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 has been 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, in print in Nature Partner Journal Quantum Information, arXiv:2006.06303 (2021).
  • The effect of data encoding on the expressive power of variational quantum machine learning models, M. Schuld, R. Sweke, J. J. Meyer, Physical Review A 103, 032430 (2021).
  • 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, Nature Communication 10, 5024 (2019).

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).

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.

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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