AI 中文总结
针对MLD计算困难问题，提出AMLD框架，回收随机译码器舍弃的样本构建自由能估计器，在两类基准码上显著缩小MWD与MLD的阈值差距并降低逻辑错误率。
AI 中文摘要
最大似然译码（MLD）在已知独立同分布（i.i.d.）泡利噪声下可实现稳定器码的最小逻辑错误率，但其精确计算是#P困难的。因此，实际流程通过最小权重译码（MWD）近似MLD，仅保留每个陪集症候中权重最低的恢复操作，舍弃陪集简并性，而最小权重搜索又由随机求解器实现。我们提出近似最大似然译码（AMLD），这是一种黑箱框架，可将随机内部译码器舍弃的候选样本回收为每类截断自由能估计器。对于候选池中表示的每个逻辑类，该估计器被证明下界为精确自由能，上界为经验最小权重。AMLD以线性经典开销返回使估计自由能最小的逻辑类。在基于模拟退火（SA）的伊辛译码器基准测试中，AMLD在环面码和颜色码的比特翻转及去极化噪声下，可弥合MWD与MLD阈值间高达83%的差距；在6.6.6颜色码的去极化噪声下，阈值提升最大，从17.28%升至18.62%。我们还在[[144,12,12]]双变量自行车码上验证了AMLD，其比特翻转译码问题具有超图结构，该应用无需基于匹配的枚举或特定于码的张量网络收缩；在p=0.05时，相较于在同一BP-OSD候选池上评估的MWD，AMLD将逻辑错误率降低了13%。
英文摘要
Maximum-likelihood decoding (MLD) achieves the minimum logical error rate of stabilizer codes under known i.i.d. Pauli noise, but its exact evaluation is \#P-hard. Practical pipelines therefore approximate MLD by minimum-weight decoding (MWD), retaining only the lowest-weight recovery per syndrome and discarding the coset degeneracy. The minimum-weight search is in turn implemented by stochastic solvers. We introduce approximate maximum-likelihood decoding (AMLD), a black-box framework that recycles the candidate samples discarded by stochastic inner decoders into a per-class truncated free-energy estimator. For every logical class represented in the candidate pool, the estimator is provably bounded below by the exact free energy and above by the empirical minimum weight. AMLD returns the logical class minimizing the estimated free energy with linear classical overhead. In SA-based Ising-decoder benchmarks, AMLD closes up to $83\%$ of the MWD--MLD threshold gap across the toric and color codes under bit-flip and depolarizing noise. The largest threshold improvement, from $17.28\%$ to $18.62\%$, occurs on the $6.6.6$ color code under depolarizing noise. We further demonstrate AMLD on the $[[144,12,12]]$ bivariate-bicycle code, whose bit-flip decoding problem has a hypergraph structure. This application requires neither matching-based enumeration nor code-specific tensor-network contraction. At $p=0.05$, AMLD reduces the logical error rate by $13\%$ relative to MWD evaluated on the same BP-OSD candidate pool.
Comments12 pages, 7 figures
