Explicit Abelian Lifts and Quantum LDPC Codes
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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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