AI 中文总结
该研究证明无权间隙团同调是$\textsf{QMA}_1^{g_2}$完全问题，表明间隙承诺而非顶点加权是间隙团同调复杂性的来源。
AI 中文摘要
King和Kohler在FOCS 2024年的研究表明，在顶点乘积加权及组合Hodge拉普拉斯算子的逆多项式谱间隙承诺下，判定给定图的团复形在给定维数上是否存在非平凡同调的问题，是$\boldsymbol{\textsf{QMA}_1}$难问题且属于$\textsf{QMA}$。顶点权重是已知证明的关键，其为谱间隙分析提供了所需的尺度分离。本文证明，当每个顶点权重为1时，该问题仍为$\boldsymbol{\textsf{QMA}_1^{g_2}}$难问题且属于$\textsf{QMA}_1^{g_2}$，其中$\textsf{QMA}_1^{g_2}$是采用通用门集$g_2=\textsf{\textsf{X},\textsf{CX},\textsf{CCX},H\boxtimes H}$的$\textsf{QMA}_1$。本文的构造用扩张替代权重：将加权复形的每个顶点爆破为一个团，团的大小编码其权重；选择块大小使得对称平均再现加权Hodge度量，因此对称部分承载加权拉普拉斯算子（差一个公共标量因子），而局部平均论证给出对称部分正交补的均匀下界，从而将加权间隙分析转移到无权团复形，且不引入额外低能态或虚假同调。属于$\textsf{QMA}_1^{g_2}$的结论源于Rudolph对稀疏整数团拉普拉斯算子的精确酉算子线性组合模拟。该结果表明，间隙承诺而非顶点加权是间隙团同调复杂性的来源。
英文摘要
Deciding whether the clique complex of a given graph has nontrivial homology in a given dimension, under vertex-product weighting and an inverse-polynomial spectral gap promise on the combinatorial Hodge Laplacian, is known to be $\mathsf{QMA}_1$-hard and contained in $\mathsf{QMA}$ by King and Kohler (FOCS 2024). The vertex weights are essential in the known proof, where they provide the scale separation needed for the spectral gap analysis. We prove that, when every vertex has weight one, the problem remains $\mathsf{QMA}_1^{g_2}$-hard and is contained in $\mathsf{QMA}_1^{g_2}$, where $\mathsf{QMA}_1^{g_2}$ is $\mathsf{QMA}_1$ with the universal gate set $g_2=\{\mathsf{X},\mathsf{CX},\mathsf{CCX},H\otimes H\}$. The construction replaces weight by expansion: each vertex of the weighted complex is blown up into a clique whose size encodes its weight. The block sizes are chosen so that symmetric averages reproduce the weighted Hodge metric. The symmetric sector therefore carries the weighted Laplacian up to a common scalar factor, while a local averaging argument gives a uniform lower bound on the orthogonal complement to the symmetric sector. Hence the weighted gap analysis transfers to an unweighted clique complex without introducing either additional low-energy states or spurious homology. Containment in $\mathsf{QMA}_1^{g_2}$ follows from Rudolph's exact linear combination of unitaries simulation of sparse integer clique Laplacians. The result shows that the gap promise, rather than vertex weighting, is the source of the complexity of gapped clique homology.
Comments25 pages, 3 figures
