Many-hypercube codes [H. Goto, Sci. Adv. 10, eadp6388 (2024)], concatenated ${[[n,n-2,2]]}$ quantum error-detecting codes ($n$ is even), have recently been proposed as high-rate quantum codes suitable for fault-tolerant quantum computing. However, the original many-hypercube codes with ${n=6}$ have large code block sizes at high concatenation levels (216 and 1296 physical qubits per block at levels 3 and 4, respectively), making not only experimental realization difficult but also logical error rates high. Toward earlier experimental realization and lower logical error rates, here we investigate smaller many-hypercube codes obtained by concatenating $[[6,4,2]]$ and/or $[[4,2,2]]$ codes, where, e.g., $D_{6,4,4}$ denotes the many-hypercube code using $[[6,4,2]]$ at level 1 and $[[4,2,2]]$ at levels 2 and 3. As a result, we found a surprising fact: $D_{6,4,4}$ ($D_{6,6,4,4}$) can achieve lower block error rates than $D_{4,4,4}$ ($D_{4,4,4,4}$), despite its higher encoding rate. Focusing on level 3, we also developed efficient fault-tolerant encoders realizing about 60% overhead reduction while maintaining or even improving the performance, compared to the original design. Using them, we numerically confirmed that $D_{6,4,4}$ also achieves the best performance for logical controlled-NOT gates in a circuit-level noise model. These results will be useful for early experimental realization of fault-tolerant quantum computing with high-rate quantum codes.

翻译：多超立方体编码 [H. Goto, Sci. Adv. 10, eadp6388 (2024)] 作为级联的 ${[[n,n-2,2]]}$ 量子错误检测码（$n$ 为偶数），近期被提出作为适用于容错量子计算的高速率量子编码。然而，原始的多超立方体编码在 ${n=6}$ 时，在高阶级联水平下具有较大的码块尺寸（在水平 3 和 4 时，每个块分别需要 216 和 1296 个物理量子比特），这不仅使得实验实现困难，还导致逻辑错误率较高。为了更早的实验实现和更低的逻辑错误率，本文研究了通过级联 $[[6,4,2]]$ 和/或 $[[4,2,2]]$ 码得到的更小多超立方体编码，其中，例如 $D_{6,4,4}$ 表示在水平 1 使用 $[[6,4,2]]$ 码，在水平 2 和 3 使用 $[[4,2,2]]$ 码的多超立方体编码。结果，我们发现了一个令人惊讶的事实：$D_{6,4,4}$（$D_{6,6,4,4}$）可以实现比 $D_{4,4,4}$（$D_{4,4,4,4}$）更低的块错误率，尽管其编码速率更高。聚焦于水平 3，我们还开发了高效的容错编码器，相比原始设计，在保持甚至提升性能的同时，实现了约 60% 的开销降低。使用这些编码器，我们通过数值模拟确认，在电路级噪声模型中，$D_{6,4,4}$ 在逻辑受控非门方面也实现了最佳性能。这些结果将有助于利用高速率量子编码早期实验实现容错量子计算。