EF1 – Extracting Dynamical Laws from Complex Data



Quantum Machine Learning

Project Heads

Jens Eisert, Klaus-Robert Müller

Project Members

Jens Eisert (FU), Frederik Wilde (FU), Klaus-Robert Müller (TU)

Project Duration

01.01.2019 – 31.12.2021

Located at

FU Berlin


One of the core tools used and developed in MATH+ is that of machine learning. This project suggests a concerted research program in a highly promising and novel kind of machine learning, that of quantum machine learning, in several flavors. Emphasis is on mathematical and conceptual method development, coordinated and in collaboration with other machine learning efforts in MATH+, taking a rigorous perspective. Results along these line of thought are improved quantum stochastic gradient methods with full recovery guarantees. However, a range of applications, ranging from communication technology to condensed-matter physics, will be explored as well.

Project Webpages

qradient – An open source package in Python which allows the efficient computation of gradients of parametrized quantum circuits by the parameter shift rule. This was used for the numerical simulations in our paper “Stochastic gradient descent for hybrid quantum-classical optimization”.


Selected Publications

  • Stochastic gradient descent for hybrid quantum-classical optimization, R. Sweke, F. Wilde, J. Meyer, M. Schuld, P. K. Fährmann, B. Meynard-Piganeau, J. Eisert, Quantum 4, 314 (2020).
  • Expressive power of tensor-network factorizations for probabilistic modeling, with applications from hidden Markov models to quantum machine learning, I. Glasser, R. Sweke, N. Pancotti, J. Eisert, J. I. Cirac, Advances in Neural Information Processing Systems 32, Proceedings of the NeurIPS 2019 Conference (2019).
  • Tensor network approaches for learning non-linear dynamical laws, A. Goeßmann, M. Götte, I. Roth, R. Sweke, G. Kutyniok, J. Eisert, arXiv:2002.12388 (2020), Proceedings of the NeurIPS 2020 Conference (2020).
  • Quantum certification and benchmarking, J. Eisert, D. Hangleiter, N. Walk, I. Roth, D. Markham, R. Parekh, U. Chabaud, E. Kashefi, arXiv:1910.06343, Nature Reviews Phys. 2, 382-390 (2020).
  • A variational toolbox for quantum multi-parameter estimation, J. Jakob Meyer, J. Borregaard, J. Eisert, arXiv:2006.06303 (2020).
  • Unifying machine learning and quantum chemistry – a deep neural network for molecular wavefunctions, K. T. Schütt, M. Gastegger, A. Tkatchenko, K. -R. Müller, R. J. Maurer, arXiv:1906.10033.

Selected Pictures

The three realms of quantum machine learning are classical data processed with quantum algorithms (CQ), classical models applied to quantum data (QC), and lastly quan- tum algorithms on quantum data (QQ).

The three realms of quantum machine learning are classical data processed with quantum algorithms (CQ), classical models applied to quantum data (QC), and lastly quantum algorithms on quantum data (QQ).

Selected Pictures

This work shows how stochastic gradients based on single-shot measurements can be transferred to the quantum regime to improve variational quantum algorithms and notions of quantum-enhanced machine learning, equipped with fully rigorous recovery guarantees.

This work shows how stochastic gradients based on single-shot measurements can be transferred to the quantum regime to improve variational quantum algorithms and notions of quantum-enhanced machine learning, equipped with fully rigorous recovery guarantees. This image shows the reduction in energy as more and more gradient-based optimization steps are performed. Scaled by the resource requirements per gradient step (lower panel) it becomes clear that fewer measurements (or shots) can accelerate this process, despite the increased stochasticity.

Selected Pictures

This picture shows various tensor networks in probabilistic modelling and in quantum-assisted machine learning that have been rigorously studied in their expressive power.

Work done in this project clarifies the precise expressive power of tensor networks – as they originate from the context of the description of quantum systems – in probabilistic modelling. The surprise is that seemingly similar tensor network structures can have unbounded separations in their expressive power to capture probability distributions in the system size.

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Mathematics in the Pandemic – Assistance in the Crisis by Berlin Mathematics

Berlin Map: Outbreak Day 17 – Interactive Visualization of Covid-19 Virus Spread Research 


How modeling and simulations by two Berlin research groups help to predict the future and support decision-makers. The approach is the same; the difference lies in the dimensions.


While the research group led by Edda Klipp from Humboldt-Universität zu Berlin focuses on a small town with complex models, the teams led by Kai Nagel from Technische Universität Berlin, and Christof Schütte and Tim Conrad from the Zuse Institute and Freie Universität Berlin refer to the big city of Berlin and mobile phone evaluations.


What happens if we all stay at home, if we just go to work, if the schools and kindergartens reopen? These are all questions that have been on our minds for a year and the effects of which we can also follow daily in the media. These questions have been incorporated into the mathematical calculations of two major Berlin research groups. The calculation models used are based on the same method: agent-based modeling. Agents are virtual avatars that move and perform activities in a computer simulation based on given rules. As Björn Goldenbogen of the Klipp Group points out, “Only an agent-based model can make realistic statements because it can represent infection dynamics via human-to-human interaction.”


Berlin is known worldwide where applied mathematics is highly concerned. This is also underlined by the Cluster of Excellence MATH+, in which all of Berlin’s mathematics is involved. Most of the scientists in the two research groups are MATH+ members.


Read more (English)

Read more (Deutsch)

16 Apr – Maria Chudnovsky: Induced subgraphs and tree decompositions (Kovalevskaya Colloquium)

Tree decompositions are a powerful tool in structural graph theory; they are traditionally used in the context of forbidden graph minors. Connecting tree decompositions and forbidden induced subgraphs has until recently remained out of reach. Tree decompositions are closely related to the existence of “laminar collections of separations” in a graph, which roughly means that the separations in the collection “cooperate” with each other, and the pieces that are obtained when the graph is simultaneously decomposed by all the separations in the collection “line up” to form a tree structure. Such collections of separations come up naturally in the context of forbidden minors.


In the case of families where induced subgraphs are excluded, while there are often natural separations, they are usually very far from forming a laminar collection. In what follows, Chudnovsky will mostly focus on families of graphs of bounded degree. It turns out that due to the bound on the degree, these collections of natural separations can be partitioned into a bounded number of laminar collections. This in turn results in a wide variety of structural and algorithmic results, which will be surveyed in this talk.


Maria Chudnovsky received her PhD from Princeton University in 2003. Currently, she is a professor at Princeton. Before returning to Princeton in 2015, she was a Veblen Research Instructor at Princeton University and the IAS, an assistant professor at Princeton, a Clay Mathematics Institute research fellow, and a Liu Family Professor of IEOR at Columbia University. Her research interests are in graph theory and combinatorics. She was also named one of the “brilliant ten” young scientists by the Popular Science magazine. In 2012, Chudnovsky received the MacArthur Foundation Fellowship. In 2014, she was an invited speaker at the International Congress of Mathematicians.

Download the poster here

MATH+ Project “Schule@DecisionTheatreLab” Wins BUA-Funding for Innovative Science Communication Format

Decision Theatre - © Tabara
Eröffnung des Schülerlabors "MathExperience" (2008) -   © Kay Herschelmann
Eröffnung des Schülerlabors "MathExperience" (2008) - © Kay Herschelmann

With its call for proposals on “Experimental Science Communication Laboratories”, the Berlin University Alliance (BUA) aimed at the development and practical implementation of innovative formats for science communication. The MATH+ project “Schule@DecisionTheatreLab” was selected and will receive funding for three years.


The successful project proposal Schule@DecisionTheatreLab, headed by Sarah Wolf (FU Berlin), was developed by a consortium of six members from the three Berlin universities and is based on the collaborative research of the Cluster of Excellence MATH+. One aim is to highlight the relevance of mathematics and modeling concerning many societal questions and problems.


Schule@DecisionTheatreLab combines two science communication formats: the School Lab Workshops, i.e., lectures and workshops at schools to teach students about the multiple real-world implications of mathematics; on the other hand, the Decision Theatre, a discussion format that visualizes potential impacts of alternative actions on screens, based on mathematical modeling, in support of discussions about societal challenges, such as Covid-19 or sustainable mobility.


Both communication formats will also be investigated from a mathematics-didactics and social science perspective and thus continuously improved.


Read more about the concept of the Schule@DecisionTheatreLab (in English or in German)


SIGEST-Award by SIAM for Team Around MATH+ Member Michael Joswig

Set of feasible solutions of a linear program (left), a logarithmic deformation (center), and the tropical limit (right). © M. Joswig/T. Brysiewicz (MPI Mathematics in the Sciences)

Michael Joswig, professor at TU Berlin, member of the Berlin Mathematics Research Center MATH+ and group leader at the Max Planck Institute for Mathematics in the Sciences, was awarded the prestigious SIGEST award by the Society of Industrial and Applied Mathematics (SIAM) for the paper “Log-barrier interior point methods are not strongly polynomial”, co-authored with Xavier Allamigeon, Pascal Benchimol and Stéphane Gaubert from École Polytechnique. The paper, which deals with a special problem for solving linear programs, is considered one of the most outstanding recent articles in the SIAM journals. 


The paper by Michael Joswig and his colleagues contributes significantly to investigating the ninth problem on the so-called Smale list. In 2000, Fields Medalist Steven Smale compiled a list of 18 mathematical problems that he believed were groundbreaking for the development of mathematics in the 21st century. The problem number 9 is about how quickly linear programs can be solved exactly. The award-winning article has now appeared in an expanded version in the SIGEST section of SIAM Review, for which one outstanding paper is selected each quarter, under the title: “What Tropical Geometry Tells Us about the Complexity of Linear Programming”.


In Berlin, as part of the Cluster of Excellence MATH+, Michael Joswig is investigating whether the tropical methods developed in the awarded paper can also contribute to the optimization of auction processes (project AA3-5 Tropical Mechanism Design with Max Klimm). Berlin mathematics has decades of great expertise in geometric methods in linear optimization (Martin Grötschel, Günter M. Ziegler). At the MPI for Mathematics in the Sciences in Leipzig, Joswig is currently focusing on the development of software for mathematical research, tropical geometry included.


Read the joint press release of MPI MiS and MATH+ in English

MPI MiS and MATH+ joint press release in German


Publication in SIAM Review 63 / 1 

Introduction to the article by the authors


DER TAGESSPIEGEL Portraits MATH+ Junior Research Group Leader Sarah Wolf from FU Berlin

© Jan-Hendrik Niemann

The Berlin newspaper DER TAGESSPIEGEL introduced Sarah Wolf who contributes to the Green Growth research process since 2012. She is part of the Biocomputing Group of Freie Universität Berlin and head of the MATH+ junior research group on “Mathematics for Sustainability Transitions” since 2019.


The little calculators are very popular on the Internet: You push a little on the controls and see right away how giving up meat or switching to public transportation would improve your individual climate footprint.


The models that Sarah Wolf, mathematician and Head of a Junior Research Group, is working on at Freie Universität Berlin function in a very similar way. They also simulate what effects certain decisions or measures would have on the economy, society, and the environment.


The only difference is that these so-called agent-based models are much larger and more complex, as Sarah Wolf explains: “For example, we represent all people in Germany as a ‘synthetic population’. Thus, it is not just about the individual climate balance, but also about interactions between the decisions of many actors.”


Read the complete TAGESSPIEGEL article (in German only)

2 Mar – Two Mathematical Talks at the BMS Days 2021!

On 1 and 2 March 2021, the Berlin Mathematical School will host its first online BMS Days, the annual open door event for BMS applicants and new students. On Tuesday, 2 March 2021 all events will be open to the MATH+ community. We are pleased to present the following two talks by two new colleagues and invite you to attend:


Gabriele Steidl (TU Berlin) – Motion and Deformation in Mathematical Imaging (11:00 am)


Dynamical imaging – the treatment of videos and multimodal images – leads to questions in optical flow, optimal transport and image metamorphosis. The talk will introduce the mathematical foundations of these techniques and their applications. For example, the concept of metamorphosis particularly endows the space of images with a nonlinear Riemannian structure, which one can use in applications such as diffeomorphism estimation by minimizing the path energies of corresponding geodesics.


Gabriele Steidl is a professor of Applied Mathematics at TU Berlin. Her research interests include applied and computational harmonic analysis, convex analysis and optimization with applications in image processing.



Gaëtan Borot (HU Berlin) – Matrix models and counting surfaces: from combinatorics to geometry (5:00 pm)


The talk will review the relation between matrix integrals and the enumeration of discrete surfaces, which was the starting point for the discovery
(in the 90s) of recursive structures solving these enumeration problem by constructing complicated surfaces from simpler ones. These ideas
of “topological recursion” now extend far beyond the realm of matrix models, apply to various problems in physics and in mathematics, and provides rigorous bridges between seemingly unrelated topics.


Gaëtan Borot is a professor of Mathematical Physics at HU Berlin. His scientific research focuses on mathematical aspects of QFT and strings


Download the poster here

Testing Potential COVID-19 Drugs by Research Team Including BMS Alumnus and MATH+ Member

Using an approach based on cloud computing, the team tested billions of molecules to see how effectively they inhibit the SARS-CoV-2 virus.


“Using the VirtualFlow software which we developed a good year ago now, we have succeeded in conducting in the shortest time the world’s largest virtual screening program of substances with a potential impact on SARS-CoV-2,” reports Dr. Christoph Gorgulla, postdoc at Harvard Medical School and alumnus of the Berlin Mathematical School (BMS). The results of this screening have now been published in the iScience open access journal.


Among the researchers involved in developing the VirtualFlow software was also Dr. Konstantin Fackeldey, associate professor at TU Berlin and co-author of the publication.


“VirtualFlow enables us to simulate the binding strength and binding affinity of specific substances with each other. In this screening, we examined the binding affinity of over one billion potential substances at 35 different binding sites of 15 SARS-CoV-2 virus proteins. These substances, also known as ligands, are mainly taken from the database of a company that produces commercial chemical molecules for the pharmaceutical industry. However, the databases of dozens of other companies were also screened.


Read the complete press release of TU Berlin

Read the publication

Jobs @ MATH+

Hier muss nichts geändert werden, dieser Post ist nur ein Platzhalter. Neue Jobangebote werden unter “Pages”->”Jobs@MATH+” eingetragen. Damit neue Jobangebote als News auf der Startseite auftauchen, muss hier das “Published on” Datum auf den aktuellen Tag gesetzt werden und gegebenenfalls das Excerpt und Featured Image angepasst werden (die beiden Sachen sieht man auf der Startseite).





The Berlin Mathematics Research Center MATH+ invites applications for positions in the new projects in the Application Areas, Emerging Fields and Transfer Unit starting in 2021. Details for the individual projects can be found under the links below.



Research position (m/f/d) @ Weierstrass Institute Berlin (WIAS)
75% until to March 2024, in the new MATH+ project “Recovery of battery ageing dynamics with multiple timescales”
Application Area 4: Energy and Markets

Application Deadline: 14 February 2021







Interview with Martin Skutella, MATH+ Chair: Talking MATH+


In an interview with MATH+, Martin Skutella, MATH+ Chair from 2019 until the end of 2020, reviews the first two years of the Cluster of Excellence. He talked about the history behind the foundation, the challenges, and the goals MATH+ wants to pursue.


“We want to do mathematical research, and we want to develop mathematics that will turn out to be important for our world in the future. We want to bring mathematics in contact with many other disciplines; we want to use mathematics to solve real-world problems and make progress on important questions of our time. However, we cannot do this as mathematicians alone; we have to cooperate with people from other disciplines, with people from industry, and politics. This is one of the big goals of MATH+; to have a relevant impact in the real world.”


The interviews were conducted online on two days in German and English as MATH+ podcasts, which are in progress.


Read more

Interview with MATH+ Hanna Neumann Fellows: Marta Panizzut and Nicole Mücke

Students, Doctorates: https://www.destatis.de/DE/Themen/Gesellschaft-Umwelt/Bildung-Forschung-Kultur/Hochschulen/
Profs: https://www.mathematik.de/images/HochschuleBeruf/dmv-mathematikprofessorinnen-professoren-download.pdf



On 11 February, the International Day for Women and Girls in Science, we are delighted to announce that Marta Panizzut and Nicole Mücke were awarded the MATH+ Hanna Neumann Fellowship 2021.


Named after the outstanding Berlin-born mathematician Hanna Neumann (1914-1971), who made key contributions in group theory, two MATH+ Hanna Neumann Fellowships are advertised and awarded annually to female postdoctoral researchers in recognition of outstanding work.


In an interview, Marta Panizzut and Nicole Mücke talked about their career paths and how to combine family and career. They described their role models and passion for mathematics.