Rounding Almost Commuting Hamiltonians

Abstract

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 motivates the study of \emph{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 new locality-preserving \emph{algorithmic rounding technique} that maps any $2$-local Hamiltonian $H=\sum_{i=1}^m h_i$ with $|[h_i,h_j]| \leq \varepsilon$ to a nearby Hamiltonian $\hat{H}$ whose terms pair-wise commute, and which is within overall distance $|H-\hat{H}| = O(m,\varepsilon^{1/6})$.

As a consequence, we show that $\delta$-approximations to the ground energy for $\varepsilon$-almost commuting $2$-local qubit Hamiltonians lie in $\mathsf{NP}$ when $\delta \gg m\varepsilon^{1/6}$, 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.