Balanced Product Quantum Codes

Nikolas P. Breuckmann; Jens N. Eberhardt

doi:10.1109/TIT.2021.3097347

paper

Balanced Product Quantum Codes

arXiv:2012.09271 · doi:10.1109/TIT.2021.3097347

Abstract

This work provides the first explicit and non-random family of $[[N, K, D]]$ LDPC quantum codes which encode $K \in Θ (N^{\frac{4}{5}})$ logical qubits with distance $D \in Ω (N^{\frac{3}{5}})$ . The family is constructed by amalgamating classical codes and Ramanujan graphs via an operation called balanced product. Recently, Hastings-Haah-O'Donnell and Panteleev-Kalachev were the first to show that there exist families of LDPC quantum codes which break the $polylog (N) N$ distance barrier. However, their constructions are based on probabilistic arguments which only guarantee the code parameters with high probability whereas our bounds hold unconditionally. Further, balanced products allow for non-abelian twisting of the check matrices, leading to a construction of LDPC quantum codes that can be shown to have $K \in Θ (N)$ and that we conjecture to have linear distance $D \in Θ (N)$ .

arXiv comment23 pages, 11 figures

Balanced Product Quantum Codes

Abstract

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