团复形中的挠检测是条件性 $QMA_1$-困难的

中文摘要 AI 辅助

用于拓扑数据分析的量子算法计算贝蒂数，即单纯复形的同调群的秩，这些秩可以从组合拉普拉斯算子的核中读出。判定一个团复形的贝蒂数是否非零是 $QMA_1$-困难的，并且在带权图上具有谱间隙保证时仍然如此。然而，积分同调包含拉普拉斯谱无法获取的信息。积分同调中出现的一个新部分是挠：即只有在遍历多次后才成为边界的循环，如射影平面或克莱因瓶中的循环。我们询问检测挠有多困难，并通过一个简单的归约来回答。我们将一个固定的 $31$ 顶点射影平面三角剖分附加到任意团复形上。输入的 $k$ 阶模 $2$ 贝蒂数随后作为高一维的 $2$-挠重新出现，而所有有理同调消失，每个组合拉普拉斯算子获得一个常数谱间隙。我们得出结论：在无权重图的团复形中检测挠是 $NP$-困难的，即使在常数间隙保证下也是如此，并且如果模 $2$ 团同调是 $QMA_1$-困难的，那么检测挠也是 $QMA_1$-困难的。

英文摘要

Quantum algorithms for topological data analysis compute Betti numbers, the ranks of the homology groups of a simplicial complex, which can be read off from the kernel of a combinatorial Laplacian. Deciding whether a Betti number of a clique complex is nonzero is $QMA_1$-hard, and remains so under a spectral gap promise on vertex-weighted graphs. Integral homology, however, contains information inaccessible to the Laplacian spectrum. A new part that appears in integral homology is \emph{torsion}: cycles that become boundaries only after being traversed several times, as in a projective plane or a Klein bottle. We provide a simple reduction that allows us to establish the first hardness results for the problem of detecting torsion. We attach to an arbitrary clique complex a fixed $31$-vertex triangulation of the projective plane. The $k$th mod-$2$ Betti number of the input then reappears as $2$-torsion two degrees up, while all rational homology disappears and every combinatorial Laplacian acquires a constant spectral gap. We conclude that detecting torsion in clique complexes of \emph{unweighted} graphs is $NP$-hard, even under a constant gap promise, and that it is $QMA_1$-hard if mod-$2$ clique homology is.

发表机构

• Department of Computer Science, University of Oxford（牛津大学计算机系）

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