Advanced Quantum Machine Learning Algorithms and Implementation
Chapter 1: Revisiting Quantum Computing and ML Foundations
Advanced Linear Algebra for Quantum States
Quantum Circuits and Universal Gate Sets Revisited
Entanglement and Non-locality Implications for QML
Density Matrices and Mixed States in QML
Classical ML Optimization Techniques Refresher
Information Geometry in Classical and Quantum Models
Complexity Theory for Classical and Quantum Algorithms
Chapter 2: Advanced Quantum Data Encoding and Feature Maps
Mathematical Foundations of Quantum Feature Maps
Higher-Order Polynomial Quantum Feature Maps
Data Re-uploading Techniques for Expressivity
Analyzing Feature Map Expressibility and Entangling Capability
Kernel Alignment and Quantum Feature Spaces
Encoding High-Dimensional Classical Data
Hands-on Practical: Implementing Custom Quantum Feature Maps
Chapter 3: Quantum Kernel Methods: Theory and Implementation
The Quantum Kernel Trick Formalism
Calculating Quantum Kernel Matrices
Properties of Quantum Kernels
Geometric Differences between Classical and Quantum Kernels
Kernel Concentration Phenomena and Mitigation
Support Vector Machines with Quantum Kernels (QSVM)
Implementation of QSVM for Complex Datasets
Hands-on Practical: Comparing Quantum Kernels
Chapter 4: Variational Quantum Algorithms for Machine Learning
The Variational Principle in Quantum Computation
Parameterized Quantum Circuits (PQC) Design Strategies
Cost Functions for QML Tasks
Gradient Calculation Methods
Advanced Classical Optimizers for VQAs
Quantum Natural Gradient Descent
Barren Plateaus Analysis and Mitigation
Hands-on Practical: Implementing a Variational Quantum Classifier
Chapter 5: Quantum Neural Networks: Architectures and Training
Models of Quantum Neurons and Layers
Quantum Circuit Born Machines (QCBMs)
Quantum Convolutional Neural Networks (QCNNs)
Quantum Graph Neural Networks (QGNNs)
Hybrid Quantum-Classical Neural Network Architectures
Training QNNs Challenges and Strategies
Overfitting and Generalization in QNNs
Practice: Building and Training a Simple QNN
Chapter 6: Quantum Generative Models
Classical Generative Models Review
Quantum Circuit Born Machines for Distribution Learning
Quantum Generative Adversarial Networks (QGANs)
Training QGANs Challenges and Architectures
Evaluating Quantum Generative Models
Sampling from Quantum Generative Models
Hands-on Practical: Implementing a Basic QGAN
Chapter 7: Hardware Considerations and Error Mitigation in QML
Characterizing Quantum Hardware Noise
Impact of Noise on QML Algorithm Performance
Quantum Error Mitigation Techniques
Circuit Optimization and Transpilation
Hardware-Efficient Ansätze Design
Benchmarking QML Algorithms on Real Quantum Devices
Practice: Applying Error Mitigation to a VQC
Higher-Order Polynomial Quantum Feature Maps
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- Supervised learning with quantum-enhanced feature spaces, Vojtěch Havlíček, Antonio D. Córcoles, Kristan Temme, Aram W. Harrow, Abhinav Kandala, Jerry M. Chow, Jay M. Gambetta, 2019 Nature, Vol. 567 (Springer Nature) DOI: 10.1038/s41586-019-0980-2 - This foundational paper introduces quantum kernel methods and proposes specific quantum feature map circuits, including those utilizing ZZ-interactions for encoding data into higher-order polynomial features.
- Power of data in quantum machine learning, Hsin-Yuan Huang, Michael Broughton, Masoud Mohseni, Ryan Babbush, Sergio Boixo, Hartmut Neven, Jarrod R. McClean, 2021 Nature Communications, Vol. 12 DOI: 10.1038/s41467-021-22539-9 - This work investigates the trainability of quantum machine learning models, addressing the 'kernel concentration' problem, where quantum kernels can become untrainable for large numbers of qubits, a consideration for higher-order feature maps.
- Quantum Machine Learning in Feature Hilbert Spaces, Maria Schuld, Nathan Killoran, 2019 Physical Review Letters, Vol. 122 (American Physical Society) DOI: 10.1103/PhysRevLett.122.040504 - This paper provides a theoretical framework for quantum machine learning by embedding classical data into a quantum Hilbert space and defines quantum kernels based on inner products in this space.
- ZZFeatureMap, Qiskit Development Team, 2023 (IBM Corp.) - Provides practical details and an implementation of a widely used quantum feature map that creates second-order polynomial features through ZZ-interactions, matching the example discussed in the section.