Quantum Computing

Quantum computing is a revolutionary subfield of computer science that leverages principles from quantum mechanics to process information in fundamentally new ways. Unlike classical computers that use bits representing either 0 or 1, quantum computers use quantum bits, or qubits, which can exist in a superposition of both states simultaneously. This property, combined with another quantum phenomenon called entanglement, allows quantum computers to perform a massive number of calculations in parallel, giving them the potential to solve certain complex problems—such as in drug discovery, materials science, and cryptography—that are intractable for even the most powerful classical supercomputers.

1. Foundations of Quantum Computing
   1. Review of Classical Computing
      1. The Bit as the Fundamental Unit of Information
         1. Binary Representation
         2. Physical Realizations of Bits
      2. Logic Gates
         1. AND Gate
         2. OR Gate
         3. NOT Gate
         4. XOR Gate
         5. NAND Gate
         6. NOR Gate
         7. Universal Gate Sets in Classical Computing
      3. Boolean Algebra
         1. Boolean Variables and Expressions
         2. Truth Tables
         3. Logic Circuit Simplification
         4. De Morgan's Laws
      4. The Turing Machine Model
         1. Components of a Turing Machine
         2. Computability and Decidability
         3. Universal Turing Machine
         4. Church-Turing Thesis
      5. Computational Complexity Classes
         1. Class P
         2. Class NP
         3. NP-Complete Problems
         4. NP-Hard Problems
         5. P vs NP Problem
         6. BQP Class
   2. Essential Concepts from Quantum Mechanics
      1. Wave-Particle Duality
         1. Double-Slit Experiment
         2. Implications for Information Processing
      2. Quantization of Energy
         1. Discrete Energy Levels
         2. Quantum Harmonic Oscillator
         3. Planck's Constant
      3. The Wave Function and Probability Amplitude
         1. Physical Interpretation
         2. Normalization Condition
         3. Complex Nature of Amplitudes
      4. Uncertainty Principle
         1. Heisenberg Uncertainty Relations
         2. Complementary Observables
         3. Implications for Measurement
      5. Hilbert Spaces
         1. Definition and Properties
         2. Finite vs Infinite Dimensional Hilbert Spaces
         3. Complete Orthonormal Bases
      6. Linear Algebra for Quantum Mechanics
         1. Vectors and Vector Spaces
            1. Basis Vectors
            2. Linear Independence
            3. Span and Dimension
         2. Bra-Ket (Dirac) Notation
            1. Bra Vectors
            2. Ket Vectors
            3. Bracket Notation
         3. Inner Products and Outer Products
            1. Properties of Inner Products
            2. Outer Product as Operator
            3. Completeness Relations
         4. Operators and Matrices
            1. Hermitian Operators
            2. Unitary Operators
            3. Matrix Representation of Operators
            4. Adjoint Operations
         5. Eigenvalues and Eigenvectors
            1. Spectral Decomposition
            2. Measurement Outcomes
            3. Degenerate Eigenspaces
         6. Tensor Products
            1. Multi-Qubit State Construction
            2. Entangled vs Separable States
            3. Tensor Product Properties
      7. The Schrödinger Equation
         1. Time-Dependent Schrödinger Equation
         2. Time-Independent Schrödinger Equation
         3. Unitary Evolution
      8. Postulates of Quantum Mechanics
         1. State Space Postulate
         2. Evolution Postulate
         3. Measurement Postulate
         4. Composite Systems Postulate
         5. Born Rule