Quantum Computing

  1. Quantum Algorithms
    1. Algorithm Classification
      1. Exact vs Approximate Algorithms
        1. Quantum Speedup Types
          1. Oracle-Based vs Non-Oracle Algorithms
          2. Simple Quantum Algorithms
            1. Deutsch's Algorithm
              1. Problem Statement
                1. Circuit Implementation
                  1. Quantum vs Classical Comparison
                  2. Deutsch-Jozsa Algorithm
                    1. Problem Solved: Determining Function Type
                      1. Constant vs Balanced Functions
                        1. Demonstration of Quantum Speedup
                          1. Circuit Implementation
                          2. Bernstein-Vazirani Algorithm
                            1. Hidden String Problem
                              1. Circuit Structure
                                1. Exponential Speedup
                              2. Search Algorithms
                                1. Grover's Algorithm
                                  1. Classical Search Complexity
                                    1. The Oracle
                                      1. Oracle Construction
                                        1. Black-Box Model
                                        2. Amplitude Amplification
                                          1. Grover Iteration
                                            1. Diffusion Operator
                                              1. Probability of Success
                                              2. Quadratic Speedup
                                                1. Optimality of Grover's Algorithm
                                                2. Variations and Extensions
                                              3. Period Finding and Factoring
                                                1. Simon's Algorithm
                                                  1. Problem Solved: Finding the Period of a Function
                                                    1. Hidden Subgroup Problem
                                                      1. Exponential Speedup over Classical Methods
                                                        1. Circuit Structure
                                                        2. The Quantum Fourier Transform (QFT)
                                                          1. Definition and Properties
                                                            1. Mathematical Formulation
                                                              1. Inverse QFT
                                                              2. Circuit Implementation
                                                                1. Decomposition into Basic Gates
                                                                  1. Circuit Depth and Complexity
                                                                  2. Comparison to Classical Discrete Fourier Transform
                                                                    1. Speedup and Applications
                                                                  3. Quantum Phase Estimation
                                                                    1. Algorithm and Circuit
                                                                      1. Steps of the Algorithm
                                                                        1. Required Gates and Resources
                                                                        2. Application in Other Algorithms
                                                                          1. Use in Shor's Algorithm
                                                                            1. Eigenvalue Problems
                                                                            2. Precision and Success Probability
                                                                            3. Shor's Algorithm
                                                                              1. Problem Solved: Integer Factorization
                                                                                1. Classical Hardness of Factoring
                                                                                2. Classical and Quantum Components
                                                                                  1. Modular Exponentiation
                                                                                    1. Quantum Order Finding
                                                                                      1. Continued Fractions
                                                                                      2. Implications for Cryptography
                                                                                        1. Impact on RSA and Public-Key Cryptography
                                                                                        2. Resource Requirements
                                                                                      3. Variational Quantum Algorithms
                                                                                        1. Hybrid Quantum-Classical Approach
                                                                                          1. Variational Quantum Eigensolver (VQE)
                                                                                            1. Ansatz Circuits
                                                                                              1. Classical Optimization Loop
                                                                                                1. Applications in Chemistry
                                                                                                  1. Barren Plateau Problem
                                                                                                  2. Quantum Approximate Optimization Algorithm (QAOA)
                                                                                                    1. Problem Mapping
                                                                                                      1. Parameter Optimization
                                                                                                        1. Applications in Combinatorial Optimization
                                                                                                          1. Performance Analysis
                                                                                                          2. Quantum Machine Learning Algorithms
                                                                                                            1. Quantum Support Vector Machines
                                                                                                              1. Quantum Neural Networks
                                                                                                                1. Quantum Principal Component Analysis

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                                                                                                            5. Core Quantum Phenomena for Computation

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                                                                                                            7. Physical Realizations of Quantum Computers