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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

Quantum Generative Models

Generative modeling focuses on learning a data distribution $p_{\text{data}}(x)$ and creating new samples $x \sim p_{\text{model}}(x)$ that resemble the original data. This chapter introduces quantum approaches to this task. Classical methods like Generative Adversarial Networks (GANs) and Variational Autoencoders (VAEs) provide a foundation, but quantum circuits offer potentially different ways to represent and sample from complex probability distributions.

You will study two primary quantum generative models:

Quantum Circuit Born Machines (QCBMs): Learn how parameterized quantum circuits can implicitly model probability distributions defined by their measurement outcomes, $p(x) = |\langle x | \psi(\theta) \rangle|^2$ .
Quantum Generative Adversarial Networks (QGANs): Understand the framework where quantum circuits serve as generators, competing against classical or quantum discriminators.

We will examine the operational principles, architectures, and specific training challenges associated with these models, such as balancing the adversarial components in QGANs. You will also learn about methods for evaluating the quality of generated samples and the techniques required to sample effectively from the trained quantum models. The chapter concludes with a practical exercise in implementing a basic QGAN.

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