使用分形拓扑码进行量子纠错
Quantum error correction with fractal topological codes
AI总结：
本文研究分形表面码作为容错量子存储器的纠错性能，改造 sweep decoder 处理比特翻转并用 MWPM 处理相位翻转，给出非零阈值及与禁闭-Higgs 相变的联系。
摘要：
最近，一类分形表面码（FSC）已在 Hausdorff 维数为 $2+ε$ 的分形晶格上构造出来，并支持容错的非 Clifford CCZ 门。我们研究了此类 FSC 作为容错量子存储器的性能。我们证明，对于 Hausdorff 维数为 $2+ε$ 的 FSC 中的比特翻转和相位翻转错误，存在具有非零阈值的解码策略。对于比特翻转错误，我们通过在分形晶格孔洞边界上设计适当修改，将为规则三维表面码中的弦状综合征开发的 sweep decoder 改造并应用于 FSC。我们为 FSC 改造的 sweep decoder 保持了其自校正和单脉冲特性。对于相位翻转错误，我们对点状综合征采用最小权完美匹配（MWPM）解码器。对于一个 Hausdorff 维数 $D_H\approx2.966$ 的特定 FSC，我们报告 sweep decoder 在唯象噪声下具有可持续容错阈值（约 $1.7\\%$），MWPM decoder 的码容量阈值下界为 $2.95\\%$。后者可映射为分形晶格上禁闭-Higgs 相变临界点的下界，而该临界点可通过 Hausdorff 维数调节。
英文摘要：
Recently, a class of fractal surface codes (FSCs), has been constructed on fractal lattices with Hausdorff dimension $2+ε$, which admits a fault-tolerant non-Clifford CCZ gate. We investigate the performance of such FSCs as fault-tolerant quantum memories. We prove that there exist decoding strategies with non-zero thresholds for bit-flip and phase-flip errors in the FSCs with Hausdorff dimension $2+ε$. For the bit-flip errors, we adapt the sweep decoder, developed for string-like syndromes in the regular 3D surface code, to the FSCs by designing suitable modifications on the boundaries of the holes in the fractal lattice. Our adaptation of the sweep decoder for the FSCs maintains its self-correcting and single-shot nature. For the phase-flip errors, we employ the minimum-weight-perfect-matching (MWPM) decoder for the point-like syndromes. We report a sustainable fault-tolerant threshold ($\sim 1.7\%$) under phenomenological noise for the sweep decoder and the code capacity threshold (lower bounded by $2.95\%$) for the MWPM decoder for a particular FSC with Hausdorff dimension $D_H\approx2.966$. The latter can be mapped to a lower bound of the critical point of a confinement-Higgs transition on the fractal lattice, which is tunable via the Hausdorff dimension.
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