(in Polish) Podstawy fizyki kwantowej na przykładzie kubitu 0800-PFKQBIT

The aim of the course is to introduce students to the fundamentals of quantum physics through the analysis of the simplest quantum system – the qubit. The course is based on the algebraic description of states and operators and on the geometric representation of a qubit state on the Bloch sphere.

Main topics:

1. Introduction to quantum physics
* the quantum world: physical scales, examples of physical systems, and the range of applicability of quantum physics,
* the concept of a physical system and its state,
* the classical bit and the qubit,
* physical examples of qubit implementations.

2. Mathematical description of a qubit state
* Hilbert space, basis, state vector, and normalization,
* quantum superposition,
* amplitude and probability,
* global and relative phase,
* interference,
* algebra of vectors and 2×2 matrices.

3. Operators and observables
* linear operators and their matrix representation,
* action of an operator on a state vector,
* Pauli matrices and their physical interpretation,
* the eigenvalue problem for operators,
* eigenvalues and eigenvectors,
* diagonalization of 2×2 matrices.

4. Quantum measurement
* the probabilistic nature of measurement,
* collapse of the state vector,
* measurement in different bases,
* expectation values of observables,
* the role of phase and interference in measurement outcomes.

5. Non-commutativity and the uncertainty principle
* commutators,
* non-commuting observables,
* Pauli matrices as an example of non-commuting observables,
* the Heisenberg uncertainty principle for a two-level system.

6. Single-qubit operations and gates
* unitary operators as quantum logic gates,
* single-qubit gates as rotations on the Bloch sphere,
* rotation matrices,
* implementation of an arbitrary transformation of a qubit state using a sequence of unitary operations,
* physical implementations of quantum logic gates.

7. Quantum dynamics
* introduction to simple differential equations,
* the Hamiltonian and the special role of the energy operator,
* the Schrödinger equation for a qubit,
* free evolution and evolution of energy eigenstates,
* the relation between energy and precession frequency,
* geometric interpretation of evolution on the Bloch sphere,
* interaction with an external system,
* Rabi oscillations and their physical implementations.

8. Multi-qubit systems
* tensor product,
* description of a pair of qubits,
* separable and entangled states,
* quantum entanglement and its measures,
* basic two-qubit quantum gates, in particular CNOT and CPHASE,
* Bell states,
* quantum teleportation,
* the BB84 protocol.

Computer laboratory
* numerical representation of states and operators,
* action of operators on quantum states,
* visualization of states on the Bloch sphere,
* simulation of transformations and quantum gates,
* simulation of time evolution,
* numerical implementation of the tensor product,
* implementation and analysis of two-qubit operations.

The scope and order of selected topics may be subject to minor modifications depending on the pace of the group and teaching needs, while maintaining the intended learning outcomes of the course.

Term 2026/27Z:

As specified in the “Basic course information” section.

Total student workload

Participation in classes: 45 hours Participation in computer laboratory classes: 15 hours Preparation for classes and individual study: 30 hours Preparation for the final assessment: 30 hours Total student workload: 120 hours

Learning outcomes - knowledge

Upon completion of the course, the student: W01 – knows the scope of applicability of quantum physics and understands the concept of a quantum state. Programme learning outcomes: K_W01, K_W05 W02 – knows the Hilbert-space formalism for a two-level system and the mathematical description of a qubit state. Programme learning outcomes: K_W04, K_W05 W03 – knows the role of linear and unitary operators in the description of observables and transformations of quantum states. Programme learning outcomes: K_W04, K_W05 W04 – understands the probabilistic nature of quantum measurement and the significance of probability amplitudes and phase. Programme learning outcomes: K_W01, K_W05 W05 – knows the significance of non-commuting observables and their relation to the uncertainty principle. Programme learning outcomes: K_W04, K_W05 W06 – knows the fundamentals of quantum dynamics generated by the Hamiltonian operator and understands the geometric interpretation of qubit evolution. Programme learning outcomes: K_W04, K_W05 W07 – knows the concepts of the tensor product and quantum entanglement, as well as basic one- and two-qubit operations and quantum gates. Programme learning outcomes: K_W01, K_W05

Learning outcomes - skills

Upon completion of the course, the student: U01 – is able to use the mathematical formalism appropriate for the description of a two-level system and to formulate basic laws of quantum mechanics using this formalism. Programme learning outcomes: K_U01, K_U04 U02 – is able to perform calculations using vectors and matrices, in particular to determine eigenvalues and eigenvectors of operators represented by 2×2 matrices and to calculate expectation values of the corresponding observables. Programme learning outcomes: K_U04 U03 – is able to analyze the state of a qubit and its evolution, including their geometric interpretation on the Bloch sphere. Programme learning outcomes: K_U01, K_U04 U04 – is able to analyze the action of basic one- or two-qubit quantum gates and to construct simple sequences of unitary operations. Programme learning outcomes: K_U01, K_U04 U05 – is able to use the QuTiP library to represent quantum states and operators and to simulate quantum dynamics and operations. Programme learning outcomes: K_U03, K_U06 U06 – is able to interpret, process, and present the results of simple computer simulations involving one- or two-qubit systems. Programme learning outcomes: K_U03, K_U04 U07 – is able to present the fundamental principles underlying quantum physics in an accessible way and explain them to people without specialist knowledge in the field. Programme learning outcomes: K_U07

Learning outcomes - social competencies

Upon completion of the course, the student: K01 – is ready to critically assess their own knowledge and to independently expand it in the field of quantum physics and its contemporary applications. Programme learning outcomes: K_K01 K02 – understands the importance of communicating and popularizing knowledge of quantum physics in a reliable and accessible manner. Programme learning outcomes: K_K04

Course coordinators

Karolina Słowik

Teaching methods

interactive lecture, computer laboratory

Expository teaching methods

- narration
- participatory lecture
- problem-based lecture
- discussion

Exploratory teaching methods

- laboratory
- classic problem-solving
- round table

Type of course

compulsory course

Prerequisites

Solid knowledge of mathematics and physics at the secondary school level.

Assessment criteria

Graded assessment based on:
Short assessment activities during the semester (class participation, problem solving, short quizzes)
Computer laboratory assessed on the basis of a project
Final examination

Criteria:
>=50% - 3.0
>=60% - 3.5
>=70% - 4.0
>=80% - 4.5
>=90% - 5.0

Practical placement

n.a.

Bibliography

The course has an original, instructor-designed format and will be taught using dedicated teaching materials prepared specifically for the course.

Supplementary literature:

1. C. Hughes, J. Isaacson, A. Perry, R. F. Sun, J. Turner, *Quantum Computing for the Quantum Curious*, Springer, 2021. Open Access.
2. A. Flarend, R. Hilborn, *Quantum Computing: From Alice to Bob*, Oxford University Press, 2022.
3. D. McMahon, *Quantum Mechanics for Beginners: With Applications to Quantum Communication and Quantum Computing*, Oxford University Press, 2020.
4. P. Kaye, R. Laflamme, M. Mosca, *An Introduction to Quantum Computing*, Oxford University Press, 2007.
5. OpenStax, *University Physics, Volume 3* – selected chapters on quantum physics.

Term 2026/27Z:

As specified in the “Basic course information” section.

Additional information

Additional information (registration calendar, class conductors, localization and schedules of classes), might be available in the USOSweb system:

Description of 0800-PFKQBIT in USOSweb