Quantum Computing: a Mathematical Introduction WM-MA-S2-E2-QK

Quantum technologies are opening a new direction in computation by exploiting principles that have no counterpart in classical information processing. Quantum computers encode and manipulate information in ways that can, for specific tasks, outperform standard machines by a dramatic margin. One widely discussed example comes from cryptography: several commonly used security schemes depend on the practical difficulty of factoring very large integers, whereas quantum methods (notably Shor’s algorithm) indicate that such problems can be handled efficiently under an appropriate quantum model.

Ongoing advances in hardware and theory, together with substantial investment from both industry and public institutions, have made quantum computing an increasingly active area of research and development. The aim of this course is to provide a mathematically grounded introduction to the subject, developing the core concepts and techniques needed to read further material and pursue more advanced study.

A background in quantum mechanics or programming is useful but not assumed and is not necessary. Students are expected to be comfortable with trigonometry, complex numbers, and linear algebra, and to be willing to work carefully with formal mathematical reasoning. Curiosity and persistence will be essential throughout the course. All necessary background will be reviewed during the course.

The format of the lectures and tutorials will be in a classroom with a blackboard and there will not be any mandatory programming involved.

(in Polish) Dyscyplina naukowa, do której odnoszą się efekty uczenia się

mathematics

(in Polish) E-Learning

(in Polish) E-Learning (pełny kurs) z podziałem na grupy

(in Polish) Grupa przedmiotów ogólnouczenianych

(in Polish) nie dotyczy

Course coordinators

Term 2026/27_L:

Dorota Dąbrowska
Omid Makhmali

Term 2025/26_L:

Omid Makhmali

(in Polish) Opis nakładu pracy studenta w ECTS

Term 2025/26_L:

For the lecture: class attendance 30 hours, participation in tests/exams outside of class 3 hours, consultations 2 hours, preparation for classes 5 hours, preparation for tests/exams 10 hours, All together 50 h, equivalent to 2 ECTS credits For the tutorials: class attendance 30 hours, consultations 5 hours, preparation for classes 20 hours, preparation for tests/exams 20 hours, All together 75 h, equivalent to 3 ECTS credits

Term 2026/27_L:

LECTURE participation in classes: 30 hours participation in exams outside of classes: 3 hours participation in consultations: 2 hours preparation for classes: 5 hrs preparation for the exam: 10 hours Total 50 hrs. 1.4 ECTS - direct contact with the teacher 0.6 ECTS - own work TUTORIALS participation in classes: 30 hrs participation in consultations: 4 hours preparation for classes: 20 hrs preparation for the pass: 21 hours Total 75 hrs. 1.4 ECTS - direct contact with the teacher 1.6 ECTS - own work

Subject level

intermediate

Learning outcome code/codes

(in Polish) wykład: MA2_W04, MA2_W05, MA2_W06, MA2_W07, MA2_W16 laboratorium: MA2_U13, MA2_U14, MA2_U15, MA2_U21, MA2_U24, MA2_K02, MA2_K08 OPIS SYMBOLI - Absolwent: wykład MA2_W04 - ma pogłębioną wiedzę w wybranej dziedzinie matematyki teoretycznej lub stosowanej, MA2_W05 - ma pogłębioną wiedzę w wybranej dziedzinie matematyki:, MA2_W06 - 2) rozumie sformułowania zagadnień pozostających na etapie badań, MA2_W07 - 3) zna powiązania zagadnień wybranej dziedziny z innymi działami matematyki teoretycznej i stosowanej, MA2_W16 - ma wiedzą adekwatną do obranej specjalności. laboratorium MA2_U13 - umie, na poziomie zaawansowanym i obejmującym matematykę współczesną, stosować oraz przedstawiać w mowie i na piśmie, metody co najmniej jednej wybranej gałęzi matematyki: analizy matematycznej i analizy funkcjonalnej, teorii równań różniczkowych i układów dynamicznych, algebry i teorii liczb, geometrii i topologii, rachunku prawdopodobieństwa i statystyki, matematyki dyskretnej i teorii grafów, logiki i teorii mnogości, MA2_U14 - w wybranej dziedzinie potrafi przeprowadzać dowody, w których stosuje w razie potrzeby również narzędzia z innych działów matematyki, MA2_U15 - potrafi określić swoje zainteresowania i je rozwijać, w szczególności jest w stanie nawiązać kontakt ze specjalistami w swojej dziedzinie, np., MA2_U21 - umie stosować metody komputerowo wspomaganego dowodzenia twierdzeń oraz logicznego wspomagania weryfikacji i specyfikacji programów, MA2_U24 - potrafi w teorii i praktyce stosować metody i techniki przynależne do wybranej specjalności, MA2_K02 - jest gotów formułować pytania, służące pogłębieniu własnego zrozumienia danego tematu lub odnalezieniu brakujących elementów rozumowania, MA2_K08 - jest gotów sprostać zadaniom społecznym związanym z wybraną specjalnością.

Type of subject

optional with unlimited choices

Preliminary Requirements

Linear Algebra

Learning outcomes

LECTURE

Student:

W1 - knows and understands the mathematical formalism of quantum computing, in particular the concepts of a qubit and the Bloch sphere, Hilbert spaces, Dirac notation, unitary and Hermitian operators, projectors, the spectral theorem, and quantum measurement (MA2_W04, MA2_W05).

W2 - knows and understands the connections between the mathematical foundations of quantum computing and linear algebra, information theory, algorithm theory, and cryptography (MA2_W07, MA2_W11).

W3 - knows and understands the mathematical foundations of selected quantum phenomena, protocols, and algorithms, in particular entanglement, the no-cloning theorem, quantum teleportation, superdense coding, quantum key distribution, and selected quantum algorithms covered in the course (MA2_W04, MA2_W11).

TUTORIALS

Student:

U1 - is able to apply linear algebraic methods to calculations involving quantum states, operators, measurements, multi-qubit systems and tensor products, and to analyse basic examples of entangled states (MA2_U10).

U2 - is able to apply the mathematical formalism of quantum computing to analyse quantum gates and circuits and selected quantum protocols and algorithms (MA2_U13, MA2_U19).

U3 - is able to provide mathematical justification for selected properties and limitations of quantum computing, in particular those concerning the no-cloning theorem, quantum teleportation, and selected properties of entanglement (MA2_U14).

U4 - is able to analyse the structure and operation of selected quantum algorithms as computational processes and, within the scope covered in the course, compare them with corresponding classical methods (MA2_U19).

Assessment criteria

For all learning outcomes, the following assessment criteria are adopted for all forms of verification:

grade 5: fully achieved (no obvious shortcomings),

grade 4.5: achieved almost fully and criteria for awarding a higher grade are not met,

grade 4: largely achieved and the criteria for a higher grade are not met,

grade 3.5: largely achieved - with a clear majority of positives - and the criteria for granting a higher grade are not met,

grade 3: achieved for most of the cases covered by the verification and criteria for a higher grade are not met,

grade 2: not achieved for most of the cases covered by the verification.

Additional information

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

Description of WM-MA-S2-E2-QK in USOSweb