A Tutorial on Formulating QUBO Models

Recent years have witnessed the remarkable discovery that the Quadratic Unconstrained Binary Optimization (QUBO) model unifies a wide variety of combinatorial optimization problems, and moreover is the foundation of adiabatic quantum computing and is the subject of study in neuromorphic computing. Through these connections, QUBO models lie at the heart of experimentation carried out with quantum computers developed by D-Wave Systems and neuromorphic computers developed by IBM and are actively being explored for their research and practical applications by Google and Lockheed Martin in the commercial realm and by Los Alamos National Laboratory, Oak Ridge National Laboratory and Lawrence Livermore National Laboratory in the public sector. Computational experience is being amassed by both the classical and the quantum computing communities that highlights not only the potential of the QUBO model but also its effectiveness as an alternative to traditional modeling and solution methodologies. This tutorial discloses the basic features of the QUBO that give it the power and flexibility to encompass the range of applications that have thrust it into prominence. We show how many different types of constraints arising in practice can be embodied within the “unconstrained” QUBO formulation in a very natural manner using penalty functions, yielding exact model representations in contrast to the approximate representations produced by customary uses of penalty functions. Each step of generating such models is illustrated in detail by simple numerical examples, to highlight the convenience of using QUBO models in numerous settings. We also describe recent innovations for solving QUBO models that offer a rich potential for integrating classical and quantum computing and for applying these models in machine learning. ECEE, College of Engineering and Applied Science, University of Colorado, Boulder, CO 80302 USA fred.glover@colorado.edu College of Business, University of Colorado at Denver, Denver, CO 80217 USA, gary.kochenberger@cudenver.edu 2 Table of