Quantum Computing

1. Theoretical Foundations
   1. Quantum Mechanics Fundamentals
      1. Wavefunction and Schrödinger Equation
         1. Definition of Wavefunction
            1. Properties and Interpretation
            2. Complex Number Representation
         2. Schrödinger Equation
            1. Time-Dependent Equation
               1. Applications and Implications
               2. Derivation and Solutions
            2. Time-Independent Equation
               1. Potential Energy
               2. Eigenstates and Eigenvalues
         3. Probability Amplitude
            1. Born Rule
            2. Conservation of Probability
      2. Uncertainty Principle
         1. Heisenberg's Uncertainty Relations
            1. Position and Momentum
            2. Energy and Time
         2. Implications for Measurement
            1. Observer Effect
            2. Limitations in Precision
         3. Applications
            1. Quantum Tunneling
            2. Atomic Scale Phenomena
      3. Quantum Entanglement
         1. EPR Paradox
            1. Einstein-Podolsky-Rosen Paradox Explanation
            2. Philosophical Implications
         2. Bell's Theorem
            1. Bell Inequalities
            2. Experimental Verification
         3. Applications of Entanglement
            1. Quantum Teleportation
            2. Quantum Cryptography
   2. Quantum Information Theory
      1. Quantum Bits and Classical Information Theory
         1. Comparison between Qubits and Bits
         2. Classical Information Theorems
            1. Shannon's Theorems
         3. Quantum Information Theorems
            1. No-Cloning Theorem
            2. Quantum Teleportation Protocol
      2. Quantum Entropy and Information
         1. Von Neumann Entropy
            1. Definition and Properties
            2. Quantum States and Entropy
         2. Classical vs Quantum Entropy
            1. Entropy as a Measure of Uncertainty
            2. Role in Quantum Communications
         3. Mutual Information
            1. Quantum vs Classical Mutual Information
            2. Importance in Quantum Communication
   3. Complexity Theory
      1. Quantum Complexity Classes
         1. BQP (Bounded-Error Quantum Polynomial Time)
            1. Definition and Importance
            2. Comparison with Classical Complexity Classes
         2. QMA (Quantum Merlin Arthur)
            1. Definition and Problem Examples
            2. Relationship to NP (Nondeterministic Polynomial Time)
         3. Other Complexity Classes
            1. QIP (Quantum Interactive Polynomial Time)
            2. QAM (Quantum Arthur-Merlin)
      2. Quantum Supremacy and Quantum Advantage
         1. Definition of Quantum Supremacy
            1. Historical Context
            2. Key Experiments and Achievements
         2. Quantum Advantage
            1. Comparison with Classical Algorithms
            2. Applications and Limitations
         3. Challenges in Demonstrating Supremacy
            1. Scalability and Noise Issues
            2. Verification and Validation Methods