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. While the original many-hypercube codes with ${n=6}$ can achieve remarkably high encoding rates (about 30% and 20% at concatenation levels 3 and 4, respectively), they have large code block sizes at high 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 per block high. Toward earlier experimental realization and lower logical error rates, here we comprehensively investigate smaller many-hypercube codes with $[[6,4,2]]$ 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 notable fact that $D_{6,4,4}$ ($D_{6,6,4,4}$) 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码率量子编码。虽然原始采用 ${n=6}$ 的多超立方体编码能够实现极高的编码率（在级联层级 3 和 4 分别达到约 30% 和 20%），但其在高层级具有较大的码块规模（在层级 3 和 4 每个码块分别需要 216 和 1296 个物理量子比特），这不仅使得实验实现困难，也导致每个码块的逻辑错误率较高。为了更早的实验实现和更低的逻辑错误率，本文全面研究了采用 $[[6,4,2]]$ 和/或 $[[4,2,2]]$ 码的较小规模多超立方体编码，其中，例如 $D_{6,4,4}$ 表示在层级 1 使用 $[[6,4,2]]$ 码，在层级 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显著事实：$D_{6,4,4}$（$D_{6,6,4,4}$）能够实现比 $D_{4,4,4}$（$D_{4,4,4,4}$）更低的块错误率，尽管其编码率更高。聚焦于层级 3，我们还开发了高效的容错编码器，相较于原始设计，在保持甚至提升性能的同时，实现了约 60% 的开销降低。利用这些编码器，我们通过数值模拟证实，在电路级噪声模型下，$D_{6,4,4}$ 在逻辑受控非门操作中也实现了最佳性能。这些结果将有助于利用高码率量子编码更早地实验实现容错量子计算。