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Cognitive technologies

QUBOs

Christian Bauckhage, Rafet Sifa

Published Nov 17, 2025• 0 citations
Publisher

Abstract

Abstract In previous chapters, we already came across an instance of a quadratic unconstrained binary optimization problem or QUBO for short. In this chapter, we will now widen our perspective on QUBOs as our major plot line in this book is that any QUBO can be solved by running a Hopfield net and that any problem that can be solved by running a Hopfield net can be solved on a quantum computer. In other words, we will have a closer look at the characteristics and structure of QUBOs and discuss some of their notable peculiarities. However, we will not yet look at practical examples nor will we discuss how to actually solve QUBOs but leave these aspects to later chapters.

DOI
10.1007/978-3-031-99402-9_4

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Abstract Having previously studied general aspects of QUBO models, we now study application specific examples and how to derive or set them up. Our application examples will consider problems in areas like combinatorial optimization, constraint satisfaction, data mining, and machine learning and are supposed to illustrate that QUBOs provide us with a surprisingly versatile and widely applicable modeling framework. Put differently, our overall goal with this chapter is to demonstrate how to rethink familiar, seemingly unrelated problems in terms of an overarching formalism that will later allow for running Hopfield nets and quantum implementations. This will largely be an exercise in applied linear algebra since we will extensively work with matrix-vector formulations of QUBOs for which we will point out and discuss simplifying modeling assumptions whenever necessary.

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Abstract In previous chapters, we already came across an instance of a quadratic unconstrained binary optimization problem or QUBO for short. In this chapter, we will now widen our perspective on QUBOs as our major plot line in this book is that any QUBO can be solved by running a Hopfield net and that any problem that can be solved by running a Hopfield net can be solved on a quantum computer. In other words, we will have a closer look at the characteristics and structure of QUBOs and discuss some of their notable peculiarities. However, we will not yet look at practical examples nor will we discuss how to actually solve QUBOs but leave these aspects to later chapters.
Abstract Having previously studied general aspects of QUBO models, we now study application specific examples and how to derive or set them up. Our application examples will consider problems in areas like combinatorial optimization, constraint satisfaction, data mining, and machine learning and are supposed to illustrate that QUBOs provide us with a surprisingly versatile and widely applicable modeling framework. Put differently, our overall goal with this chapter is to demonstrate how to rethink familiar, seemingly unrelated problems in terms of an overarching formalism that will later allow for running Hopfield nets and quantum implementations. This will largely be an exercise in applied linear algebra since we will extensively work with matrix-vector formulations of QUBOs for which we will point out and discuss simplifying modeling assumptions whenever necessary.
We show that the fundamental tasks of sorting lists and building search trees or heaps can be modeled as quadratic unconstrained binary optimization problems (QUBOs). The idea is to understand these tasks as permutation problems and to devise QUBOs whose solutions represent appropriate permutation matrices. We discuss how to construct such QUBOs and how to solve them using Hopfield nets or adiabatic) quantum computing. In short, we show that neurocomputing methods or quantum computers can solve problems usually associated with abstract data structures.
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.
Abstract In this chapter, we finally have our first look at quantum computing. In particular, we consider adiabatic quantum computing which is a paradigm tailored to the general problem of bipolar QUBO solving. Given our study up to this point, its basic principles will be easy to understand since Hopfield nets for problem solving and adiabatic quantum computing are closely related concepts. Indeed, we already know about tensor products, Hopfield energies and their representation in terms of Hamiltonians, annealing, the axioms of quantum mechanics, the Schrödinger equation, and quantum superposition. All these concepts will now be jointly put to use and brought together with yet another quantum mechanical principle, namely the adiabatic theorem. Based on two examples of simple QUBO problems, we demonstrate how adiabatic quantum computers can explore their exponentially large search spaces in parallel and thus achieve quadratic speedup over classical exhaustive search procedures.
Abstract Having spent almost an entire book on working out that Hopfield nets and adiabatic quantum computing are closely related concepts, this chapter departs from our main story line of problem solving as energy minimization and concludes our study with an outlook to the possibilities of quantum gate computing. Indeed, the central ideas behind quantum gate computing differ from most of what we studied so far but our study still has us prepared for getting to grips with the basics of this paradigm. As this paradigm posits that quantum algorithms are sequences of unitary operators which each act on the joint state of a qubit system, we will look at the circuit model of quantum computing and see that quantum circuits consist of quantum gates which are interconnected by quantum wires where the latter represent the states of qubits on which the former perform unitary operations. As a practical example, we discuss how such circuits could compute univariate Boolean functions. This then seamlessly allows us to finally study Deutsch’s problem and algorithm and therefore to see an instance of a computational problem for which quantum computing algorithms can be exponentially faster than their classical counterparts.
The quadratic unconstrained binary optimization (QUBO) problem arises in diverse optimization applications ranging from Ising spin problems to classical problems in graph theory and binary discrete optimization. The use of preprocessing to transform the graph representing the QUBO problem into a smaller equivalent graph is important for improving solution quality and time for both exact and metaheuristic algorithms and is a step towards mapping large scale QUBO to hardware graphs used in quantum annealing computers. In an earlier paper (Lewis and Glover, 2016) a set of rules was introduced that achieved significant QUBO reductions as verified through computational testing. Here this work is extended with additional rules that provide further reductions that succeed in exactly solving 10% of the benchmark QUBO problems. An algorithm and associated data structures to efficiently implement the entire set of rules is detailed and computational experiments are reported that demonstrate their efficacy.