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Quantum Computing from Hopfield Nets

A Textbook with Python Code Examples

  • Open Access
  • 2026
  • Open Access
  • Buch
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SUCHEN

Über dieses Buch

Dieses Open-Access-Buch ist als Lehrbuch für Informatik-Studenten gedacht, die einen sanften Einstieg in die Welt der Quantencomputertechnologie suchen. Genauer gesagt ist es für Leser geschrieben, die über grundlegende Kenntnisse der künstlichen Intelligenz (KI) und des maschinellen Lernens (ML) verfügen und mit Suchalgorithmen, Optimierungstechniken und neuronalen Netzwerken vertraut sind. Das liegt nicht daran, dass die Autoren an Quantum AI oder Quantum ML interessiert wären, sondern daran, dass sie von der Grundannahme ausgehen, dass es eine konzeptionelle Brücke zwischen bestimmten KI / ML-Modellen und bestimmten Quantencomputermodellen gibt. Der Zweck dieses Buches besteht daher darin, 1) diese KI / ML-Modelle und ihre Anwendungen zu überdenken und 2) auf dieser vertrauten Grundlage in die Erforschung der Quantencomputertechnologie und ihrer möglichen Anwendungsfälle einzusteigen. Die Präsentation ist technisch, aber pragmatisch und praxisorientiert. Die Autoren behandeln die Theorie im notwendigen Umfang, gehen aber weitgehend beispielhaft vor. Die meisten Beispiele betreffen die kombinatorische Optimierung und betrachten Probleme, die als quadratische, uneingeschränkte binäre Optimierungsprobleme betrachtet werden können. Zahlreiche Python / Numpy / Scipy-Codes unterstützen die mathematische Diskussion und zeigen, wie man Theorie in die Praxis umsetzt, begleitet von Übungen für jedes Kapitel. Teile des Materials wurden aus langjährigen Vorlesungen zur Mustererkennung, zu den Grundlagen der Quantencomputertechnologie und zu Quantencomputeralgorithmen übernommen, die von den Autoren im Masterstudiengang Informatik an der Universität Bonn gelehrt werden.

Inhaltsverzeichnis

Frontmatter

Preliminaries

Frontmatter

Open Access

Chapter 1. Setting the Stage
Abstract
This chapter tries to motivate why 21st century computer scientists should know about quantum computing. To this end, we look at an important yet easily understood combinatorial optimization problem, namely the subset sum problem or SSP for short. As it is combinatorial, the SSP is generally difficult even for modern artificial intelligence models. We discuss why this is and why quantum computers may have an edge. To be able to see this, we show how to set up the SSP for quantum processing. This involves a first look at binary vectors and the notion of quadratic unconstrained binary optimization problems (QUBOs). In passing, we thus already encounter much of the terminology and concepts used in later chapters.
Christian Bauckhage, Rafet Sifa

Open Access

Chapter 2. Boolean Domains, Numbers, and Vectors
Abstract
If we want to firmly understand the behavior of Hopfield nets and many quantum computing algorithms, we first need to understand quite a few properties of vectors whose entries can only assume one of two values. In this chapter, we therefore look at sets of bivalent numbers and vectors and study their characteristics from various different angles. These include Boolean logic, (multi-)liner algebra, probability theory and statistics, geometry, and graph theory. The content of this chapter is therefore dense and technical but its seemingly unrelated topics turn out to be interlinked and fundamental to many of the upcoming chapters. In short, this chapter equips us with a very broad perspective on the apparently innocuous ideas of bivalent numbers and vectors and introduces important terminology and concepts which we will need later on.
Christian Bauckhage, Rafet Sifa

Open Access

Chapter 3. Kronecker Products
Abstract
Tensor products or, more specifically, Kronecker products play a fundamental role in quantum computing and can also help with modeling certain combinatorial problems. In this chapter, we therefore discuss Kronecker products and their properties. To segue into this discussion, we first introduce the important notion of pseudo Boolean functions and look at their relations to multilinear algebra. Given this use case, we can easily understand the role of different kinds of tensor products and investigate algebraic properties especially of Kronecker products.
Christian Bauckhage, Rafet Sifa

Open Access

Chapter 4. QUBOs
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.
Christian Bauckhage, Rafet Sifa

Open Access

Chapter 5. QUBO Models
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.
Christian Bauckhage, Rafet Sifa

Hopfield Nets

Frontmatter

Open Access

Chapter 6. Hopfield Nets
Abstract
In this chapter, we finally look at Hopfield nets which are rather simple recurrent neural networks. First, we ever so briefly contrast them to other kinds of neural networks and then list and discuss their defining (structural) properties. We then introduce the important notion of the energy function of a neural network and discuss synchronous and asynchronous update mechanisms for running Hopfield nets. For the latter, we then carefully prove the important facts that, if the neurons of a Hopfield net update their individual states one at a time, then the overall state of the network will converge to a stable state of low energy and that this convergence will happen within a finite number of steps. In all of this, we will concentrate on classical Hopfield nets and ignore their modern variants.
Christian Bauckhage, Rafet Sifa

Open Access

Chapter 7. Hopfield Nets in Action
Abstract
In this chapter, we introduce the notion of the gradients of Hopfield energy functions and discuss how to employ them for efficient informed asynchronous updates of a Hopfield net. Armed with the corresponding algorithm, we will then demonstrate the use of Hopfield nets as problem solvers. In particular, we will consider a constraint satisfaction problem and a combinatorial optimization problem, show how to set up respective Hopfield net parameters, and then practically evaluate the the problem solving capabilities of Hopfield nets under different initializations and parameterizations. In preparation for things still to come, we finally discuss the characteristics of local search procedures and how more global procedures may overcome some of their limitations.
Christian Bauckhage, Rafet Sifa

Open Access

Chapter 8. Hopfield Nets and Statistical Mechanics
Abstract
Having familiarized ourselves with the basic theory and practice of Hopfield nets, we can now widen our perspective. In this chapter, we therefore look at Hopfield nets from the point of view of physics or, more specifically, statistical mechanics. The basic theme will be to understand a Hopfield net as an ensemble of interacting microscopic entities and to apply statistical techniques to reason about their collective behavior or macroscopic state. First, we discuss Ising models of spin systems and relations to Hopfield nets to answer the still open question of why we may talk about the energy of a Hopfield net. Based on these foundations, we then look at stochastic Hopfield nets or Boltzmann machines and the idea of stochastic asynchronous updates. Finally, we will get to know completely different ways of running Hopfield nets and study stochastic simulated annealing and mean field annealing.
Christian Bauckhage, Rafet Sifa

Open Access

Chapter 9. Hopfield Nets and Quantum Mechanics
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.
Christian Bauckhage, Rafet Sifa

Quantum Computing

Frontmatter

Open Access

Chapter 10. Quantum Mechanics in a Nutshell
Abstract
With this chapter, we begin to transit from digital information processing to quantum information processing. However, to be able to master this transition, we first need to familiarize ourselves with quantum mechanical principles and the mathematics used to describe them. Since we already know about abstract state spaces, tensor products, and Hamiltonian operators and their eigenvectors and eigenvalues, this should not be too difficult or too alien. And yet, there remain a few challenges: Mathematical descriptions of quantum states usually involve the Dirac notation, require complex linear algebra in high dimensional spaces, and encode unintuitive probabilistic interpretations. Rising to these challenges, we briefly study the notion of Hilbert spaces, introduce the Dirac notation, and discuss the axioms of quantum mechanics and some of their consequences. All this will admittedly be dense and abstract but prepare us for everything that is still to follow. Our main goal is to emphasize that quantum mechanics and therefore quantum computing is just complex linear algebra in very high dimensional spaces so that quantum computing needs a different kind of thinking than we are used to in digital computing.
Christian Bauckhage, Rafet Sifa

Open Access

Chapter 11. Exploring Quantum Weirdness
Abstract
In this chapter, we familiarize ourselves with certain aspects of quantum weirdness, namely energy quantization, quantum tunneling, and quantum superposition. To this end, we consider three didactic examples of quantum systems, solve the respective scenario specific Schrödinger equations, and (mathematically and graphically) analyze and interpret the results we obtain therefrom. Despite of the utter simplicity of our scenarios, two of them already defy analytic solutions. Since this is not uncommon when trying to solve Schrödinger equations for specific settings, we also discuss simple numerical methods for Schrödinger equation solving and present code examples which implement those.
Christian Bauckhage, Rafet Sifa

Open Access

Chapter 12. 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.
Christian Bauckhage, Rafet Sifa

Open Access

Chapter 13. An Outlook to Quantum Gate Computing
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.
Christian Bauckhage, Rafet Sifa
Backmatter
Titel
Quantum Computing from Hopfield Nets
Verfasst von
Christian Bauckhage
Rafet Sifa
Copyright-Jahr
2026
Electronic ISBN
978-3-031-99402-9
Print ISBN
978-3-031-99401-2
DOI
https://doi.org/10.1007/978-3-031-99402-9

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