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Quantum message-passing algorithm for optimal and efficient decoding
arXiv - CS - Information Theory Pub Date : 2021-09-16 , DOI: arxiv-2109.08170
Christophe Piveteau, Joseph M. Renes

Recently, one of us proposed a quantum algorithm called belief propagation with quantum messages (BPQM) for decoding classical data encoded using a binary linear code with tree Tanner graph that is transmitted over a pure-state CQ channel [Renes, NJP 19 072001 (2017)]. This algorithm presents a genuine quantum counterpart to decoding based on classical belief propagation, which has found wide success in classical coding theory when used in conjunction with LDPC or Turbo codes. More recently Rengaswamy et al. [npj Quantum Information 7 97 (2021)] numerically observed that, for a small example code, BPQM implements the optimal decoder for determining the entire input codeword. Here we significantly expand the understanding, formalism, and applicability of the BPQM algorithm with the following contributions. First, we prove analytically that BPQM realizes optimal decoding for any binary linear code with tree Tanner graph. We also provide the first formal description of the BPQM algorithm in full detail and without any ambiguity. In so doing, we identify a key flaw overlooked in the original algorithm and subsequent works which implies quantum circuit realizations will be exponentially large in the code size. Although BPQM passes quantum messages, other information required by the algorithm is processed globally. We remedy this problem by formulating a truly message-passing algorithm which approximates BPQM and has circuit complexity $\mathcal{O}(\text{poly } n, \text{polylog } \frac{1}{\epsilon})$, where $n$ is the code length and $\epsilon$ is the approximation error. Finally, we also propose a novel method for extending BPQM to factor graphs containing cycles by making use of approximate cloning. We show some promising numerical results that indicate that BPQM on factor graphs with cycles can significantly outperform the best possible classical decoder.

中文翻译:

用于优化和高效解码的量子消息传递算法

最近,我们中的一个人提出了一种称为带有量子消息的置信传播 (BPQM) 的量子算法,用于解码使用二进制线性代码和树 Tanner 图编码的经典数据,该代码通过纯状态 CQ 通道传输 [Renes, NJP 19 072001 (2017) )]。该算法为基于经典置信传播的解码提供了真正的量子对应物,当与 LDPC 或 Turbo 码结合使用时,该算法在经典编码理论中取得了广泛的成功。最近 Rengaswamy 等人。[npj Quantum Information 7 97 (2021)] 从数值上观察到,对于一个小的示例代码,BPQM 实现了用于确定整个输入码字的最佳解码器。在这里,我们通过以下贡献显着扩展了 BPQM 算法的理解、形式和适用性。第一的,我们通过分析证明了 BPQM 实现了对任何带有树 Tanner 图的二进制线性码的最优解码。我们还完整详细地提供了 BPQM 算法的第一个正式描述,没有任何歧义。在这样做的过程中,我们发现了原始算法和后续工作中被忽视的一个关键缺陷,这意味着量子电路实现的代码大小将呈指数级增长。虽然 BPQM 传递量子消息,但算法所需的其他信息是全局处理的。我们通过制定一个真正的消息传递算法来解决这个问题,该算法近似于 BPQM 并具有电路复杂度 $\mathcal{O}(\text{poly } n, \text{polylog } \frac{1}{\epsilon})$,其中 $n$ 是代码长度,$\epsilon$ 是近似误差。最后,我们还提出了一种通过使用近似克隆将 BPQM 扩展到包含循环的因子图的新方法。我们展示了一些有希望的数值结果,表明具有循环的因子图上的 BPQM 可以显着优于最好的经典解码器。
更新日期:2021-09-20
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