Quantum Computing

The objective of this course is to familiarize students with the fundamentals of quantum computing. In this course, students will learn the basic principles of quantum mechanics, how information is stored and processed in quantum systems, and fundamental algorithms in this field. Students will also become familiar with concepts such as quantum computational complexity, quantum information theory, and quantum cryptography.

1. Basic Concepts
   • Introduction and history of quantum computing, concept of qubit
2. Linear Algebra Review
   • Vector space, inner product, dual space, tensor product
   • Linear operators, adjoint, Singular Value Decomposition (SVD)
3. Principles of Quantum Mechanics
   • Schrödinger equation, Hamiltonian mechanics, physical quantities
   • State space, time evolution, measurement, composite systems, entanglement
4. Principles of Quantum Computing
   • Quantum measurement, teleportation, superdense coding
   • Density matrix, partial trace, ensemble
   • Purification, EPR paradox, Bell's theorem
5. Quantum Circuits
   • Quantum gates, components of quantum circuits, measurement postponement principle
6. Quantum Algorithms
   • Simon's algorithm
   • Shor's factorization algorithm
   • Grover's search algorithm and its applications
7. Quantum Computational Complexity
   • Quantum complexity classes, relation to classical classes, comparison with probabilistic computations
8. Dynamics of Quantum Systems
   • Environmental interaction, measurement as dynamics, positive and trace-preserving maps
9. Quantum Cryptography
   • Private key cryptography, quantum key distribution, security of quantum key distribution

• Assignments: Three theoretical assignments and three programming assignments (40% of total grade)
• Exam: Final exam (60% of total grade)