# Research

The group works on the interface between quantum mechanics and other areas of applied mathematics.

Our main emphasis lies on the application of rigorous mathematical methods to problems in quantum information theory and many-body theory.

Conversely, we aim to use methods originating in quantum physics to classical problems, e.g. in machine learning theory.

This page is currently under construction.

### Quantum algorithms

Under construction.

### Quantum causal structures

Under construction.

### Quantum device characterization

Under construction.

### Variational Quantum Algorithms

Currently available quantum computers are limited to a small number of qubits and flawed - or noisy - operations. Variational quantum algorithms make use of today's quantum devices in tandem with a classical computer: To encode a problem of interest, one chooses a cost function that can be implemented efficiently on the noisy device and that depends on a set of control parameters for the circuit executed in the device. The classical computer then uses the evaluations on the quantum device to minimize the cost function. General questions arise, for example: Which optimization routines are suitable for this hybrid working mode? Do they behave similar to classical optimization problems? And how does the optimization behaviour depend on the specifications of the encoding circuit and the problem of interest itself? At the same time, very concrete technical challenges are of interest, like: If a gradient-based optimization routine is used on the classical computer, how can one measure the gradient of the cost function on the quantum computer? How should one distribute measurements across the optimization epochs? How do measurement statistics for the cost function influence the hyperparameters of algorithms?

### The Stabilizer Formalism and Discrete Phase Space Methods

The stabilizer formalism deals with a finite set of highly symmetric states and unitaries---the stabilizer states and Clifford unitaries. This formalism has a rich algebraic and geometric structure that makes it useful in a variety of scenarios: quantum error correction, simulation of quantum computing, quantum device characterization, etc. Our group works on several projects whose goals are, roughly speaking, to characterize the mathematical structures arising from this formalism.

###### Tensor power Cliffords

Tensor power representations arise rather naturally in quantum information---notably, they are closely related to the construction of quantum t-designs which are useful when characterising quantum devices or when modelling chaotic quantum systems. Through a series of projects, our group has characterized tensor power representations of the Clifford group and the closely-related oscillator representation. This way, we have extended the ideas of the mathematical subjects of *Schur-Weyl duality* and the *Theta correspondence* (also known as *Howe duality*) to the Clifford group. Moreover, we have used these findings to an efficient construction of unitary t-designs whose circuits require an amount of non-Clifford gates which is independent of the number of qubits.

###### Symmetries of the stabilizer polytope

Under construction.

###### Quantum computing with magic states

Under construction.