量子码的局部概率译码
Local Probabilistic Decoding of a Quantum Code
AI总结：
针对flip译码器应用于量子码时存在不可纠正恒权错误的问题，研究发现三维环面码的环状校验子下flip存在阈值，结合置信传播与概率型flip的局部译码策略在唯象噪声下阈值约5.5%，运行效率远高于现有最优方案，可推广至其他低密度奇偶校验码。
摘要：
flip是一种极为简单且最大局部化的经典译码器，在特定类别经典码中已取得极佳应用效果。将其应用于量子码时，存在一些恒权错误（例如半个稳定子）是该译码器无法纠正的，因此过往研究已对flip的改进版本展开探讨，有时还会将其与其他译码器结合使用。我们提出这一做法并非始终必要，并给出数值证据表明：将flip应用于立方格点上三维环面码的环状校验子（syndrome）时，存在译码阈值。该结果可归因于以下事实：对该译码器而言，权重最低的不可纠正错误与可纠正错误的（汉明距离）更近，而非与其他不可纠正错误更近，因此在后续码周期中，经额外噪声作用发生变换后，这类错误很可能变得可纠正。向译码器中引入随机性可使其以有限概率纠正这类“不可纠正”错误，对于结合置信传播（belief propagation）与概率型flip的译码策略，我们观测到其在唯象噪声下的阈值约为5.5%。这一结果与该码已知的最佳阈值（约7.1%）相当，后者是通过置信传播结合有序统计译码[Higgott和Breuckmann，2022]实现的；该策略的运行时间为O(n³)，而我们的局部译码器运行时间为O(n)（并行化后为O(1)）。我们预计该策略可推广至其他低密度奇偶校验码中并取得良好效果，也希望这些结果能推动对其他此前被忽视的译码器的研究。
英文摘要：
flip is an extremely simple and maximally local classical decoder which has been used to great effect in certain classes of classical codes. When applied to quantum codes there exist constant-weight errors (such as half of a stabiliser) which are uncorrectable for this decoder, so previous studies have considered modified versions of flip, sometimes in conjunction with other decoders. We argue that this may not always be necessary, and present numerical evidence for the existence of a threshold for flip when applied to the looplike syndromes of a three-dimensional toric code on a cubic lattice. This result can be attributed to the fact that the lowest-weight uncorrectable errors for this decoder are closer (in terms of Hamming distance) to correctable errors than to other uncorrectable errors, and so they are likely to become correctable in future code cycles after transformation by additional noise. Introducing randomness into the decoder can allow it to correct these "uncorrectable" errors with finite probability, and for a decoding strategy that uses a combination of belief propagation and probabilistic flip we observe a threshold of $\sim5.5\%$ under phenomenological noise. This is comparable to the best known threshold for this code ($\sim7.1\%$) which was achieved using belief propagation and ordered statistics decoding [Higgott and Breuckmann, 2022], a strategy with a runtime of $O(n^3)$ as opposed to the $O(n)$ ($O(1)$ when parallelised) runtime of our local decoder. We expect that this strategy could be generalised to work well in other low-density parity check codes, and hope that these results will prompt investigation of other previously overlooked decoders.
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