.01149.
  20. Tali Kaufman and Avi Wigderson. Symmetric LDPC codes and local testing. In Andrew Chi-Chih Yao, editor, Innovations in Computer Science - ICS 2010, Tsinghua University, Beijing, China, January 5-7, 2010. Proceedings, pages 406-421. Tsinghua University Press, 2010. URL: http://conference.iiis.tsinghua.edu.cn/ICS2010/content/papers/32.html.
  21. Huaan Li, Baoming Bai, Xijin Mu, Ji Zhang, and Hengzhou Xu. Algebra-assisted construction of quasi-cyclic LDPC codes for 5G new radio. IEEE Access, 6:50229-50244, 2018. URL: https://doi.org/10.1109/ACCESS.2018.2868963.
  22. Alexander Lubotzky, R. Phillips, and Peter Sarnak. Ramanujan graphs. Combinatorica, 8:261-277, 1988. URL: https://doi.org/10.1007/BF02126799.
  23. G. A. Margulis. Explicit group-theoretic constructions of combinatorial schemes and their applications in the construction of expanders and concentrators. Problemy Peredachi Informatsii, 24(1):51-60, 1988. URL: http://mi.mathnet.ru/eng/ppi686.
  24. Sidhanth Mohanty, Ryan O'Donnell, and Pedro Paredes. Explicit near-ramanujan graphs of every degree. In Proceedings of the 52nd ACM Symposium on Theory of Computing, pages 510-523. ACM, 2020. URL: https://doi.org/10.1145/3357713.3384231.
  25. M. Morgenstern. Existence and explicit constructions of q + 1 regular Ramanujan graphs for every prime power q. J. Comb. Theory Ser. B, pages 44-62, September 1994. URL: https://doi.org/10.1006/jctb.1994.1054.
  26. Alon Nilli. On the second eigenvalue of a graph. Discrete Mathematics, 91(2):207-210, 1991. URL: https://doi.org/10.1016/0012-365X(91)90112-F.
  27. R. O'Donnell and X. Wu. Explicit near-fully X-Ramanujan graphs. In Proceedings of the 61st IEEE Symposium on Foundations of Computer Science, pages 1045-1056, 2020. URL: https://doi.org/10.1109/FOCS46700.2020.00101.
  28. Pavel Panteleev and Gleb Kalachev. Quantum LDPC Codes with Almost Linear Minimum Distance. IEEE Transactions on Information Theory, December 2021. http://arxiv.org/abs/2012.04068, URL: https://doi.org/10.1109/TIT.2021.3119384.
  29. Shravas Rao. A Hoeffding inequality for Markov chains. Electronic Communications in Probability, 24:1-11, 2019. URL: https://doi.org/10.1214/19-ECP219.
  30. O. Reingold, S. Vadhan, and A. Wigderson. Entropy waves, the zig-zag graph product, and new constant-degree expanders and extractors. In Proceedings of the 41st IEEE Symposium on Foundations of Computer Science, 2000. URL: https://doi.org/10.1109/SFCS.2000.892006.
  31. Tom Richardson and Ruediger Urbanke. Modern coding theory. Cambridge university press, 2008. URL: https://doi.org/10.1017/CBO9780511791338.
  32. Andrew Steane. Multiple-particle interference and quantum error correction. Proceedings of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences, 452(1954):2551-2577, November 1996. URL: https://doi.org/10.1098/rspa.1996.0136.
  33. Salil P. Vadhan. Pseudorandomness. Foundations and Trends® in Theoretical Computer Science, 7(1–3):1-336, 2012. URL: https://doi.org/10.1561/0400000010.