高性能线性速率LDPC量子码的单次解码
Single-Shot Decoding of Linear Rate LDPC Quantum Codes with High Performance
AI总结：
本文基于四维双曲流形镶嵌构造了一类线性速率LDPC量子码，通过Coxeter群表示和拓扑覆盖实现构造，证明编码速率下界为13/72，并展示了低复杂度并行解码方案在含综合征噪声的单次解码中达到约4%阈值。
摘要：
我们构造并分析了一类具有线性编码速率、多项式缩放距离和高效解码方案的低密度奇偶校验（LDPC）量子码。该码族基于闭合四维双曲流形的镶嵌，这一思路最早由Guth和Lubotzky提出。本工作的主要贡献是通过Coxeter群的有限表示、它们在Galois域上的线性表示以及拓扑覆盖来构造合适的流形。我们建立了编码速率k/n的下界为13/72 = 0.180...，并证明该下界对于我们所构造的实例是紧的。数值模拟给出证据表明，低计算复杂度的可并行化解码方案足以获得高性能。这些解码方案能够处理综合征噪声，因此无需重复进行奇偶校验测量即可完成解码。我们的数据与在单次机制下含综合征噪声的唯象噪声模型中约4%的阈值相一致。
英文摘要：
We construct and analyze a family of low-density parity check (LDPC) quantum codes with a linear encoding rate, polynomial scaling distance and efficient decoding schemes. The code family is based on tessellations of closed, four-dimensional, hyperbolic manifolds, as first suggested by Guth and Lubotzky. The main contribution of this work is the construction of suitable manifolds via finite presentations of Coxeter groups, their linear representations over Galois fields and topological coverings. We establish a lower bound on the encoding rate~k/n of~13/72 = 0.180... and we show that the bound is tight for the examples that we construct. Numerical simulations give evidence that parallelizable decoding schemes of low computational complexity suffice to obtain high performance. These decoding schemes can deal with syndrome noise, so that parity check measurements do not have to be repeated to decode. Our data is consistent with a threshold of around 4% in the phenomenological noise model with syndrome noise in the single-shot regime.
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