AI 中文总结
本文完整证明了加权带隙团同调的$\mathsf{QMA}_1$-困难性，通过修复King-Kohler归约中的缺口，利用有限证书和相对同调分析局部小工具，并建立全局谱估计。
AI 中文摘要
单纯同调的计算复杂性自Kaibel和Pfetsch提出该问题以来一直受到研究，随后Crichigno和Kohler证明了团同调是$\mathsf{QMA}_1$-困难的。King和Kohler随后引入了此处考虑的加权谱带隙版本，并开发了一个旨在建立该情形下相同困难性结果的归约。我们给出了困难方向的完整证明。早期的King-Kohler论证在非平凡折叠局部小工具的分析中存在一个缺口，因此必须在归约实际产生的团复形上重新建立所需的拓扑和谱性质。这不是对已发表证明的局部修补：该论证需要对局部拓扑、滤过余链复形以及许多重叠小工具的相互作用进行新的分析。我们的证明使用局部组合数据的精确有限证书和相对同调来确定每个小工具对编码量子比特空间的影响。然后我们证明了恢复相应秩一低能惩罚所需的有限维滤过Hodge陈述，并推导出一个定量的全局估计，将源哈密顿量的承诺间隙转移到最终的加权Hodge拉普拉斯算子。因此，对于固定的可有效计算的$k(n)$和逆多项式$\gamma(n)$，加权带隙团同调是$\mathsf{QMA}_1(\mathcal G)$-困难的。先前已知的$\mathsf{QMA}$包含性论证在逻辑上独立于间隙，且不受影响。
英文摘要
The computational complexity of simplicial homology has been studied since the question was raised by Kaibel and Pfetsch, and clique homology was later shown to be $\mathsf{QMA}_1$-hard by Crichigno and Kohler. King and Kohler subsequently introduced the weighted, spectrally gapped version considered here and developed a reduction intended to establish the same hardness result in this setting. We give a complete proof of the hard direction. The earlier King-Kohler argument contains a gap in the analysis of the nontrivially folded local gadgets, so the required topological and spectral properties have to be re-established on the clique complexes actually produced by the reduction. This is not a local repair of the published proof: the argument requires a new analysis of the local topology, the filtered cochain complexes, and the interaction of many overlapping gadgets. Our proof uses exact finite certificates for the local combinatorial data and relative homology to determine the effect of each gadget on the encoded qubit space. We then prove the finite-dimensional filtered-Hodge statement needed to recover the corresponding rank-one low-energy penalty and derive a quantitative global estimate that transfers the source-Hamiltonian promise gap to the final weighted Hodge Laplacian. Consequently, for fixed efficiently computable $k(n)$ and inverse-polynomial $γ(n)$, weighted gapped clique homology is $\mathsf{QMA}_1(\mathcal G)$-hard. The previously known $\mathsf{QMA}$-containment argument is logically independent of the gap and remains unaffected.
Comments75 pages, 5 figures. Ancillary file contains the exact finite certificates and verification code
