Rounding Almost Commuting Hamiltonians

2026-05-25

AI Summary

This work addresses the theoretical and computational challenges arising from the approximate commutativity of near-commuting Hamiltonians by proposing a locality-preserving, algorithmic rounding procedure that efficiently approximates two-local qubit Hamiltonians with strictly commuting ones. Leveraging operator norm estimates and commutator control, the method establishes—for the first time—an effective approximation framework that transforms near-commuting Hamiltonians into exactly commuting counterparts while preserving locality and achieving a total error bound of $O(m\varepsilon^{1/6})$. As a key contribution, this framework places the ground-state energy approximation problem within the complexity class NP and enables significant applications in Gibbs sampling and fast Hamiltonian simulation, thereby substantially broadening the scope of tractable quantum many-body problems.

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 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.

Problem

• almost commuting Hamiltonians
• quantum many-body physics
• ground energy approximation
• computational complexity
• 2-local qubit Hamiltonians

Innovation

• almost commuting Hamiltonians
• algorithmic rounding
• locality-preserving
• ground energy approximation
• Hamiltonian simulation