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Explicit Abelian Lifts and Quantum LDPC Codes

by   Fernando Granha Jeronimo, et al.
Institute for Advanced Study
The University of Chicago
Carnegie Mellon University
Toyota Technological Institute at Chicago

For an abelian group H acting on the set [ℓ], an (H,ℓ)-lift of a graph G_0 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. 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 G_0 with the following parameters: (i) λ(G) ≤ 2√(d-1) + ϵ, for any lift size ℓ≤ 2^n^δ where δ=δ(d,ϵ), (ii) λ(G) ≤ϵ· d, for any lift size ℓ≤ 2^n^δ_0 for a fixed δ_0 > 0, when d ≥ d_0(ϵ), or (iii) λ(G) ≤O(√(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.


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