## Quantum Machine Learning: What Quantum Computing Means to Data Mining

Machine learning has become indispensable in discovering patterns in large data sets, and the theory is at the core of a larger set of tools known as data mining. It is a mature field with an astonishing array of practical applications.

Quantum computing has the potential of taking machine learning to the next level. While hardware implementations of quantum computing systems are still in an initial phase, recent theoretical developments hint at the benefits of applying quantum methods to learning algorithms.

Computational complexity can be reduced exponentially in some cases, whereas we see quadratic reduction in others. Yet, improved learning time is just one part of the equation. Through nonconvex objective functions, quantum machine learning algorithms are more robust to noise and outliers, which makes their generalization performance better than many known classical algorithms. Examples include quantum support vector machines, learning a function by quantum process tomography, quantum neural networks, and adiabatic quantum optimization.

Quantum Machine Learning: What Quantum Computing Means to Data Mining explains the most relevant concepts of machine learning, quantum mechanics, and quantum information theory, and contrasts classical learning algorithms to their quantum counterparts.

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## Table of Contents

### Part I: Fundamental Concepts

**1 Introduction**

1.1 Learning Theory and Data Mining

1.2 Why Quantum Computers

1.3 A Heterogeneous Model

1.4 An Overview of Quantum Machine Learning

1.5 Quantum-Like Learning on Classical Computers

**2 Machine Learning**

2.1 Data-Driven Models

2.2 Feature Space

2.3 Supervised and Unsupervised Learning

2.4 Generalization Performance

2.5 Model Complexity

2.6 Ensembles

2.7 Data Dependencies and Computational Complexity

**3 Quantum Mechanics**

3.1 States and Superposition

3.2 Density Matrix Representation and Mixed States

3.3 Composite Systems and Entanglement

3.4 Evolution

3.5 Measurement

3.6 Uncertainty Relations

3.7 Tunneling

3.8 Adiabatic Theorem

3.9 No-Cloning Theorem

**4 Quantum Computing**

4.1 Qubits and the Bloch Sphere

4.2 Quantum Circuits

4.3 Adiabatic Quantum Computing

4.4 Quantum Parallelism

4.5 Grover's Algorithm

4.6 Complexity Classes

4.7 Quantum Information Theory

### Part II: Classical Learning Algorithms

**5 Unsupervised Learning**

5.1 Principal Component Analysis

5.2 Manifold Embedding

5.3 *K*-Means and *K*-Medians Clustering

5.4 Hierarchical Clustering

5.5 Density-Based Clustering

**6 Pattern Recognition and Neural Networks**

6.1 The Perceptron

6.2 Hopfield Networks

6.3 Feedforward Networks

6.4 Deep Learning

6.5 Computational Complexity

**7 Supervised Learning and Support Vector Machines**

7.1 *K*-Nearest Neighbors

7.2 Optimal Margin Classifiers

7.3 Soft Margins

7.4 Nonlinearity and Kernel Functions

7.5 Least-Squares Formulation

7.6 Generalization Performance

7.7 Multiclass Problems

7.8 Loss Functions

7.9 Computational Complexity

**8 Regression Analysis**

8.1 Linear Least-Squares

8.2 Nonlinear Regression

8.3 Nonparametric Regression

8.4 Computational Complexity

**9 Boosting**

9.1 Weak Classifiers

9.2 AdaBoost

9.3 A Family of Convex Boosters

9.4 Nonconvex Loss Functions

### Part III: Quantum Computing and Machine Learning

**10 Clustering Structure and Quantum Computing**

10.1 Quantum Random Access Memory

10.2 Calculating Dot Products

10.3 Quantum Principal Component Analysis

10.4 Towards Quantum Manifold Embedding

10.5 Quantum *K*-Means

10.6 Quantum *K*-Medians

10.7 Quantum Hierarchical Clustering

10.8 Computational Complexity

**11 Quantum Pattern Recognition**

11.1 Quantum Associative Memory

11.2 The Quantum Perceptron

11.3 Quantum Neural Networks

11.4 Physical Realizations

11.5 Computational Complexity

**12 Quantum Classification**

12.1 Nearest Neighbors

12.2 Support Vector Machines with Grover's Search

12.3 Support Vector Machines with Exponential Speedup

12.4 Computational Complexity

**13 Quantum Process Tomography**

13.1 Channel-State Duality

13.2 Quantum Process Tomography

13.3 Groups, Compact Lie Groups, and the Unitary Group

13.4 Representation Theory

13.5 Parallel Application and Storage of Unitary

13.6 Optimal State for Learning

13.7 Applying the Unitary

**14 Boosting and Adiabatic Quantum Computing**

14.1 Quantum Annealing

14.2 Quadratic Unconstrained Binary Optimization

14.3 Ising Model

14.4 QBoost

14.5 Nonconvexity

14.6 Sparsity and Generalization Performance

14.7 Mapping to Hardware

14.8 Computational Complexity

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