Syllabus

1. Mathematical Foundations:
Linear Algebra: Linear Vector Space, Normed Space, Inner Product Space, Linear Independence, Completeness, Orthogonality, Rank, Hilbert Space.
Complex Matrices: Hermitian and Unitary Matrices.
Eigenvalue Problem: Spectral Theorem.
Probability and Statistics: Basics of Statistics, Axioms of Probabilities, Probability Distributions, Central Limit Theorem.

2. Quantum Mechanics Basics:
Postulates of Quantum Mechanics.
Density Operator Formalism: Pure and Mixed States.
Superposition and Entanglement in Quantum Mechanics.

3. Axiomatic Quantum Theory:
Quantum States, Observables, and Measurement.
Hilbert Space and Unitary Transformations.
Schrodinger Equation and Unitary Evolution.
No Cloning Theorem.

4. Qubits vs Classical Bits:
Spin-Half Systems and Photon Polarizations.
Trapped Atoms, Ions, and Semiconducting Quantum Dots.
Artificial Atoms Using Circuits.
Single and Two Qubit Gates: Solovay-Kitaev Theorem.

5. Pure and Mixed States:
Density Matrices and Quantum Evolution.
Superoperators and Kraus Operators.
Positive and Completely Positive Trace-Preserving Maps.

6. Quantum Correlations:
Entanglement and Bell's Theorems.
Review of Turing Machines and Classical Computational Complexity.
Reversible Computation.
Universal Quantum Logic Gates and Circuits.

7. Quantum Algorithms:
Deutsch and Deutsch-Josza Algorithms.
Bernstein-Vazirani Algorithm.
Database Search: Grover's Algorithm.
Quantum Fourier Transform and Prime Factorization: Shor's Algorithm.

8. Error Correction and Fault-Tolerance:
Introduction to Error Correction.
Fault-Tolerance Techniques.
Simple Error Correcting Codes.

9. Survey of Current Status and Future Roadmap:
NISQ Era Processors.
Quantum Advantage Claims.
Roadmap for Future Quantum Technologies.

To build a strong mathematical foundation essential for understanding quantum mechanics and quantum computing principles.

To explore practical implementation techniques involving quantum gates, algorithms, and circuits.

To introduce advanced topics such as quantum error correction, fault-tolerance, and the current status of quantum technologies.

To familiarize participants with axiomatic quantum theory and the conceptual differences between classical and quantum systems.

Understand and apply linear algebra concepts relevant to quantum mechanics, including Hilbert spaces, complex matrices, and eigenvalue problems.
Analyze and interpret the postulates of quantum mechanics, superposition, entanglement, and quantum measurement processes.
Differentiate between classical and quantum computing paradigms and understand qubit implementations using physical systems.
Develop and implement key quantum algorithms such as Deutsch, Grover's, and Shor's algorithms.
Evaluate error correction methods, fault-tolerant designs, and the future roadmap of quantum computing technologies.

Nielsen, M. A., & Chuang, I. L., Quantum Computation and Quantum Information, Cambridge University Press

Rieffel, E., & Polak, W., Quantum Computing: A Gentle Introduction, MIT Press

Kitaev, A. Y., Shen, A. H., & Vyalyi, M. N., Classical and Quantum Computation, AMS

Benenti, G., Casati, G., & Strini, G., Principles of Quantum Computation and Information, World Scientific