Fermionic Independent Set and Laplacian of an independence complex are QMA-hard

TL;DR

This paper introduces a fermionic variant of the Independent Set problem, proving it and related Laplacian eigenvalue problems are QMA-hard, thus highlighting their computational complexity in quantum settings.

Contribution

It defines a fermionic generalization of the Independent Set problem and proves QMA-hardness for this and related topological data analysis problems.

Findings

Fermionic Independent Set is QMA-hard.

QMA-hardness of the Laplacian eigenvalue problem is established.

First natural topological data analysis problem shown to be QMA-hard.

Abstract

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.