Balanced Product Quantum Codes

• URL: http://arxiv.org/abs/2012.09271v3
• Date: Wed, 28 Jul 2021 10:58:06 GMT
• Title: Balanced Product Quantum Codes
• Authors: Nikolas P. Breuckmann and Jens N. Eberhardt
• Abstract summary: This work provides the first explicit and non-random family of $[[N,K,D]]$ LDPC quantum codes. The family is constructed by amalgamating classical codes and Ramanujan graphs via an operation called balanced product.
• Score: 5.33024001730262
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: This work provides the first explicit and non-random family of $[[N,K,D]]$ LDPC quantum codes which encode $K \in \Theta(N^\frac{4}{5})$ logical qubits with distance $D \in \Omega(N^\frac{3}{5})$. 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 $K\in \Theta(N)$ and that we conjecture to have linear distance $D\in \Theta(N)$.

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