The Ising framework maps the decoding problem in quantum error correction onto ground-state optimization of a classical Hamiltonian, in which $X$-$Z$ error correlations enter as cross terms. Under phenomenological depolarizing noise, the exact joint formulation contains up to 8-body interactions for the toric code and 10-body for the $6.6.6$ color code. These high-order terms degrade solver convergence, inflate runtime, and raise the auxiliary spin overhead when embedding into native 2-body Ising hardware. In this work, we propose the iterative low-order decoding (ILOD) algorithm, which alternates between $X$- and $Z$-type sub-Hamiltonians, approximating cross-type correlations through Bayesian priors that reweight each type's couplings using the other type's inferred error configuration. This halves the maximum body count of interaction terms in the Hamiltonian, accelerating the solver, restoring convergence at larger code distances, and reducing the total spin count for 2-body embedding by a factor of $2.5$. For the toric code, ILOD attains a threshold of $4.73%$ versus $4.83%$ for the joint formulation, with the empirical runtime ratio scaling as $(0.81)^d$. For the $6.6.6$ color code, their thresholds agree within statistical uncertainty for small code distances, and ILOD remains convergent for larger distances where the joint formulation fails to converge despite a larger annealing budget.

翻译：Ising框架将量子纠错中的解码问题映射为经典哈密顿量的基态优化问题，其中$X$-$Z$误差关联以交叉项形式出现。在现象学去极化噪声下，精确联合公式对于环面码包含最多8体相互作用，对于$6.6.6$色码则包含10体相互作用。这些高阶项会降低求解器的收敛性、增加运行时间，并在嵌入到本征2体Ising硬件时提高辅助自旋开销。本文提出迭代式低阶解码(ILOD)算法，该算法在$X$型和$Z$型子哈密顿量之间交替进行，通过贝叶斯先验来近似交叉类型关联——利用另一种类型推断出的误差配置重新加权每种类型的耦合项。这使哈密顿量中相互作用项的最大体数减半，从而加速求解器、在更大编解码距离下恢复收敛性，并将2体嵌入的总自旋数降低至原值的$2.5$分之一。对于环面码，ILOD达到$4.73\%$的阈值，而联合公式为$4.83\%$，经验运行时间比缩放为$(0.81)^d$。对于$6.6.6$色码，两者阈值在小编解码距离下在统计不确定度内一致，且在更大距离下——尽管联合公式即使采用更大的退火预算也无法收敛——ILOD仍能保持收敛。