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arXiv:1811.11538

A Tutorial on Formulating and Using QUBO Models

Fred Glover, Gary A. Kochenberger, Yu Du

Published Nov 13, 2018• 164 citations
arXivHTMLPDF

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.

DOI
10.48550/arxiv.1811.11538

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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.
The field of Combinatorial Optimization (CO) is one of the most important areas in the general field of optimization, with important applications found in every industry, including both the private and public sectors. It is also one of the most active research areas pursued by the research communities of Operations Research, Computer Science, and Analytics as they work to design and test new methods for solving real world CO problems. In recent years, we have discovered that a mathematical formulation known as QUBO, an acronym for a Quadratic Unconstrained Binary Optimization problem, can embrace a large variety of important CO problems found in industry and government. Through special reformulation techniques that are easy to apply, the power of QUBO solvers can be used to efficiently solve many important problems once they are put into the QUBO framework. This two-step process of first re-casting an original model into the form of a QUBO model and then solving it with appropriate software enables the QUBO model to become a unifying framework for combinatorial optimization. The alternative path that results for effectively modeling and solving many important problems is a new development in the field of combinatorial optimization. The materials provided in subsequent sections illustrate the process of reformulating important optimization problems as QUBO models through a series of explicit examples. Collectively these examples highlight the application breadth of the QUBO model. We do not focus here on modeling a given problem in the form classically adopted in the optimization community. Rather, our examples disclose the methods that can be used to recast a given problem, perhaps existing in a classical mathematical form, into an equivalent QUBO model.
This tutorial offers a quick, hands-on introduction to solving Quadratic Unconstrained Binary Optimization (QUBO) models on currently available quantum computers and their simulators. We cover both IBM and D-Wave machines: IBM utilizes a gate-circuit architecture, and D-Wave is a quantum annealer. We provide examples of three canonical problems and two models from practical applications. The tutorial is structured to bridge the gap between theory and practice: we begin with an overview of QUBOs, explain their relevance and connection to quantum algorithms, introduce key quantum computing concepts, provide the foundations for two quantum heuristics, and provide detailed implementation guides. An associated GitHub repository provides the codes in five companion notebooks. In addition to reaching undergraduate and graduate students in computationally intensive disciplines, this article aims to reach working industry professionals seeking to explore the potential of near-term quantum applications. As our title indicates, this tutorial is intended to be a starting point in a journey towards solving more complex QUBOs on quantum computers.
Quantum computing promises to have a tremendous advantage over its classical counterpart for solving computationally hard problems, yet remains in a relatively early stage for practical applications owing to the limited capabilities of today’s quantum computers. The approach to the special purposes of quantum computers by exploiting the special-purpose physics-inspired or quantum-inspired computers is emerging as a novel alternative to its quantum counterpart in tackling hard problems in high-performance computing. Inspired by physics, the Ising machine — a type of special-purpose computer that implements or emulates physics or quantum effects of the Ising model to speed up finding solutions to optimization problems — has recently become an active research area in the field of combinatorial optimization. This paper is to address the key enabling software and hardware technology underlying physics-inspired optimization using Ising machines in the unified quadratic unconstrained binary optimization (QUBO) framework for modeling and solving computationally hard combinatorial optimization problems, and with an aim to shed some light on the challenges and opportunities associated with the ever-growing landscape of this novel high-performance computing.
This paper investigates several Quadratic Unconstrained Binary optimisation (QUBO) formulations for the parallel machine scheduling problem P m | | C max with the goal of assessing their suitability for quantum annealing and related heuristic optimisation methods. Three QUBO models are constructed, ranging from formulations derived from classical integer and one-hot encodings to a more compact load-balancing based representation. Extensive computational experiments using Simulated Annealing (SA) and Quantum Annealing (QA) reveal substantial differences in scalability and solution quality among the formulations. The alternative load-balanced model consistently yields near-optimal makespans, typically within 1%–3% of the theoretical lower bound, highlighting its stability and effectiveness compared to the more penalty sensitive formulations, whose performance deteriorates for larger instances. Sensitivity analyses further reveal that the appropriate selection of penalty weights and problem-specific parameter tuning significantly influences both solution accuracy and computational effort in QUBO formulations. While current quantum hardware limitations restrict experimentation to small problem sizes, the findings illustrate how compact QUBO designs can better exploit quantum optimisation frameworks. The insights gained in this study provide practical modelling guidelines for future quantum scheduling research and lay the groundwork for extending QUBO-based approaches to more complex scheduling environments. • Three QUBO formulations are compared for identical parallel machine scheduling. • A compact load-balancing QUBO avoids auxiliary variables. • Load balancing yields makespans on average within 1–3%. • Both simulated and quantum annealing produce near-optimal schedules. • Penalty selection and encoding strongly affect scalability and solution quality.
Quantum optimization shows significant potential in tackling complex combinatorial challenges, prompting our research exploration into the domain of quantum computing to address the Bin Packing Problem (BPP) through Quadratic Unconstrained Binary Optimization (QUBO) formulations. This work-in-progress paper outlines our proposal for a thorough study that encompasses classical bin packing formulations, initiating with the Mixed Integer Linear Program, followed by a comparative analysis of the corresponding QUBO representations, and adopting a structured approach of Quantum Annealing on the D-Wave machine. While our work is still in progress, continuous comparative analyses are being conducted to evaluate various QUBO formulations, derived from diverse classical bin packing models, aiming to identify the most efficient QUBO for addressing the BPP. On another front, efforts are dedicated to refining the solution process further, with a focus on enhancing the precision of the selected QUBO formulation.
Quantum annealing computers are designed to produce high-quality solutions to optimization problems that can be formulated as quadratic unconstrained binary optimization (QUBO) problems. While most of the well known NP-hard problems can easily be represented as quadratic binary problems, such formulations usually contain constraints that have to be added as penalties to the objective function in order to obtain QUBOs. In this paper, we propose a method based on finite automaton representation of the constraints for generating penalty implementations for them, which uses fewer qubits than the alternatives and is general enough to be applied to a whole class of constraints.