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 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 new locality-preserving algorithmic rounding technique that maps any $2$-local Hamiltonian H=∑_i=1^m h_i with |[h_i,h_j]| ≤ ε to a nearby Hamiltonian Ĥ whose terms pair-wise commute, and which is within overall distance |H−Ĥ| = O(m ε^1/3). As a consequence, we show that δ-approximations to the ground energy for ε-almost commuting $2$-local qubit Hamiltonians lie in NP when δ≫ mε^1/3, 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.

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BibTeX RIS

BibTeX

@misc{faisal2026rounding,
  title = {{Rounding Almost Commuting Hamiltonians}},
  author = {Islam Faisal and Anand Natarajan and Alexander Poremba},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2605.26096},
  eprint = {2605.26096},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2605.26096},
  note = {CC BY 4.0},
}

RIS

TY  - UNPB
TI  - Rounding Almost Commuting Hamiltonians
AU  - Islam Faisal
AU  - Anand Natarajan
AU  - Alexander Poremba
PY  - 2026
DA  - 2026/10/06
PB  - arXiv
DO  - 10.48550/arxiv.2605.26096
UR  - https://arxiv.org/abs/2605.26096
AB  - 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 new locality-preserving algorithmic rounding technique that maps any $2$-local Hamiltonian H=∑_i=1^m h_i with |[h_i,h_j]| ≤ ε to a nearby Hamiltonian Ĥ whose terms pair-wise commute, and which is within overall distance |H−Ĥ| = O(m ε^1/3). As a consequence, we show that δ-approximations to the ground energy for ε-almost commuting $2$-local qubit Hamiltonians lie in NP when δ≫ mε^1/3, 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.
ER  -

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