发表机构
Yau Mathematical Sciences Center, Tsinghua University; State Key Laboratory of Low Dimensional Quantum Physics, Department of Physics, Tsinghua University; Joint Center for Quantum Information and Computer Science, University of Maryland; Shanghai Institute for Mathematics and Interdisciplinary Sciences (SIMIS)（清华大学丘成桐数学科学中心; 清华大学物理系低维量子物理国家重点实验室; 马里兰大学量子信息与计算机联合中心; 上海数学与交叉学科研究院）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本研究证明一般量子LDPC码对局部相干噪声具有非零码容量阈值，通过簇重求和技术，在距离为对数阶时实现逻辑恢复误差随码距离指数衰减，为容错量子计算提供鲁棒性保证。
AI 中文摘要
量子低密度奇偶校验（qLDPC）码的一个关键吸引力在于其能够在非零阈值以下抑制随机泡利噪声。相干错误本质上是不同的：它们产生错误模式的叠加，这些模式的振幅即使在综合征测量之后也可能发生干涉。对相干错误的严格理解仍然有限。在这里，我们证明一般的qLDPC码对局部相干噪声以及更一般的局部信道噪声具有非零的码容量阈值。对于任何距离为$d=\Omega(\log n)$的qLDPC码族，我们证明存在一个恒定的噪声强度，低于该强度时，金刚石距离下的逻辑恢复误差随码距离指数衰减。该结果对最优恢复以及最小权重解码器均成立。关键技术要素是我们所称的“簇重求和”：我们不是逐一限制叠加的错误配置，而是在信道展开中隔离一个大的连通错误簇，并在取范数之前精确重求和所有与之不相连的错误。然后，标准的簇计数产生指数抑制。这项工作解决了容错理论中一个长期存在的挑战，为qLDPC码针对相干噪声提供了普遍的鲁棒性保证，并为未来容错量子技术的研究奠定了严格的基础。
英文摘要
A key appeal of quantum low-density parity check (qLDPC) codes is their ability to suppress stochastic Pauli noise below nonzero thresholds. Coherent errors are fundamentally different: they produce superpositions of error patterns whose amplitudes can interfere even after syndrome measurement. Rigorous understanding of coherent errors remains limited. Here we show that general qLDPC codes admit a nonzero code capacity threshold against local coherent noise and more generally local channel noise. For any family of qLDPC codes with distance $d=Ω(\log n)$, we show that there is a constant noise strength below which the logical recovery error in diamond distance decays exponentially with the code distance. The result is established for optimal recovery as well as the minimum-weight decoder. The key technical ingredient is what we call a \emph{cluster resummation}: rather than bounding superposed error configurations one by one, we isolate a large connected error cluster in the channel expansion and exactly resum all errors disconnected from it before taking norms. Standard cluster counting then yields exponential suppression. This work resolves a longstanding challenge in fault tolerance theory, providing general robustness guarantees for qLDPC codes against coherent noise and laying a rigorous foundation for future studies of fault-tolerant quantum technologies.
Comments19 pages, 2 figures
