Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

International Joint Conference on Neural Networks (IJCNN)

Problem Solving with Hopfield Networks and Adiabatic Quantum Computing

Christian Bauckhage, Ramsés J. Sánchez, Rafet Sifa

Published Jul 1, 2020• 18 citations
Publisher

Abstract

Our goal with this paper is to elucidate the close connection between Hopfield networks and adiabatic quantum computing. Focusing on their use in problem solving, we point out that the energy functions minimized by Hopfield networks are essentially identical to those minimized by adiabatic quantum computers. To practically illustrate this, we consider a simple textbook problem, namely the k-rooks problem, and discuss how to set it up for solution via a Hopfield network or adiabatic quantum computing.

DOI
10.1109/ijcnn48605.2020.9206916

Related papers

Adiabatic Quantum Computing

Christian Bauckhage, Rafet Sifa

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.

Cognitive technologies•Nov 17, 2025•0 citations

An Outlook to Quantum Gate Computing

Christian Bauckhage, Rafet Sifa

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.

Cognitive technologies•Nov 17, 2025•0 citations

Adiabatic Quantum Computing for Binary Clustering

Christian Bauckhage, Eduardo Barreto Brito, Kostadin Cvejoski +3 more

Quantum computing for machine learning attracts increasing attention and recent technological developments suggest that especially adiabatic quantum computing may soon be of practical interest. In this paper, we therefore consider this paradigm and discuss how to adopt it to the problem of binary clustering. Numerical simulations demonstrate the feasibility of our approach and illustrate how systems of qubits adiabatically evolve towards a solution.

arXiv:1706.05528•Jun 17, 2017•4 citations

Hopfield Nets and Quantum Mechanics

Christian Bauckhage, Rafet Sifa

Abstract Previously, we viewed Hopfield nets from the point of view of statistical mechanics. In this chapter, we widen our perspective one again and view them from the point of view of quantum mechanics. This requires us to introduce completely different representations of the states and energy functions of Hopfield nets and relate them to the kind of tensor product spaces we studied earlier. At this point, this different perspective on Hopfield nets is of purely theoretical interest because it generally defies practical implementations on digital computers. However, it establishes a connection between Hopfield nets and important quantum mechanical concepts. In particular, we will get to know the notion of the Hamiltonian of a physical system, its eigenstates, and eigenvalues. Since the material in this chapter is more abstract and less intuitive than in previous chapters, we develop our ideas by looking at an exemplary simple Hopfield net and support our technical claims through practical computations and code examples.

Cognitive technologies•Nov 17, 2025•0 citations

QUBOs

Christian Bauckhage, Rafet Sifa

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.

Cognitive technologies•Nov 17, 2025•0 citations

Adiabatic Quantum Computing for Max-Sum Diversification

Christian Bauckhage, Rafet Sifa, Stefan Wrobel

The combinatorial problem of max-sum diversification asks for a maximally diverse subset of a given set of data. Here, we show that it can be expressed as an Ising energy minimization problem. Given this result, max-sum diversification can be solved on adiabatic quantum computers and we present proof of concept simulations which support this claim. This, in turn, suggests that quantum computing might play a role in data mining. We therefore discuss quantum computing in a tutorial like manner and elaborate on its current strengths and weaknesses for data analysis.

Society for Industrial and Applied Mathematics eBooks•Jan 1, 2020•13 citations
Our goal with this paper is to elucidate the close connection between Hopfield networks and adiabatic quantum computing. Focusing on their use in problem solving, we point out that the energy functions minimized by Hopfield networks are essentially identical to those minimized by adiabatic quantum computers. To practically illustrate this, we consider a simple textbook problem, namely the k-rooks problem, and discuss how to set it up for solution via a Hopfield network or adiabatic quantum computing.
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.
Abstract Previously, we viewed Hopfield nets from the point of view of statistical mechanics. In this chapter, we widen our perspective one again and view them from the point of view of quantum mechanics. This requires us to introduce completely different representations of the states and energy functions of Hopfield nets and relate them to the kind of tensor product spaces we studied earlier. At this point, this different perspective on Hopfield nets is of purely theoretical interest because it generally defies practical implementations on digital computers. However, it establishes a connection between Hopfield nets and important quantum mechanical concepts. In particular, we will get to know the notion of the Hamiltonian of a physical system, its eigenstates, and eigenvalues. Since the material in this chapter is more abstract and less intuitive than in previous chapters, we develop our ideas by looking at an exemplary simple Hopfield net and support our technical claims through practical computations and code examples.
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.
The combinatorial problem of max-sum diversification asks for a maximally diverse subset of a given set of data. Here, we show that it can be expressed as an Ising energy minimization problem. Given this result, max-sum diversification can be solved on adiabatic quantum computers and we present proof of concept simulations which support this claim. This, in turn, suggests that quantum computing might play a role in data mining. We therefore discuss quantum computing in a tutorial like manner and elaborate on its current strengths and weaknesses for data analysis.