## MTAT.05.008 Fou Math

## Mathematical Foundations for Computer Science

### Role of the course

Students who want to specialize in Theoretical Computer Science occasionally lack some of the math background. That is particularly true for those who want to specialize in Quantum Computer Science (quantum computing, quantum crypto), where advanced linear algebra is needed. This course is meant to fill in math background for 1st semester MSc students who want to specialize in Quantum Computing, Quantum Cryptography or Theoretical Computer Science.

The biggest hurdle here is Linear Algebra: Outside of quantum computer science, here seems to be no other reason for a computer science student to know, e.g., complex numbers or even spectral theory (crazy, right?), so CS students' background there is often too weak for quantum stuff. Even math undergrads have learned less about tensor products of vector spaces than what is needed for quantum computer science (but, of course, they can catch on quickly).

Fou Math (or undergrad degree in math) is compulsory requirement for Quantum Algorithms (Qualms MTAT.05.118), all quantum error correction courses, and all "quantum physics for quantum computer scientists" (Q𝚽4QCS) courses (currently QIP *Light*).

Apart from Linear Algebra, the course also fills in gaps in modular arithmetic and probability.

##### Quantum Computing

For students who want to specialize in quantum computing, it is ** highly** recommended to also take the course

➜

*LTFY.04.012 Fundamentals of Quantum Computing (FunQ)*

at the Physics Institute this fall. There, the focus is on programming quantum computers (e.g., IBM Quantum Experience), and all required quantum mechanics background will be taught.

### Syllabus

This is the fourth run of the course. Again, we have learned from the feedback of the students, and made changes. The choice of content is now more focused, the homework load has been reduced. With the possibility to point interested students to future Q4𝚽QCS courses, in Fou Math, quantum mechanics appears only where it motivates the math.

The content consist of three fields:

**I. Linear Algebra**

- Complex numbers
- Exponential function, and more complex numbers
- Real and complex Hilbert spaces, ONBs, tensor products
- Operator concept, adjoint, matrices & operators, basis change
- Commuting operators, spectral theorem and applications
- 🆕 Basics of infinite dimensional Hilbert space

**II. Probability****III. Modular arithmetic**

The coverage of II and III depends on the extent in which gaps in knowledge exist: Probability covering what's in the "MathWiki" and modular arithmetic up to the Chinese Remainder Theorem are standard content of standard compulsory 1st year undergrad CS courses universally (e.g., chapters 4 & 7 in "The Rosen").

### Organization

PARTICIPATION

- All sessions will be streamed online, presence in the lecture hall is not required
- Everybody who is present in the lecture hall must wear a mask (except when speaking)

SPACETIME

- Class times: Mon 12:15-14:00, Thu 10:15-12:00
- Online class meetings: We will share a video conference link by email
- First class meeting: Thu, Sep 2

ONLINE

- Slides and homework assignments: github.com/dojt/Fou-Math-2021 (only for course participants)
- When you have a question:

### Instructor team

- Assoc. Prof. Dirk Oliver Theis
- Evgenii Dolzhkov