## Course: “Mathematical Physics I – compact groups and their representations”

**Lecturer (2h per week): **

Adam Sawicki (CTP PAS)

**Tutorial (1h per week): **

Oskar Słowik (CTP PAS)

**Description:**

Groups are key ingredients in many modern developments in quantum physics and quantum information theory. In particular compact semisimple Lie groups play an important role in quantum computing and entanglement theory. The course is a comprehensive introduction to Lie groups theory and representation theory.

**During the course we will focus on the following subjects:**

- Structure of classical Lie groups
- Highest-Weight theory
- Schur-Weyl duality
- Young diagrams
- Peter Weyl theorem
- Tensor products of representations
- Character theory
- Gelfand Tsetlin patterns

**In order to show how to use the above concepts in practise the lecture will be accompanied by the problem classes.**

**Lectures**: every Thursday at 1pm

**Tutorial**: every Friday at 12 pm

If you want to join the course email: a.sawicki@cft.edu.pl

## Course: “Modern trends in quantum computing”

**Lecturer (2h per week): **

Adam Sawicki (CTP PAS)

Michał Oszmaniec (CTP PAS)

**Description:**

The course is a comprehensive and fast introduction to quantum computing

**During the course we will focus on the following subjects:**

- Definition of Turing Machines, Computable Functions
- Why the Halting problem cannot be solved by Turing machines?
- Time and space complexity. Time hierarchy theorem (main idea)
- Complexity classes: P, NP, PSPACE, EXP (with examples)
- Polynomial reductions. NP-hardness vs NP-completeness
- Boolean circuits. Universal classical gates. Relation to Turing Machines (basic idea)
- Definition and explanation of complexity classes BPP
- Power of reversible classical computation
- Comparison of classical randomized and computation classical computation. Definition of BQP
- Definition of universality + examples of universal sets
- Clifford group and its generators
- Proof of universality of Clifford + T for one qubit
- Cartan decomposition of semisimple real Lie algebra
- Cartan decomposition of SU(4) and implementation of 2-qbit gates
- Iterative KAK decomposition of SU(2^n)
- CNOT-gate count in implementation of quantum gates using KAK
- Givens rotations and implementation of N-mode beamsplitters with 2-modes beamsplitters
- Shrinking lemma of the Solovay-Kiatev theorem
- Iterative procedure of the Solovay-Kiatev and formula for circuit depth
- Time efficiency of the Solovay-Kiatev algorithm
- Fidelity of a channel and entanglement fidelity
- 2-designs and their applications for assessment of quantum gates fidelity
- Formula for twirl wrt to Haar measure of any superoperator
- Proof that Clifford group is a 2-design
- Relationship between twirl operation for Pauli and Clifford groups
- Symplectic inner product for n qubit Pauli operators
- Connection between t-designs and epsilon-nets
- Description of a basic version of a randomized benchmarking protocol. The role of 2-design property in this protocol.
- Discrimination of unknown quantum state via a single measurement
- Vectorisation and Choi-Jamiołkowski isomorphism in the context of unitary designs
- Convergence of random quantum circuits to t-designs in terms of moment operators
- Local random quantum circuits and spectral gaps of local hamiltonians
- Anticoncentration in approximate unitary 2-designs

**Lectures**: every Tuesday at 11am

If you want to join the course email: a.sawicki@cft.edu.pl