Mathematics for Quantum Informatics

Course guarantor:

Štěpán Starosta

Lecturer:

Štěpán Starosta

Tutor:

Štěpán Starosta

Supervisor:

Department of Applied Mathematics

Synopsis:

Linear algebra on finite dimensional spaces with scalar product, Hilbert spaces, Dirac's bra-ket formalism, normal, Hermitian and unitary operators, operator spectrum, orthonormalization, diagonalization, matrix exponential, tensor product of vector spaces and operators. Discrete Fourier transform and fast Fourier transform.

Requirements:

Syllabus of lectures:

1. Complex numbers, vector spaces, scalar product.

2. Geometry of Hilbert spaces of finite dimension: orthonormal basis, Fourier expansion, Parseval's equality, Schwarz inequality.

3. Linear operators on Hilbert spaces of finite dimension, matrix representation of an operator.

4. Dirac bra-ket notation. Hermitian adjoint operator (matrix). Dual space of a Hilbert space of finite dimension and Riesz theorem.

5. Eigenvalues and eigenvectors of an operator (matrix), spectrum of an operator, diagonalization.

6. Normal operators: projectors, Hermitian operators, unitary operators.

7. Properties of normal operators, especially spectral properties. Spectral decomposition of operators (matrices).

8. Tensor product of vector spaces and operators (matrices).

9. Matrix exponential, relation of Hermitian and unitary operators (matrices).

10. Operator trace and its properties. Density matrix.

11. Discrete Fourier Transform (DFT) as a unitary operator (matrix).

12. Properties of Discrete Fourier transforms.

13. DFT implementation using Fast Fourier Transform (FFT).

Syllabus of tutorials:

1. Complex numbers, vector spaces, scalar product.

2. Geometry of Hilbert spaces of finite dimension: orthonormal basis, Fourier expansion, Parseval's equality, Schwarz inequality.

3. Linear operators on Hilbert spaces of finite dimension, matrix representation of an operator.

4. Dirac bra-ket notation. Hermitian adjoint operator (matrix). Dual space of a Hilbert space of finite dimension and Riesz theorem.

5. Eigenvalues and eigenvectors of an operator (matrix), spectrum of an operator, diagonalization.

6. Normal operators: projectors, Hermitian operators, unitary operators.

7. Properties of normal operators, especially spectral properties. Spectral decomposition of operators (matrices).

8. Tensor product of vector spaces and operators (matrices).

9. Matrix exponential, relation of Hermitian and unitary operators (matrices).

10. Operator trace and its properties. Density matrix.

11. Discrete Fourier Transform (DFT) as a unitary operator (matrix).

12. Properties of Discrete Fourier transforms.

13. DFT implementation using Fast Fourier Transform (FFT).

Study Objective:

Study materials:

1. Strang, G.: Introduction to Linear Algebra, 5th Edition

Wellesley-Cambridge Press 2016, ISBN 978-0980232776

2. Lay, D.C., Lay S. R., McDonald, J. J.: Linear Algebra and Its Applications, 5th Edition, Pearson 2015, ISBN 978-0321982384

3. Lipton, R. J., Regan, K. W.: Introduction to Quantum Algorithms via Linear Algebra, 2nd Edition, MIT Press 2021, ISBN 9780262045254

Note:

Information about the course and teaching materials can be found at https://courses.fit.cvut.cz/QNI-MQI

Further information:

https://courses.fit.cvut.cz/QNI-MQI

Time-table for winter semester 2025/2026:

Time-table is not available yet

Time-table for summer semester 2025/2026:

Time-table is not available yet

The course is a part of the following study plans:

• Quantum Informatics (compulsory course in the program)