平衡乘积量子码
Balanced Product Quantum Codes
摘要：
本工作提供了第一个明确的和非随机的$[[N,K,D]]$ LDPC量子码族，其编码具有距离$D \in Ω(N^\frac{3}{5})$的$K \in Θ(N^\frac{4}{5})$逻辑量子比特。该家族是通过融合经典代码和拉马努金图通过一个称为平衡产品的操作构建的。最近，Hastings-Haah-O 'Donnell和Panteleev-Kalachev首次表明存在打破$\operatorname{polylog}(N)\sqrt{N}$距离障碍的LDPC量子码族。然而，他们的建设是基于概率的参数，只保证代码参数具有高概率，而我们的界限无条件地举行。此外，平衡乘积允许校验矩阵的非阿贝尔扭曲，导致可以被示出为具有$K\in Θ(N)$并且我们推测具有线性距离$D\in Θ(N)$的LDPC量子码的构造。
英文摘要：
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 $D\in Θ(N)$.
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