Commuting Hamiltonians lie at the boundary between classical constraint satisfaction and quantum many-body physics, exhibiting rich quantum structure while remaining more tractable than general noncommuting models. In contrast, physical Hamiltonians are rarely exactly commuting, which naturally motivates the study of almost commuting Hamiltonians. Despite their relevance, the implications of approximate commutation are only poorly understood. In this work, we show how to efficiently approximate any almost commuting $2$-local qubit Hamiltonian by a commuting one: we give a locality-preserving algorithmic rounding technique that maps any $2$-local Hamiltonian $H=\sum_{i=1}^m h_i$ with $\|[h_i,h_j]\| \leq ε$ to a nearby Hamiltonian $\hat{H}$ whose terms pair-wise commute, and which is within overall distance $\|H-\hat{H}\| = O(m\,ε^{1/6})$. As a consequence, we show that $δ$-approximations to the ground energy for $ε$-almost commuting $2$-local qubit Hamiltonians lie in $\mathsf{NP}$ when $δ\gg mε^{1/6}$, extending the classical containment well beyond the commuting setting. Finally, we present two applications of our rounding framework: Gibbs sampling and fast Hamiltonian simulation for almost commuting systems.

翻译：交换哈密顿量位于经典约束满足与量子多体物理的边界，在保持比一般非交换模型更易处理的同时展现出丰富的量子结构。然而，物理哈密顿量很少严格交换，这自然引出了对几乎交换哈密顿量的研究。尽管其具有重要性，但近似交换的深层含义仍未被充分理解。在本工作中，我们展示了如何高效地将任意几乎交换的$2$-局域量子比特哈密顿量近似为交换哈密顿量：我们提出了一种保持局域性的算法取整技术，可将任意满足$\|[h_i,h_j]\| \leq ε$的$2$-局域哈密顿量$H=\sum_{i=1}^m h_i$映射到近邻哈密顿量$\hat{H}$，其所有项两两交换，且全局距离满足$\|H-\hat{H}\| = O(m\,ε^{1/6})$。由此我们证明，当$δ\gg mε^{1/6}$时，对于$ε$-几乎交换的$2$-局域量子比特哈密顿量，其基态能量的$δ$-近似属于$\mathsf{NP}$类，将经典包含性从交换情形显著扩展至非交换领域。最后，我们展示了取整框架的两项应用：几乎交换系统的吉布斯采样与快速哈密顿量模拟。