量子消息传递收敛性与随机LDPC码的块错误概率消失
Quantum Message Passing Convergence and Vanishing Block-Error Probability for Random LDPC Codes
- Duke University（杜克大学）
- IBM Research Europe – Zurich（IBM 欧洲苏黎世研究中心）
- ETH Zurich（苏黎世联邦理工学院）
机构由 AI 辅助整理，请以论文原文为准。
AI总结：
本文为随机LDPC码构造两阶段BPQM解码器，证明其块错误概率随块长消失，支持DQI及Regev归约算法的解码步骤。
AI中文摘要：
带量子消息的信度传播（BPQM）是一种量子算法，用于解码通过经典-量子信道传输的经典码。在纯态经典-量子信道上，它能在树因子图上实现最优解码。然而，这种基于树的分析并不能确保具有环的LDPC Tanner图的块错误概率消失。在本工作中，我们为对称q元纯态信道上的随机q元LDPC码（其中q为素数）构建了一个两阶段BPQM解码器，并证明其系综平均块错误概率随着块长N趋于无穷而消失。对于d_v≥3的正则系综，保真度界在BPQM成功区域内给出平均符号错误概率的双指数衰减。我们对具有树邻域的坐标应用深度为ℓ的BPQM，并将剩余坐标视为擦除。通过选择合适的ℓ=Θ(log log N)，非交换联合界控制了BPQM解码错误，而最小距离性质保证了擦除恢复。我们还将分析扩展到有限支撑的不规则系综。这些结果与基于Regev归约的量子算法相关，其中相干解码取消计算码字寄存器。解码量子干涉测量（DQI）使用一个密切相关的基于傅里叶的框架，将稀疏最大LINSAT优化问题归约为纯态信道上的LDPC解码问题。我们的结果证明了在DQI的解码步骤以及基于Regev归约的编码理论算法中使用BPQM的合理性，只要码是从本文分析的一个随机LDPC系综中抽取的，且诱导的无记忆对称纯态信道位于BPQM成功区域内。
英文摘要：
Belief propagation with quantum messages (BPQM) is a quantum algorithm that decodes classical codes transmitted over classical--quantum channels. It realizes optimal decoding on tree factor graphs over pure-state classical-quantum channels. However, this tree-based analysis does not ensure vanishing block-error probability for LDPC Tanner graphs with cycles. In this work, we construct a two-stage BPQM decoder for random $q$-ary LDPC codes over symmetric $q$-ary pure-state channels, where $q$ is prime, and prove that its ensemble-average block-error probability vanishes as the blocklength $N$ tends to infinity. For regular ensembles with $d_v\geq3$, fidelity bounds yield double-exponential decay of the average symbol-error probability throughout the BPQM success region. We apply depth-$\ell$ BPQM to coordinates with tree neighbourhoods and treat the remaining coordinates as erasures. With a suitable $\ell=Θ(\log\log N)$, a noncommutative union bound controls the BPQM decoding errors, while the minimum-distance property guarantees erasure recovery. We also extend the analysis to finite-support irregular ensembles. These results are relevant to quantum algorithms based on Regev's reduction, where coherent decoding uncomputes a codeword register. Decoded quantum interferometry (DQI) uses a closely related Fourier-based framework that reduces sparse max-LINSAT optimization problems to LDPC decoding problems on pure-state channels. Our results justify the use of BPQM in the decoding step of DQI and of coding-theoretic algorithms based on Regev's reduction whenever the code is drawn from one of the random LDPC ensembles analyzed here and the induced memoryless symmetric pure-state channel lies in the BPQM success region. Curiously, our numerical results indicate that DQI+BPQM achieves a satisfaction ratio that closely matches that of simulated annealing.
