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The Fermionic Cohomology Problem on the Full Fock Space Is QMA1-Complete

  • Yibin Wang
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Abstract

Fermionic cohomology detects zero-energy states of supersymmetric Hamiltonians. Previous work showed that deciding nonzero cohomology in an input-specified particle-number sector is QMA1-hard and belongs to QMA. We study total cohomology on the unrestricted full Fock space, where a NO instance must exclude zero-energy states in every sector and in their superpositions. We prove that this global problem is QMA1-complete. The input differential is an operator that raises fermion number by one and squares to zero on the full Fock space. It is given as an exact list of local fermionic monomials, each involving at most $41$ modes. The reduction encodes each data site by one fermion in a block of modes. Every sector violating this occupation rule has energy at least one. An exact quantum verifier accepts a suitable witness with certainty on every YES instance, establishing containment with perfect completeness. We also prove QMA1-completeness for the problem in an input-specified particle-number sector. Its hard instances use $30$-mode terms and admit a one-dimensional block-chain realization.

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Cite

BibTeXRIS
BibTeX
@misc{wang2026fermionic,
  title = {{The Fermionic Cohomology Problem on the Full Fock Space Is $\mathrmQMA_1$-Complete}},
  author = {Yibin Wang},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.00694},
  eprint = {2609.00694},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.00694},
}
RIS
TY  - UNPB
TI  - The Fermionic Cohomology Problem on the Full Fock Space Is QMA_1-Complete
AU  - Yibin Wang
PY  - 2026
DA  - 2026/09/08
PB  - arXiv
DO  - 10.48550/arxiv.2609.00694
UR  - https://arxiv.org/abs/2609.00694
AB  - Fermionic cohomology detects zero-energy states of supersymmetric Hamiltonians. Previous work showed that deciding nonzero cohomology in an input-specified particle-number sector is QMA_1-hard and belongs to QMA. We study total cohomology on the unrestricted full Fock space, where a NO instance must exclude zero-energy states in every sector and in their superpositions. We prove that this global problem is QMA_1-complete. The input differential is an operator that raises fermion number by one and squares to zero on the full Fock space. It is given as an exact list of local fermionic monomials, each involving at most $41$ modes. The reduction encodes each data site by one fermion in a block of modes. Every sector violating this occupation rule has energy at least one. An exact quantum verifier accepts a suitable witness with certainty on every YES instance, establishing containment with perfect completeness. We also prove QMA_1-completeness for the problem in an input-specified particle-number sector. Its hard instances use $30$-mode terms and admit a one-dimensional block-chain realization.
ER  -