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Paper 1

Balanced Product Quantum Codes

Nikolas P. Breuckmann, Jens N. Eberhardt

Year

2020

Journal

arXiv preprint

DOI

arXiv:2012.09271

arXiv

2012.09271

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 $\operatorname{polylog}(N)\sqrt{N}$ 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 $D\in Θ(N)$.

Paper 2

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