Testing the presence of balanced and bipartite components in a sparse graph is QMA1-hard

TL;DR

This paper proves that testing for balanced and bipartite components in sparse graphs is QMA1-hard, linking spectral graph properties to quantum complexity and emphasizing their computational difficulty.

Contribution

It establishes the QMA1-hardness of spectral property testing in graphs by mapping complex simplicial complex problems to graph Laplacians, revealing computational hardness.

Findings

Testing balanced components in signed graphs is QMA1-hard.

Testing bipartite components in unsigned graphs is QMA1-hard.

Transformations preserve efficient sparse access to Laplacians.

Abstract

Determining whether an abstract simplicial complex, a discrete object often approximating a manifold, contains multi-dimensional holes is a task deeply connected to quantum mechanics and proven to be QMA1-hard by Crichigno and Kohler. This task can be expressed in linear algebraic terms, equivalent to testing the non-triviality of the kernel of an operator known as the Combinatorial Laplacian. In this work, we explore the similarities between abstract simplicial complexes and signed or unsigned graphs, using them to map the spectral properties of the Combinatorial Laplacian to those of signed and unsigned graph Laplacians. We prove that our transformations preserve efficient sparse access to these Laplacian operators. Consequently, we show that key spectral properties, such as testing the presence of balanced components in signed graphs and the bipartite components in unsigned graphs, are QMA1-hard. These properties play a paramount role in network science. The hardness of the bipartite test is relevant in quantum Hamiltonian complexity, as another example of testing properties related to the eigenspace of a stoquastic Hamiltonians are quantumly hard in the sparse input model for the graph.