The Complexity of Single-Interaction Hamiltonians
Abstract
We study the complexity of Local Hamiltonian problems under the restriction that every local term is an unweighted occurrence of the same fixed interaction. In the resulting Single-Interaction Hamiltonian (SIH) model, once the interaction is fixed, the Hamiltonian is specified entirely by its ordered interaction hypergraph. Our main technical tool is an exact compiler that converts any fixed finite alphabet of local interactions into a single interaction. On a designated auxiliary subspace, the compiled Hamiltonian reproduces the source Hamiltonian exactly, while the complete spectrum below a chosen separation scale is preserved with multiplicity. Combined with a finite-alphabet normalization, this gives a universal polynomial-time reduction from k-Local Hamiltonian to SIH3k+3 up to a known global energy scale and inverse-polynomial approximation. Sharper constructions give QMA-completeness of SIH3 and of geometrically local SIH8 with bounded degree and interaction range. In the stoquastic setting, we obtain StoqMA-completeness of Stoq-SIH3 and QMA-completeness of 1-Pinned Stoq-SIH4. We also establish hardness results for uniformly system-size-dependent interactions and derive single-interaction formulations of the Hamiltonian quantum PCP conjecture, frustration-free, exponentially precise, and guided Local Hamiltonian problems. These results show that independently chosen local matrices and independently tunable coupling strengths are not necessary for a broad range of Hamiltonian-complexity phenomena.
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BibTeX
@misc{hecker2026complexity,
title = {{The Complexity of Single-Interaction Hamiltonians}},
author = {Elad Hecker},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.39592},
eprint = {2609.39592},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.39592},
}
RIS
TY - UNPB
TI - The Complexity of Single-Interaction Hamiltonians
AU - Elad Hecker
PY - 2026
DA - 2026/09/30
PB - arXiv
DO - 10.48550/arxiv.2609.39592
UR - https://arxiv.org/abs/2609.39592
AB - We study the complexity of Local Hamiltonian problems under the restriction that every local term is an unweighted occurrence of the same fixed interaction. In the resulting Single-Interaction Hamiltonian (SIH) model, once the interaction is fixed, the Hamiltonian is specified entirely by its ordered interaction hypergraph. Our main technical tool is an exact compiler that converts any fixed finite alphabet of local interactions into a single interaction. On a designated auxiliary subspace, the compiled Hamiltonian reproduces the source Hamiltonian exactly, while the complete spectrum below a chosen separation scale is preserved with multiplicity. Combined with a finite-alphabet normalization, this gives a universal polynomial-time reduction from k-Local Hamiltonian to SIH_3k+3 up to a known global energy scale and inverse-polynomial approximation. Sharper constructions give QMA-completeness of SIH_3 and of geometrically local SIH_8 with bounded degree and interaction range. In the stoquastic setting, we obtain StoqMA-completeness of Stoq-SIH_3 and QMA-completeness of 1-Pinned Stoq-SIH_4. We also establish hardness results for uniformly system-size-dependent interactions and derive single-interaction formulations of the Hamiltonian quantum PCP conjecture, frustration-free, exponentially precise, and guided Local Hamiltonian problems. These results show that independently chosen local matrices and independently tunable coupling strengths are not necessary for a broad range of Hamiltonian-complexity phenomena.
ER -