NLTM Hamiltonians from gauged sheaf quantum locally testable codes

@misc{wei2026nltm,
  title = {{NLTM Hamiltonians from gauged sheaf quantum locally testable codes}},
  author = {Fuchuan Wei and Zhengyi Han and Zimu Li and Zi-Wen Liu},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.40220},
  eprint = {2609.40220},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.40220},
}

TY  - UNPB
TI  - NLTM Hamiltonians from gauged sheaf quantum locally testable codes
AU  - Fuchuan Wei
AU  - Zhengyi Han
AU  - Zimu Li
AU  - Zi-Wen Liu
PY  - 2026
DA  - 2026/09/30
PB  - arXiv
DO  - 10.48550/arxiv.2609.40220
UR  - https://arxiv.org/abs/2609.40220
AB  - Understanding the complexity of low-energy quantum states is a central problem in quantum complexity theory and many-body physics. Here we prove the no low-energy trivial magic (NLTM) conjecture, constructing a family of local Hamiltonians for which every state within a positive energy density window requires circuit depth Ω(log n) to prepare from arbitrary stabilizer state that may themselves be highly entangled. Notably, our result is intrinsic in the sense that the same Hamiltonian family and energy density threshold apply to arbitrary qudit stabilizer inputs with any bounded local dimensions. Our construction builds on non-Abelian gauged sheaf codes obtained using the cup product structure on good quantum locally testable codes. We establish constant operator soundness for the associated Hamiltonians, enabling their protected logical structure to constrain all states at sufficiently low energy density. By excluding shallow stabilizer-based classical witnesses, Our result substantially strengthens NLTS and advances our understanding of low-energy complexity relevant to quantum PCP.
ER  -