## Representation Theory in Quantum Information Science

### PhD School at the Faculty of SCIENCE at University of Copenhagen

**Aim and content**

Quantum information theory has seen much development over the past 30 years, and is becoming ever more important with the upcoming access to intermediate-scale quantum hardware.

Representation theory is a very successful branch of mathematics that has been extensively studied over the years. It finds many applications in quantum theory, where physical operations correspond linear transformations and thus has naturally played a very influential role in quantum mechanics.

Recently, interest in representation theory has skyrocketed, because it is an important tool to elegantly describe various tasks at the intersection of many-body quantum physics, quantum information, and quantum computing. Key examples are tasks involving independent copies of the same state (e.g., state tomography), the study of averaged properties of random states and/or circuits, which appear in a multitude of settings from black hole physics to cryptography, and randomized benchmarking of quantum devices, which is of key practical interest.

The aim of this course is to present these developments in a coherent way, identifying the key ingredients from the representation theory of the unitary and Clifford groups, as well as showcasing some of the many interesting applications in quantum information theory.

The topic of the school is highly relevant to SCIENCE, because (i) it is one of the most exciting and active fields at the intersection of mathematics, quantum information and physics, with (ii) substantial contributions by many members of QMATH, and at the same time (iii) direct relevance to (interdisziplinary) breakthroughs in the field, such as characterizing entanglement growth in black holes, non-local games, quantum complexity of (sampling from) random circuits, efficient benchmarking of quantum devices, quantum learning theory, among many others.

**Formel requirements**

Background in quantum theory, quantum information, or quantum computing.

**Learning outcome**

Knowledge:

• Obtain knowledge of representation theory of the symmetric, unitary and Clifford groups, with a focus on Schur-Weyl duality.

• Be acquainted with basic properties of the Weingarten calculus and their application in random circuits & learning theory

• Understand sample-complexity bounds for key tasks such as quantum state tomography, and deciding whether a state is a stabilizer state.

Skills:

• Be able to derive concrete asymptotic bounds on the complexity of quantum processing tasks.

• Translate a conceptual physical problem to a representation theory problem.

• Derive key results about random circuits and random states.

Competences:

• Be able to identify opportunities for applying representation-theoretic tools in the context of quantum information theory.

• Understand which sort of questions could be solved using representation theory combined with the ability to find and apply the relevant results in the literature.

• A broad overview over the topics that have been studied in representation theory in connection with quantum information science.

**Target group**

PhD students from mathematics, physics, and computer science with an interest in quantum information theory.

**Lecturers**

Daniel Stilck França: ENS de Lyon & INRIA faculty. Daniel is a pioneer in the field of benchmarking, learning, tomography, and random circuits, and will lecture on how representation theory comes into play in these cutting-edge topics.

David Gross: professor at University of Cologne. David is among the most influential researchers in quantum information theory and has contributed to many topics, and in particular those relevant to the school, such as quantum state tomography, random circuits, and multi-party entanglement theory. He will give lectures on fundamental aspects of representation theory and their applications in entanglement characterization.

Michael Walter: professor at Ruhr University Bochum. In his work he has related tensors, copies of tensors, and tensor networks to complexity theory, black hole physics, error correction, and entanglement. He will give lectures on the cutting edge of this field.

**Remarks**

Students need to write a brief paragraph detailing their motivation.

Registration for this course will open in the autumn 2024.