Asymptotic Analysis of Problem Formulations for Quantum Annealers

Quantum annealing is a promising technique to solve NP-hard problems more efficiently than state-of-the-art classical computing algorithms. Current generation quantum annealers require users to formulate problems following specific models: either as an Ising Hamiltonian, or as a quadratic unconstrained binary optimization (QUBO) polynomial. In this paper, we show that a problem may be formulated in multiple, fundamentally different ways using these models, and that the choice of problem formulation drastically impacts the scope of problem instances that the quantum annealer is able to solve. Using Boolean satisfiability (SAT) solving as an example, we investigate which characteristics of problem formulation are determinant in solution viability. Drawing inspiration from classical algorithmic asymptotic analysis, we establish a framework for measuring the complexity of any given formulation using the QUBO or Ising models, which we argue can be useful for practitioners to evaluate and compare the different ways one may tackle a problem using quantum annealers.