Quantum Computing

11.

Advanced Topics and Future Directions

11.1.

Quantum Information Theory

11.1.1.

Quantum Entropy Measures

11.1.1.1.

Von Neumann Entropy

11.1.1.2.

Relative Entropy

11.1.1.3.

Conditional Entropy

11.1.2.

Quantum Channels

11.1.2.1.

Channel Capacity

11.1.2.2.

11.1.2.3.

Channel Fidelity

11.1.3.

Quantum Communication Protocols

11.1.3.1.

Quantum Teleportation

11.1.3.1.1.

11.1.3.1.2.

Resource Requirements

11.1.3.1.3.

Fidelity Analysis

11.1.3.2.

Superdense Coding

11.1.3.2.1.

11.1.3.2.2.

Information Capacity

11.1.3.3.

Quantum Secret Sharing

11.1.4.

Entanglement Theory

11.1.4.1.

Entanglement Measures

11.1.4.2.

Entanglement Distillation

11.1.4.3.

Entanglement of Formation

11.1.5.

Quantum Complexity Theory

11.1.5.1.

BQP and Other Quantum Complexity Classes

11.1.5.2.

Quantum Interactive Proofs

11.1.5.3.

11.2.

Quantum Supremacy and Quantum Advantage

11.2.1.

Defining the Milestone

11.2.1.1.

Quantum vs Classical Computation

11.2.1.2.

Computational vs Practical Advantage

11.2.2.

Experimental Demonstrations

11.2.2.1.

Google's Sycamore Experiment

11.2.2.2.

IBM's Response and Classical Improvements

11.2.2.3.

Limitations and Criticisms

11.2.3.

Random Circuit Sampling

11.2.4.

11.2.5.

Future Prospects for Quantum Advantage

11.3.

Quantum Networking

11.3.1.

Quantum Internet Architecture

11.3.1.1.

Network Protocols

11.3.1.2.

Quantum Repeaters

11.3.1.3.

Entanglement Swapping

11.3.2.

Entanglement Distribution

11.3.2.1.

Long-Distance Entanglement

11.3.2.2.

Satellite-Based QKD

11.3.3.

Distributed Quantum Computing

11.3.3.1.

Multi-Node Quantum Algorithms

11.3.3.2.

Quantum Cloud Computing

11.3.4.

Quantum Network Security

11.4.

Quantum Sensing and Metrology

11.4.1.

Quantum-Enhanced Measurement

11.4.1.1.

Shot Noise Limit

11.4.1.2.

Heisenberg Limit

11.4.1.3.

Quantum Fisher Information

11.4.2.

Applications in Precision Sensing

11.4.2.1.

11.4.2.2.

11.4.2.3.

Gravitational Wave Detection

11.4.2.4.

Medical Imaging

11.4.3.

Quantum Radar and Lidar

11.4.4.

Quantum-Enhanced Imaging

11.5.

Emerging Quantum Technologies

11.5.1.

Quantum Batteries

11.5.2.

Quantum Heat Engines

11.5.3.

Quantum Thermodynamics

11.5.4.

Quantum Biology

11.5.4.1.

11.5.4.2.

Avian Navigation

11.5.4.3.

Enzyme Catalysis

11.6.

Future Research Directions

11.6.1.

Fault-Tolerant Quantum Computing Timeline

11.6.2.

Quantum-Classical Hybrid Algorithms

11.6.3.

Quantum Advantage in Near-Term Applications

11.6.4.

Novel Quantum Computing Paradigms

11.6.5.

Quantum Computing Education and Workforce Development

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10. Quantum Software and Programming

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1. Foundations of Quantum Computing