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Topological Quantum Computation: A Fault-Tolerant Quantum Framework

Table of Contents Introduction Motivation and Importance Basics of Topology in Physics What is Topological Quantum Computation (TQC)? Anyons and Their Role in TQC Braiding Anyons as Quantum Gates Fusion Rules...

Measurement-Based Quantum Computation: The One-Way Quantum Computer

Table of Contents Introduction Quantum Computation Models: A Brief Recap What Is Measurement-Based Quantum Computation (MBQC)? History and Origins of MBQC The One-Way Quantum Computer Cluster States and Graph States Preparing...

Adiabatic Quantum Computation: Theory, Power, and Practicality

Table of Contents Introduction Quantum Computation Paradigms Overview What Is Adiabatic Quantum Computation (AQC)? Historical Background The Quantum Adiabatic Theorem Basic AQC Model and Formal Description Initial and Final Hamiltonians Ground State...

Quantum Cellular Automata: Local Rules and Global Quantum Dynamics

Table of Contents Introduction Classical Cellular Automata Recap What Are Quantum Cellular Automata (QCA)? Motivation and Applications Basic Definitions and Formalism Hilbert Space Representation Local Unitarity and Global Evolution Translation-Invariance and Causality Partitioned...

Quantum Communication Complexity: Fundamentals and Frontiers

Table of Contents Introduction Classical Communication Complexity Recap Why Quantum Communication Complexity? Basic Model: Alice and Bob Communication Cost and Protocol Goals Types of Quantum Communication Protocols Quantum vs Classical: Key...

Quantum Turing Machines: Theoretical Foundation of Quantum Computation

Table of Contents Introduction Classical Turing Machines Recap Need for a Quantum Analog Definition of Quantum Turing Machines (QTM) Configuration and Tape Alphabet Quantum States and Hilbert Space Transition Function and...

Interactive Quantum Computation: Models and Complexity

Table of Contents Introduction What is Interactive Computation? Classical Interactive Proof Systems (IP) Motivation for Quantum Interactivity Quantum Interactive Proof Systems (QIP) One-Round vs Multi-Round QIP Formal Model of QIP Verifier and...

Quantum Finite Automata: Structure, Power, and Limitations

Table of Contents Introduction Classical Finite Automata: A Brief Recap Motivation for Quantum Automata Basic Structure of Quantum Finite Automata (QFA) 1-Way Quantum Finite Automata (1QFA) Measure-Once 1QFA (MO-1QFA) Measure-Many 1QFA...

Quantum Advice and BQP/qpoly: Power and Limits

Table of Contents Introduction to Quantum Advice Classical vs Quantum Advice Formal Definition of BQP/qpoly Role of Advice Strings in Computation BQP/qpoly vs BQP/poly The Power of Nonuniform Quantum Computation Encoding...

Quantum Lower Bounds: Foundations and Techniques

Table of Contents Introduction Why Quantum Lower Bounds Matter Types of Quantum Lower Bounds The Quantum Black-Box Model Polynomial Method: A Formal Introduction Approximate Degree and Boolean Functions Symmetric Function Lower...

Quantum Query Complexity: A Deep Dive

Table of Contents Introduction Classical Computation and Query Models Oracle Access in Classical and Quantum Settings What is Query Complexity? Classical Query Complexity: Deterministic and Randomized Quantum Query Model: The...

Quantum Query Complexity: A Deep Dive

Table of Contents Introduction Classical vs Quantum Query Models The Oracle Model in Quantum Computation Query Complexity: Definition and Significance Classical Query Complexity Recap Quantum Query Algorithms6.1. Deutsch-Jozsa Algorithm6.2. Grover’s...

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