用于最优且高效译码的量子消息传递算法
Quantum message-passing algorithm for optimal and efficient decoding
AI总结：
本文证明 BPQM 对树状 Tanner 图二元线性码实现最优译码，指出其电路指数膨胀缺陷，并提出多项式复杂度的真正量子消息传递近似算法及基于近似克隆的含环因子图扩展。
摘要：
最近，Renes 提出了一种称为带量子消息的置信传播（BPQM）的量子算法，用于对通过纯态 CQ 信道传输、采用具有树状 Tanner 图的二元线性码编码的经典数据进行译码 [Renes, NJP 19 072001 (2017)]。该算法呈现了基于经典置信传播算法的译码的真正量子对应物；经典置信传播与 LDPC 码或 Turbo 码结合使用时，已在经典编码理论中取得广泛成功。更近一些，Rengaswamy 等人 [npj Quantum Information 7 97 (2021)] 观察到 BPQM 在一个小型示例码上实现了最优译码器。在此，我们通过以下贡献显著扩展对 BPQM 算法的理解、形式化和适用性。首先，我们通过解析证明，BPQM 对任何具有树状 Tanner 图的二元线性码都能实现最优译码。我们还首次完整、详细且毫无歧义地给出了 BPQM 算法的形式化描述。在此过程中，我们发现了原始算法及后续工作忽视的一个关键缺陷，该缺陷意味着量子电路实现的规模将随码维数呈指数增长。尽管 BPQM 传递量子消息，但算法所需的其他信息是全局处理的。我们通过构造一种真正的消息传递算法来解决这个问题，该算法近似 BPQM，并且量子电路复杂度为 $\mathcal{O}(\text{poly } n, \text{polylog } \frac{1}ε)$，其中 $n$ 是码长，$ε$ 是近似误差。最后，我们还提出了一种利用近似克隆将 BPQM 扩展到含环因子图的新方法。我们展示了一些有前景的数值结果，这些结果表明，在含环因子图上运行的 BPQM 可以显著优于可能的最佳经典译码器。
英文摘要：
Recently, Renes 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)]. The algorithm presents a genuine quantum counterpart to decoding based on the classical belief propagation algorithm, 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)] observed that BPQM implements the optimal decoder on a small example code. 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 dimension. 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 quantum circuit complexity $\mathcal{O}(\text{poly } n, \text{polylog } \frac{1}ε)$, where $n$ is the code length and $ε$ 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.
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