发表机构
University of Illinois at Urbana-Champaign; Korea Institute for Advanced Study（伊利诺伊大学厄巴纳-香槟分校; 韩国高等科学研究院）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文为群值量子纠错码提出设计原则，引入框架宽度参数，构造低权重全正则逻辑基的LP和PP码，并给出距离认证方法，实现高距离与低权重的平衡。
AI 中文摘要
高效的容错量子计算架构受益于低权重的逻辑算子。然而，已报道的高速率、高距离码的规范基可能比码距离重几倍，且基本障碍可能阻止最小权重逻辑算子形成完整的规范基。为解决此问题，我们为群值量子纠错码开发了具有低权重全正则规范逻辑基的设计原则，这些基通过平移一组种子算子生成。我们引入框架宽度 $f$，即允许完整规范逻辑基的最小权重截止值，作为除距离 $d$ 和校验权重 $w$ 之外的附加码参数。将这些参数与显式逻辑基一起编目，可以为资源估计和容错逻辑操作的编译提供信息。首先，我们确定一个结构障碍：对于奇数阶群上的完全填充二元单项式CSS校验，全正则性强制距离为二，而对于二的幂阶群，在将每个群元素替换为 $1$ 后，两个校验矩阵的满行秩保证全正则性。其次，我们通过显式的高速率提升积（LP）和对划分（PP）码展示我们的原则：一个 $[[1088,128,22]]$ LP码，$f=22$，以及一个 $[[1024,256,24]]$ PP码，$f \leq 27$，两者均在 $w=10$。在 $w=11$ 时，我们的PP构造达到 $d=25$ 且 $f \leq 27$，或 $d=24$ 且证明最优的 $f=25$，展示了这些参数之间的权衡以及距离与框架宽度之间的严格分离。最后，我们提出一种结合对称约简和基于稳定器的剪枝的方法，用于穷举距离认证，并使用它高效地建立超过一千个量子比特的LP示例的精确距离。
英文摘要
Efficient fault-tolerant quantum computing architectures benefit from low-weight logical operators. However, reported canonical bases for high-rate, high-distance codes can be several times heavier than the code distance, and fundamental obstructions can prevent minimum-weight logical operators from forming a complete canonical basis. To address this issue, we develop design principles for group-valued quantum error-correcting codes with low-weight full regular canonical logical bases, generated by translating a set of seed operators. We introduce frame width $f$, the smallest weight cutoff that permits a complete canonical logical basis, as an additional code parameter alongside distance $d$ and check weight $w$. Cataloguing these parameters together with explicit logical bases can inform resource estimates and the compilation of fault-tolerant logical operations. First, we identify a structural obstruction: for fully populated binary monomial CSS checks over a group of odd order, full regularity forces distance two, while for groups of power-of-two order, full row rank of both check matrices after replacing every group element by $1$ guarantees full regularity. Second, we demonstrate our principles through explicit high-rate lifted-product (LP) and pair-partition (PP) codes: an $[[1088,128,22]]$ LP code with $f=22$ and an $[[1024,256,24]]$ PP code with $f \leq 27$, both at $w=10$. At $w=11$, our PP constructions attain either $d=25$ with $f \leq 27$ or $d=24$ with the proved optimum $f=25$, demonstrating both a tradeoff among these parameters and a strict separation between distance and frame width. Finally, we propose a method combining symmetry reduction and stabilizer-based pruning for exhaustive distance certification, and use it to efficiently establish exact distances for LP examples with more than a thousand qubits.
