$o(1)$ 时间内的稀疏开花解码
Sparse-Blossom Decoding in $o(1)$ Time
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中文总结 AI 辅助
针对量子纠错中匹配解码的加速问题，提出并行稀疏开花算法，证明其与串行版本等价，并证明平均并行运行时间为 $o(1)$，为可扩展并行解码奠定基础。
中文摘要 AI 辅助
基于匹配的解码广泛应用于量子纠错，加速解码是实现快速且可扩展的容错量子计算的关键。最小权重完美匹配（MWPM）解码为错误抑制提供了严格的保证，而稀疏开花算法则使其在适度问题规模下得以实际实现。然而，现有稀疏开花实现的运行时间不可避免地随问题规模增加而增长，这促使我们提出一个严格的并行化框架，以保证正确性且运行时间短于综合征提取的时间尺度。在此，我们提出这样一个框架，并证明由此产生的并行稀疏开花算法产生与原始非并行稀疏开花算法相同的修正结果。对于码距为 $d$ 且物理错误率低于有限阈值的旋转表面码，我们证明 $O(d)$ 轮综合征提取的解码平均并行运行时间上界为 $d$ 的拟多对数函数。对于 $d$ 轮解码窗口，这意味着每轮的平均并行运行时间为 $o(1)$。我们还进行了数值模拟，以确定并行每轮运行时间随码距增加而减少的条件。这些结果表明，增加码距不一定导致更长的并行解码时间，为可扩展的基于匹配的并行解码奠定了基础。
英文摘要
Matching-based decoding is widely used in quantum error correction, and accelerating it is key to enabling fast and scalable fault-tolerant quantum computation. Minimum-weight perfect matching (MWPM) decoding provides rigorous guarantees for error suppression, while sparse blossom enables its practical implementation at modest problem sizes. However, the runtime of existing sparse-blossom implementations unavoidably increases with problem size, motivating a rigorous parallelization framework that guarantees correctness and a runtime shorter than the syndrome-extraction timescale. Here, we present such a framework and prove that the resulting parallel sparse-blossom algorithm produces the same correction as the original, non-parallel sparse blossom. For the rotated surface code with code distance $d$ and physical error rates below a finite threshold, we prove that the average parallel runtime of decoding for $O(d)$ rounds of syndrome extraction is upper bounded by a quasi-polylogarithmic function of $d$. For a $d$-round decoding window, this implies that the average parallel runtime per round is $o(1)$. We also perform numerical simulation to identify conditions under which the parallel runtime per round decreases with increasing code distance. These results suggest that increasing code distance need not lead to longer parallel decoding times, providing a foundation for scalable parallel matching-based decoding.
发表机构
- The University of Tokyo（东京大学）
机构由 AI 辅助整理，请以论文原文为准。
