拓扑量子纠错中的罕见事件与格里菲斯相
Rare Events and Griffiths Phases in Topological Quantum Error Correction
AI总结：
该研究探讨了拓扑量子纠错中非均匀错误率的影响，发现1D重复码存在两种解码相，而2D扭码则因罕见事件导致解码失败。
摘要：
量子纠错（QEC）码的性能通常在假设空间-时间均匀错误率的情况下进行研究。另一方面，实验实现几乎总是产生异质性错误率，这可能是由于不完美的制造和/或宇宙射线等效应导致的。因此，了解这些异质性错误率是否以及如何以定性方式影响QEC性能非常重要。在本工作中，我们研究了非均匀错误率在1D重复码和2D扭码的代表性例子中的影响，重点是当它们扩展了空间-时间相关性时的情况；这些可能源于罕见事件（如宇宙射线）暂时提高整个编码区域的错误率。这些效果可以在相应的解码统计力学模型中描述，其中错误率的长程相关性导致了更弱耦合的扩展罕见区域。对于1D重复码，其中罕见区域是线性的，我们发现两种不同的可解码相：一种是传统有序相，其中逻辑失败率随编码距离指数衰减；另一种是受罕见区域主导的格里菲斯相，其中失败率参数化更大且衰减为伸展指数。特别是，后者相存在当罕见区域的错误率高于体阈值时。对于2D扭码，其中罕见区域是平面的，我们发现没有可解码的格里菲斯相：提升错误率超过体阈值的罕见事件导致渐近阈值损失和解码失败。解析失败机制表明，抑制扩展的重复罕见事件序列（否则将具有高概率统计存在）的技术对于使用扭码的QEC至关重要。
英文摘要：
The performance of quantum error correcting (QEC) codes are often studied under the assumption of spatio-temporally uniform error rates. On the other hand, experimental implementations almost always produce heterogeneous error rates, in either space or time, as a result of effects such as imperfect fabrication and/or cosmic rays. It is therefore important to understand if and how their presence can affect the performance of QEC in qualitative ways. In this work, we study effects of non-uniform error rates in the representative examples of the 1D repetition code and the 2D toric code, focusing on when they have extended spatio-temporal correlations; these may arise, for instance, from rare events (such as cosmic rays) that temporarily elevate error rates over the entire code patch. These effects can be described in the corresponding statistical mechanics models for decoding, where long-range correlations in the error rates lead to extended rare regions of weaker coupling. For the 1D repetition code where the rare regions are linear, we find two distinct decodable phases: a conventional ordered phase in which logical failure rates decay exponentially with the code distance, and a rare-region dominated Griffiths phase in which failure rates are parametrically larger and decay as a stretched exponential. In particular, the latter phase is present when the error rates in the rare regions are above the bulk threshold. For the 2D toric code where the rare regions are planar, we find no decodable Griffiths phase: rare events which boost error rates above the bulk threshold lead to an asymptotic loss of threshold and failure to decode. Unpacking the failure mechanism implies that techniques for suppressing extended sequences of repeated rare events (which, without intervention, will be statistically present with high probability) will be crucial for QEC with the toric code.
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