Computer Science 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.
1.1.
Review of Classical Computing
1.1.1.
The Bit as the Fundamental Unit of Information
1.1.1.1. Binary Representation
1.1.1.2. Physical Realizations of Bits
1.1.2.
Logic Gates
1.1.2.7. Universal Gate Sets in Classical Computing
1.1.3.
Boolean Algebra
1.1.3.1. Boolean Variables and Expressions
1.1.3.3. Logic Circuit Simplification
1.1.4.
The Turing Machine Model
1.1.4.1. Components of a Turing Machine
1.1.4.2. Computability and Decidability
1.1.4.3. Universal Turing Machine
1.1.4.4. Church-Turing Thesis
1.1.5.
Computational Complexity Classes
1.1.5.3. NP-Complete Problems
1.2.
Essential Concepts from Quantum Mechanics
1.2.1.
Wave-Particle Duality
1.2.1.1. Double-Slit Experiment
1.2.1.2. Implications for Information Processing
1.2.2.
Quantization of Energy
1.2.2.1. Discrete Energy Levels
1.2.2.2. Quantum Harmonic Oscillator
1.2.2.3. Planck's Constant
1.2.3.
The Wave Function and Probability Amplitude
1.2.3.1. Physical Interpretation
1.2.3.2. Normalization Condition
1.2.3.3. Complex Nature of Amplitudes
1.2.4.
Uncertainty Principle
1.2.4.1. Heisenberg Uncertainty Relations
1.2.4.2. Complementary Observables
1.2.4.3. Implications for Measurement
1.2.5.
Hilbert Spaces
1.2.5.1. Definition and Properties
1.2.5.2. Finite vs Infinite Dimensional Hilbert Spaces
1.2.5.3. Complete Orthonormal Bases
1.2.6.
Linear Algebra for Quantum Mechanics
1.2.6.1. Vectors and Vector Spaces
1.2.6.1.2. Linear Independence
1.2.6.1.3. Span and Dimension
1.2.6.2. Bra-Ket (Dirac) Notation
1.2.6.2.3. Bracket Notation
1.2.6.3. Inner Products and Outer Products
1.2.6.3.1. Properties of Inner Products
1.2.6.3.2. Outer Product as Operator
1.2.6.3.3. Completeness Relations
1.2.6.4. Operators and Matrices
1.2.6.4.1. Hermitian Operators
1.2.6.4.2. Unitary Operators
1.2.6.4.3. Matrix Representation of Operators
1.2.6.4.4. Adjoint Operations
1.2.6.5. Eigenvalues and Eigenvectors
1.2.6.5.1. Spectral Decomposition
1.2.6.5.2. Measurement Outcomes
1.2.6.5.3. Degenerate Eigenspaces
1.2.6.6.1. Multi-Qubit State Construction
1.2.6.6.2. Entangled vs Separable States
1.2.6.6.3. Tensor Product Properties
1.2.7.
The Schrödinger Equation
1.2.7.1. Time-Dependent Schrödinger Equation
1.2.7.2. Time-Independent Schrödinger Equation
1.2.7.3. Unitary Evolution
1.2.8.
Postulates of Quantum Mechanics
1.2.8.1. State Space Postulate
1.2.8.2. Evolution Postulate
1.2.8.3. Measurement Postulate
1.2.8.4. Composite Systems Postulate