The Fermionic Cohomology Problem on the Full Fock Space Is ${\mathrm{QMA}}_1$-Complete

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 ${\mathrm{QMA}}_1$-hard and belongs to $\mathrm{QMA}$. 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 ${\mathrm{QMA}}_1$-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 ${\mathrm{QMA}}_1$-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.

21 pages. Revised abstract and exposition; clarified coefficient encoding and locality bounds. Main results unchanged