# 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 Computing

###### Classical Simulation and Resources for QC

Classical simulation algorithms for quantum circuits are clearly desirable from a practical point of view since they allow us to certify the outcomes of near-term devices. However, the fact that these algorithms usually scale exponentially in the input size is one of the reason why it is believed that quantum computers are able to outperform classical computers. Thus, understanding the difficulty in simulating them is closely related to understanding the necessary resources for quantum speed-ups.

### 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 which have a particularly easy description. Nevertheless, they have proven to be of major importance in various fields of quantum information. In quantum computing, Clifford unitaries serve a basic building blocks for quantum computers and so-called stabilizer codes are the best studied family of quantum error correction codes. The mathematics of the formalism is based on group theory and can be represented on a classical "phase space" which enables a systematic analysis of the underlying structures and simplifies explicit computations.

###### Tensor-powers and Howe Duality

Tensor powers of Clifford unitaries appear rather naturally in many areas of quantum information, notably in the construction of quantum t-designs. Extending the ideas of Schur-Weyl duality to the case of the Clifford group leads to what is known as *Howe duality* in the mathematics community. Howe duality is a one-to-one correspondence between representations of the orthogonal group and the symplectic group. In this area we're focusing both on using insights from Howe duality to understand Clifford tensor-powers better, and the converse: using insights from quantum information theory to better understand Howe duality.

### t-designs and Representation Theory

The notion of a *t*-design refers to a subset of an ensemble which approximates the ensemble well in the sense that it shows the same first *t* moments. Recently, so-called unitary *t*-designs have become an important tool in quantum information theory. These *t*-designs are subsets of the unitary group and reproduce the first *t* moments of the Haar measure. There are plenty of constructions that only depend on lower moments where drawing unitaries from a *t*-designs is sufficient.

###### Representation Theory

Although there are many known constructions of t-designs, most of these require that you control the quantum system to an unrealistic precision. An idea to overcome this difficulty is to construct t-designs using finite groups. Since this approach is closely related to representation theory, one may import this valuable set of tools.