The Independent Set is a well known NP-hard optimization problem. In this work, we define a fermionic generalization of the Independent Set problem and prove that the optimization problem is QMA-hard in a $k$-particle subspace using perturbative gadgets. We discuss how the Fermionic Independent Set is related to the problem of computing the minimum eigenvalue of the $k^{\text{th}}$-Laplacian of an independence complex of a vertex weighted graph. Consequently, we use the same perturbative gadget to prove QMA-hardness of the later problem resolving an open conjecture from arXiv:2311.17234 and give the first example of a natural topological data analysis problem that is QMA-hard.

翻译：独立集是一个众所周知的NP困难优化问题。在本工作中，我们定义了独立集问题的费米子推广，并利用微扰工具证明了该优化问题在$k$粒子子空间中是QMA困难的。我们讨论了费米子独立集与计算顶点加权图独立复形的$k^{\text{th}}$-拉普拉斯算符最小特征值问题之间的关系。因此，我们使用相同的微扰工具证明了后一问题的QMA困难性，从而解决了arXiv:2311.17234中提出的一个开放猜想，并给出了首个自然拓扑数据分析问题是QMA困难的实例。