# Course syllabus - Quantum Computing and Information 7.5 credits

### Kvantberäkningar och information

 Course code: MAA509 Valid from: Autumn semester13 Autumn semester14 Level of education: Second cycle Subject: Mathematics Main Field(s) of Study: Mathematics/Applied Mathematics, In-Depth Level: A1N (Second cycle, has only first-cycle course/s as entry requirements), School: UKK Ratification date: 2013-02-01

## Objectives

The aim of the course is to introduce the fundamental ideas, concepts and principles that are used to analyze information and quantum information through algorithms based on information theory and quantum mechanical mathematical principles.
Additionally the course will generally develop the student's ability to assimilate and communicate mathematical theory and use it in conjunction with other areas, such as information technology and computer science, to build a new generation of efficient algorithms and to solve problems theoretically and using the computer. In addition the course shall enhance students' knowledge of mathematical programming and computational techniques, and their abilities for independent analysis of mathematical models and solution of mathematical problems.

## Learning outcomes

After completing the course, students should be able to

- explain and compare main examples and fundamentals for classical- and quantum information
- explain ideas, principles and mathematical models which underlie secure and reliable transmission of information and classical and quantum cryptography
- describe and use different types of quantum gates in their matrix representation and input-output signal system presentation, as well as understand principles for construction and presentation of quantum circuits and quantum algorithms in matrix and in system forms
- provide and explain the examples and basics of the theory of quantum operations, and apply techniques of linear algebra, geometry of linear transformations and factorizations of matrices for construction of more complex quantum operations from the elementary quantum operations
- explain the use of matrix norms and vector norms for estimating computational errors in quantum circuits as well as explain basic ideas, theory and examples of error correcting codes
- explain the difference in complexity between quantum algorithms and classical algorithms, as well as explain why some quantum algorithms are significantly more efficient than the known classical ones
- explain the basic ideas as well as the mathematical principles behind quantum cryptography algorithms, and describe the most important examples of applications of quantum cryptography

## Course content

Main examples and fundamentals for classical- and quantum information, as well as mathematical and information theoretical ideas, models, and basic principles which underlie secure and reliable transmission of information and algorithms for classical and quantum cryptography.
Qubits and quantum gates described as input-output systems, as vectors and matrices and as geometrical objects and transformations in multidimensional spaces. Linear transformations of density matrices as a mathematical basis for quantum operations and as a framework for description of quantum mechanical systems.
Elementary quantum gates as quantum operations and their matrix representations and geometrical interpretations. Quantum circuits: construction from elementary quantum gates as input-output systems with control and as matrix factorizations.
The Kronecker product and its use as a tool in constructing quantum circuits.
Universal and error-correcting codes on matrix form and circuit form. Matrix norms and their use for error estimation and error minimization when constructing quantum circuits.
The most well-known quantum algorithms and their mathematical meaning: Shor's algorithm for prime integer factorization, calculation of discrete logarithms, quantum Fourier transform as an analogue of the discrete Fourier transform. Classical algorithms and quantum algorithms for computation of the quantum Fourier transform, Grover's search algorithm, the period-finding algorithm for functions and algorithms for solving the hidden subgroup problem.
The concept of mathematical complexity. Comparisons between classical algorithms and quantum algorithms with respect to complexity. Simulation of quantum circuits and quantum algorithms on classical computers.
Current and future quantum computers: updated overview of basic principles, new and future technologies for the quantum computers of today and tomorrow.

## Teaching methods

Lectures, and classes with work individually and in group.

## Examination

Project (PRO1), 4.5 credits, marks Pass (G) or Pass with distinction (VG)
Seminars (SEM1), 3 credits, marks Pass (G) or Pass with distinction (VG)

A student who has a certificate from MDH regarding a disability has the opportunity to submit a request for supportive measures during written examinations or other forms of examination, in accordance with the Rules and Regulations for Examinations at First-cycle and Second-cycle Level at Mälardalen University (2016/0601). It is the examiner who takes decisions on any supportive measures, based on what kind of certificate is issued, and in that case which measures are to be applied.

Suspicions of attempting to deceive in examinations (cheating) are reported to the Vice-Chancellor, in accordance with the Higher Education Ordinance, and are examined by the University’s Disciplinary Board. If the Disciplinary Board considers the student to be guilty of a disciplinary offence, the Board will take a decision on disciplinary action, which will be a warning or suspension.

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## Enviromental aspects

No specific environmental aspects are included in this course.

Course literature is preliminary until 3 weeks before the course starts. Literature may be valid over several terms.

Valid from: Autumn semester13

Decision date: 2013-08-19

Last update: 2013-11-07

## Books

Quantum computing explained

ISBN: 9780470096994 (cloth) LIBRIS-ID: 10934077

xviii, 332 s.