Investigating the Relation Between Problem Hardness and QUBO Properties

Combinatorial optimization problems, integral to various scientific and industrial applications, often vary significantly in their complexity and computational difficulty. Transforming such problems into Quadratic Unconstrained Binary Optimization (QUBO) has regained considerable research attention in recent decades due to the central role of QUBO in Quantum Annealing. This work aims to shed some light on the relationship between the problems' properties. In particular, we examine how the spectral gap of the QUBO formulation correlates with the original problem, since it has an impact on how efficiently it can be solved on quantum computers. We analyze two well-known problems from Machine Learning, namely Clustering and Support Vector Machine (SVM) training, regarding the spectral gaps of their respective QUBO counterparts. An empirical evaluation provides interesting insights, showing that the spectral gap of Clustering QUBO instances positively correlates with data separability, while for SVM QUBO the opposite is true.