[Paper Review] A Tutorial on Formulating and Using QUBO Models

Fred Glover, Gary Kochenberger+1 more|arXiv (Cornell University)|Nov 13, 2018

Simulation Techniques and Applications68 references177 citations

TL;DR

The paper explains how to formulate a wide range of problems as QUBO models, using penalty functions to encode constraints exactly and illustrating with simple numerical examples, while also discussing modern solving innovations and links to quantum and neuromorphic computing.

ABSTRACT

The Quadratic Unconstrained Binary Optimization (QUBO) model has gained prominence in recent years with the discovery that it unifies a rich variety of combinatorial optimization problems. By its association with the Ising problem in physics, the QUBO model has emerged as an underpinning of the quantum computing area known as quantum annealing and has become a 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. 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 model that give it the power and flexibility to encompass the range of applications that have thrust it onto center stage of the optimization field. We show how many different types of constraining relationships 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 fertile avenue for integrating classical and quantum computing and for applying these models in machine learning.

Motivation & Objective

• Motivate the QUBO model as a unifying framework for combinatorial optimization.
• Show how constrained relationships can be encoded in unconstrained QUBO form using penalties.
• Provide step-by-step guidance and simple numerical examples to generate QUBO representations.
• Discuss recent innovations in solving QUBO models and their integration with classical, quantum, and machine learning methods.

Proposed method

• Describe the basic features of the QUBO model and its power to encompass diverse applications.
• Demonstrate how various constraints can be embodied in an unconstrained QUBO formulation using penalty functions.
• Show that penalty-based QUBO representations can be exact, in contrast to approximate traditional penalties.
• Illustrate the model-building process with simple numerical examples to highlight practicality.
• Discuss recent innovations for solving QUBO models and their potential for integrating classical/quantum computing and machine learning.

Experimental results

Research questions

• RQ1 How can a wide range of constrained combinatorial problems be represented as unconstrained QUBO models?
• RQ2 In what ways can constraints be encoded exactly using penalty functions within QUBO formulations?
• RQ3 What is the step-by-step process to generate QUBO representations from practical problems?
• RQ4 What recent solving innovations exist for QUBO models, and how can they bridge classical, quantum, and machine learning approaches?

Key findings

• QUBO models can unify many combinatorial optimization problems through unconstrained formulations.
• Penalty functions can yield exact representations of constraints within QUBO, enabling precise modeling.
• The tutorial provides detailed, simple numerical examples to illustrate QUBO construction.
• Recent solving innovations offer opportunities to integrate classical and quantum computing and machine learning with QUBO models.