Quantum error correction has recently been shown to benefit greatly from specific physical encodings of the code qubits. In particular, several researchers have considered the individual code qubits being encoded with the continuous variable GottesmanKitaev-Preskill (GKP) code, and then imposed an outer discrete-variable code such as the surface code on these GKP qubits. Under such a concatenation scheme, the analog information from the inner GKP error correction improves the noise threshold of the outer code. However, the surface code has vanishing rate and demands a lot of resources with growing distance. In this work, we concatenate the GKP code with generic quantum low-density parity-check (QLDPC) codes and demonstrate a natural way to exploit the GKP analog information in iterative decoding algorithms. We first show the noise thresholds for two lifted product QLDPC code families, and then show the improvements of noise thresholds when the iterative decoder – a hardware-friendly min-sum algorithm (MSA) – utilizes the GKP analog information. We also show that, when the GKP analog information is combined with a sequential update schedule for MSA, the scheme surpasses the well-known CSS Hamming bound for these code families. Furthermore, we observe that the GKP analog information helps the iterative decoder in escaping harmful trapping sets in the Tanner graph of the QLDPC code, thereby eliminating or significantly lowering the error floor of the logical error rate curves. Finally, we discuss new fundamental and practical questions that arise from this work on channel capacity under GKP analog information, and on improving decoder design and analysis.

中文翻译：

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超越 CSS 汉明界的有限速率 QLDPC-GKP 编码方案

量子纠错最近被证明可以从代码量子比特的特定物理编码中受益匪浅。特别是，一些研究人员已经考虑使用连续变量 GottesmanKitaev-Preskill (GKP) 代码对单个代码量子比特进行编码，然后在这些 GKP 量子比特上施加外部离散变量代码，例如表面代码。在这种级联方案下，来自内层 GKP 纠错的模拟信息提高了外层码的噪声阈值。然而，表面代码具有消失率，并且随着距离的增加需要大量资源。在这项工作中，我们将 GKP 代码与通用量子低密度奇偶校验 (QLDPC) 代码连接起来，并展示了一种在迭代解码算法中利用 GKP 模拟信息的自然方法。我们首先展示了两个提升产品 QLDPC 代码系列的噪声阈值，然后展示了迭代解码器（一种硬件友好的最小和算法（MSA））利用 GKP 模拟信息时噪声阈值的改进。我们还表明，当 GKP 模拟信息与 MSA 的顺序更新计划相结合时，该方案超过了这些代码系列众所周知的 CSS Hamming 界限。此外，我们观察到 GKP 模拟信息有助于迭代解码器逃避 QLDPC 码的 Tanner 图中的有害陷阱集，从而消除或显着降低逻辑错误率曲线的错误底限。最后，我们讨论了 GKP 模拟信息下关于信道容量的这项工作产生的新的基本和实际问题，