AI 中文总结
该研究针对量子LDPC码译码难题，构建两种译码器，其中采样译码器性能优于主流BP+OSD且可对多数决策提供认证，区域译码器可实现特定码的精确简并最大似然译码。
AI 中文摘要
量子低密度奇偶校验（qLDPC）码可将容错量子计算的量子比特开销降低一个数量级，但其译码难度高于经典对应方案：由于许多物理错误在稳定子意义上等价，简并最大似然（ML）译码器必须比较整个错误等价类（即配分函数）的概率，而非单个错误的概率。主流译码器BP+OSD以启发式方式规避简并性，且不提供任何保证。我们将简并译码视为无向图模型中的概率推断：每个逻辑类的概率是码的校验变量上无约束、严格正的马尔可夫随机场的配分函数，该构造将表面码的随机键伊辛映射推广至任意CSS码，以及存在测量误差和电路级噪声的时空译码。基于该模型，我们构建了两种译码器：第一种通过带公共随机数的退火重要采样估计所有类配分函数，并为每个决策附加最优性证书——配对自助检验，或结合WISH等常数因子估计器时的精确最优性证明；第二种基于区域：其偏差在类间抵消的贝特自由能，在所有测试的表面码实例上以毫秒级成本重现精确ML译码，将区域扩展至消除簇可使[[72,12,6]]双变量自行车码的精确简并ML译码成为可能。在表面码及双变量自行车码[[72,12,6]]、[[144,12,12]]上，针对码容量、唯象及电路级噪声，采样译码器的性能与BP+OSD相当或更优，同时为大部分决策提供认证，且该证书能准确标记任何快速译码器应被不信任的症候群。
英文摘要
Quantum low-density parity-check (qLDPC) codes reduce the qubit overhead of fault-tolerant quantum computation by an order of magnitude, but their decoding is harder than its classical counterpart: because many physical errors are equivalent up to stabilizers, the degenerate maximum-likelihood (ML) decoder must compare the probabilities of entire equivalence classes of errors, that is, partition functions, rather than single errors. The workhorse decoder BP+OSD sidesteps degeneracy heuristically and offers no guarantees. We treat degenerate decoding as probabilistic inference in an undirected graphical model: the probability of each logical class is the partition function of an unconstrained, strictly positive Markov random field over the code's check variables, a construction that generalizes the random-bond Ising mapping of the surface code to arbitrary CSS codes and to spacetime decoding with measurement errors and circuit-level noise. On this model we build two decoders. The first estimates all class partition functions by annealed importance sampling with common random numbers and attaches to every decision a certificate of optimality: a paired bootstrap test, or, composed with constant-factor estimators such as WISH, an exact optimality proof. The second is region-based: the Bethe free energy, whose bias cancels between classes, reproduces exact ML decoding on every tested surface-code instance at millisecond cost, and enlarging the regions to elimination clusters makes exact degenerate ML decoding of the [[72,12,6]] bivariate bicycle code feasible. Across surface codes and the bivariate bicycle codes [[72,12,6]] and [[144,12,12]], under code-capacity, phenomenological, and circuit-level noise, the sampling decoder matches or exceeds BP+OSD while certifying the bulk of its decisions, and the certificate flags exactly the syndromes on which any fast decoder should be distrusted.
