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量子机器学习中数据挖掘的量子计算方法

作者:维特克(Wittek P.)

出版社:哈尔滨工业大学出版社

出版年: 2016-01-01

页数:163

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内容简介

Machine learning is a fascinating area to work in: from detecting anomalous events in live streams of sensor data to identifying emergent topics involving text collection,exciting problems are never too far away.

Quantum information theory also teems with excitement. By manipulating particles at a subatomic level, we are able to perform Fourier transformation exponentially faster, or search in a database quadratically faster than the classical limit. Superdense coding transmits two classical bits using just one qubit. Quantum encryption is unbreakable-at least in theory.

 

目录

Preface

Notations

Part One FundamentaIConcepts

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 Algorithms

1.5 Quantum—Like Learning on Classical Computers

2 Machine Learning

2.1 Data—DrivenModels

2.2 FeatureSpace

2.3 Supervised and Unsupervised Learning

2.4 GeneralizationPerformance

2.5 ModeIComplexity

2.6 Ensembles

2.7 Data Dependencies and ComputationalComplexity

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 UncertaintyRelations

3.7 Tunneling

3.8 Adiabatic Theorem

3.9 No—CloningTheorem

4 Quantum Computing

4.1 Qubits and the Bloch Sphere

4.2 QuantumCircuits

4.3 Adiabatic Quantum Computing

4.4 QuantumParallelism

4.5 Grover's Algorithm

4.6 ComplexityClasses

4.7 QuantumInformationTheory


Part Two ClassicalLearning Algorithms

5 Unsupervised Learning

5.1 Principal Component Analysis

5.2 ManifoldEmbedding

5.3 K—Means and K—Medians Clustering

5.4 HierarchicalClustering

5.5 Density—BasedClustering

6 Pattern Recogrution and Neural Networks

6.1 ThePerceptron

6.2 HopfieldNetworks

6.3 FeedforwardNetworks

6.4 DeepLearning

6.5 ComputationalComplexity

7 Supervised Learning and Support Vector Machines

7.1 K—NearestNeighbors

7.20ptimal Margin Classifiers

7.3 SoftMargins

7.4 Nonlinearity and KemelFunctions

7.5 Least—SquaresFormulation

7.6 Generalization Performance

7.7 Multiclass Problems

7.8 Loss Functions

7.9 ComputationalComplexity

8 Regression Analysis

8.1 Linear Least Squares

8.2 NonlinearRegression

8.3 NonparametricRegression

8.4 ComputationalComplexity

9 Boosting

9.1 WeakClassifiers

9.2 AdaBoost

9.3 A Family of Convex Boosters

9.4 Nonconvex Loss Functions


Part Three 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 Toward Quantum Manifold Embedding

10.5 QuantumK—Means

10.6 QuantumK—Medians

10.7 Quantum Hierarchical Clustering

10.8 ComputationalComplexity

11 Quantum Pattern Recognition

11.1 Quantum Associative Memory

11.2 The Quantum Perceptron

11.3 Quantum Neural Networks

11.4 PhysicaIRealizations

11.5 ComputationalComplexity

12 QuantumClassification

12.1 Nearest Neighbors

12.2 Support Vector Machines with Grover's Search

12.3 Support Vector Machines with Exponential Speedup

12.4 ComputationalComplexity

13 Quantum Process Tomography and Regression

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 the Unitary

13.6 Optimal State for Learning

13.7 Applying the Unitary and Finding the Parameter for the Input State

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, Bit Depth, and Generalization Performance

14.7 Mapping to Hardware

14.8 ComputationalComplexity

Bibliography


前言/序言

Machine learning is a fascinating area to work in: from detecting anomalous events in live streams of sensor data to identifying emergent topics involving text collection,exciting problems are never too far away.

Quantuminformation theory also teems with excitement. By manipulating particles at a subatomic level, we are able to perform Fourier transformation exponentially faster, or search in a database quadratically faster than the classical limit. Superdense coding transmits two classical bits using just one qubit. Quantum encryption is unbreakable-at least in theory.

The fundamental question of this monograph is simple: What can quantum computing contribute to machine learning? We naturally expect a speedup from quantum methods, but what kind of speedup? Quadratic? Or is exponential speedup possible? It is natural to treat any form of reduced computational complexity with suspicion. Are there tradeoffs in reducing the complexity?

Execution time is just one concern of learning algorithms. Can we achieve higher generalization performance by turning to quantum computing? After all, training error is not that difficult to keep in check with classical algorithms either: the real problem is finding algorithms that also perform well on previously unseen instances. Adiabatic quantum optimization is capable of finding the global optimum of nonconvex objective functions. Grover's algorithm finds the global minimum in a discrete search space. Quantum process tomography relies on a double optimization process that resembles active learning and transduction. How do we rephrase learning problems to fit these paradigms?

Storage capacity is also of interest. Quantum associative memories, the quantum variants of Hopfield networks, store exponentially more patterns than their classical counterparts. How do we exploit such capacity efficiently?

These and similar questions motivated the writing of this book. The literature on the subject is expanding, but the target audience of the articles is seldom the academics working on machine learning, not to mention practitioners. Coming from the other direction, quantum information scientists who work in this area do not necessarily aim at a deep understanding oflearning theory when devising new algorithms.

This book addresses both of these communities: theorists of quantum computing and quantum information processing who wish to keep up to date with the wider context of their work, and researchers in machine learning who wish to benefit from cutting-edge insights into quantum computing.

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