发表机构
Graduate School of Mathematics, Nagoya University（名古屋大学大学院数学研究科）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
该研究证明全福克空间上的费米子上同调问题为QMA₁完全，配套证明指定次数的对应问题也为QMA₁完全，为相关量子复杂度问题提供了关键结论。
AI 中文摘要
费米子上同调刻画与费米子微分相关的超对称哈密顿量的零能态。已有工作表明，限定为输入指定的粒子数 sector 的该问题是QMA₁难的，且属于QMA类。我们研究全局问题，即未指定任何sector，上同调可出现在全福克空间的任意位置。该形式化直接对应全空间基态问题：指定次数的NO实例可能在其他sector存在零能态，而全局NO承诺排除所有sector及其叠加态中的零能态。我们证明，当微分以局部单项式的精确列表给出时，该全福克问题是QMA₁完全的，即使每个单项式最多作用于41个模式。作为配套结果，我们证明指定次数问题也是QMA₁完全的，其30模式项的困难实例可实现为一维块链结构。
英文摘要
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.
Comments21 pages. Revised abstract and exposition; clarified coefficient encoding and locality bounds. Main results unchanged
