Open Quantum Neural Networks: From Fundamental Concepts to Implementations with Atoms and PhotonsCopyright: Markus Müller
Classical neural networks, originally inspired by the neural structure of the brain, have become a powerful and ubiquitous information processing paradigm in our every day’s life. Neural-network-based algorithms and software are used with impressive success for tasks as diverse as image and speech recognition, machine learning, the analysis of ‘big data’ and ‘deep learning’. Driven by the hope of combining properties such as massive parallel information processing in neural networks with advantages like the computational speedup promised by quantum computers, there have been efforts in several directions to develop quantum neural networks.
The overarching goal of this theoretical research program, which started in October 2019, is to establish and explore a conceptual framework for quantum neural networks, based on concepts from open many-body quantum systems, and to identify quantum optical physical building blocks for the practical implementation of such a quantum processor paradigm. Thereby, the project aims at laying the foundation for quantum neuromorphic engineering of quantum neural hardware in state-of-the-art and newly emerging experimental systems, including trapped ions, Rydberg atoms and hybrid platforms.
In a recent work  we proposed a new framework to understand how quantum effects may impact on the dynamics of neural networks. In particular, we developed an open-system quantum generalisation of the celebrated Hopfield neural network, a simple toy model of associative memory, which allowed us to treat thermal and quantum coherent effects on the same footing.
 Open quantum generalisation of Hopfield neural networks, P. Rotondo, M. Marcuzzi, J. P. Garrahan, I. Lesanovsky, and M. Müller, Journal of Physics A: Mathematical and Theoretical 51, 115301 (2018).
This research is supported by ERC Starting Grant QNets.
Advanced quantum computing with trapped ions 'AQTION'Copyright: Markus Müller
Ion-trap quantum information processors are one of the leading platforms to realise scalable quantum computers. Here, quantum information is encoded in meta-stable electronic states of electromagnetically trapped ions, which can be manipulated with high precision using laser fields to implement single- and multi-qubit gate operations. The overall goal of this project is to develop and exploit a robust, compact ion-trap quantum computing demonstrator with up to 50 qubits, based on scalable quantum hardware and industry standards. Besides our theory group at Aachen and Jülich, the project involves leading academic experimental and theoretical teams (Innsbruck University, Mainz University, ETH Zurich, Swansea University, Oxford University) who join forces with commercial leaders in the field of laser technology (TOPTICA Photonics AG), optical design (Fraunhofer Gesellschaft, IOF), and computer architectures (Bull SAS) to realize the first ion-trap quantum computer that will be able to tackle problems beyond the capabilities of classical computers. Specifically, on the theory side, we are particularly interested in developing realistic protocols for scalable topological quantum error correction and fault-tolerant quantum computing, and in investigating quantum algorithms which might enable the demonstration of a quantum advantage over their classical counterparts in medium-scale quantum processors.
Encoded Qubit Alive (eQual)Copyright: Markus Müller
Quantum processors are notoriously vulnerable to perturbations, with the resulting decoherence and noise preventing to date the construction of large-scale fault-tolerant quantum computers. Today, one of the most realistic and promising approaches to overcome this problem are topological quantum error correcting codes, where quantum information is stored in interacting 2D or 3D many-body quantum systems. This approach offers the highest known error thresholds, which are already today within reach of the experimental accuracy in state-of-the-art setups.
One of the outstanding challenges in quantum error correction is the realisation of logical encoded qubits, which outperform their physical constituent qubits. In this project, we develop and theoretically explore new extensible and fault-tolerant schemes to achieve such robust logical qubits [1,2], with a focus on state-of-the-art experimental segmented and multi-species ion-trap architectures. Our work on this project is realised in an international collaboration with leading experimental and theoretical groups in Innsbruck, Zurich, Mainz, Sydney, and Oxford. Here, we are interested in exploring realistic routes towards reliable operation and coupling of logical qubits. Furthermore, we are working on the microscopic modelling of experimental building blocks and the development and on the development of efficient numerical techniques to simulate logical qubits of increasing sizes. The latter tools allow one to quantitatively study the expected performance of quantum error correction protocols under realistic noise models, including e.g. quantum gate imperfections, qubit leakage and loss, and crosstalk errors, and thereby guide experimental efforts towards reliable quantum processors based on trapped ions and other physical platforms.
 Assessing the progress of trapped-ion processors towards fault-tolerant quantum computation, A. Bermudez, X. Xu, R. Nigmatullin, J. O'Gorman, V. Negnevitsky, P. Schindler, T. Monz, U. G. Poschinger, C. Hempel, J. Home, F. Schmidt-Kaler, M. Biercuk, R. Blatt, S. Benjamin, M. Müller, Physical Review X 7, 041061 (2017).
 Transversality and lattice surgery: exploring realistic routes towards coupled logical qubits with trapped-ion quantum processors, M. Gutiérrez, M. Müller, and A. Bermudez, Phys. Rev. A. 99, 022330 (2019).
This research is done in collaboration with Encoded Qubit Alive (eQual).
Certified Topological Quantum Computation (CETO)Copyright: Markus Müller
Studying the fundamental physics of many-particle quantum systems, as well as the practical implementation of engineered quantum systems such as quantum computers and simulators, both require the ability to accurately characterise the underlying quantum states and processes. However, even for quantum systems of very moderate sizes of only a few qubits, the required resources (time, number of measurements etc.) of standard methods to characterise quantum processors, such as quantum state and process tomography, grow exponentially so that these methods become practically inefficient.
The CETO collaboration, which involves besides our theory team the experimental ion-trap group at Innsbruck (Austria) and theory groups at Sydney (Australia), Waterloo (Canada) and Madrid (Spain), aims developing and implementing new tools to characterise and validate physical and logical gate operations. Here, we are developing techniques for the efficient detection, characterization and quantification of quantum correlations , as well as of noise and imperfections, such as spatial and temporal noise correlations  or leakage or loss of qubits. Complementary, we are interested in understanding and finding new ways to mitigate the impact of such imperfections , e.g. on quantum information and error correction protocols .
 Estimating localizable entanglement from witnesses, D. Amaro, M. Müller, A. K. Pal, New Journal of Physics 20, 063017 (2018).
 Experimental quantification of spatial correlations in quantum dynamics, L. Postler, A. Rivas, P. Schindler, A. Erhard, R. Stricker, D. Nigg, T. Monz, R. Blatt, M. Müller, Quantum 2, 90 (2018).
 Iterative Phase Optimisation of Elementary Quantum Error Correcting Codes, M. Müller, A. Rivas, E. A. Martínez, D. Nigg, P. Schindler, T. Monz, R. Blatt, M. A. Martin-Delgado, Physical Review X 6, 031030 (2016).
 Twins Percolation for Qubit Losses in Topological Color Codes, D. Vodola, D. Amaro, M.A. Martin-Delgado, M. Müller, Phys. Rev. Lett. 121, 060501 (2018).
This research is done in collaboration with Certified Topological Quantum Computation (CETO) .
Quantum Simulation of Complex and Open Many-Body Quantum SystemsCopyright: Markus Müller
Simulating the general dynamics of interacting many-body systems can be a notoriously hard problem, even for the most powerful classical supercomputers. The idea of quantum simulators is to map the dynamics of complex, interacting many-particle quantum systems of interest onto other, controlled quantum devices, to study their properties and time evolution in a more accessible and controlled way. This provides a promising route to study otherwise intractable interacting many-body systems from various fields ranging from condensed matter, quantum chemistry, to high-energy physics and potentially even biological systems where quantum effects might play a role. Furthermore, an open-system quantum simulator  enables the simulation of dynamics in open quantum systems, by engineering the coupling to a tailored environment. This opens new possibilities for the study of the competition of coherent and dissipative dynamics in driven, open many-body systems . Recently, along this line of research we have been interested in developing new schemes for quantum simulation of topological many-body phases such as lattice gauge theories  and of models of interest in high-energy physics , using e.g. lattices of interacting Rydberg atoms  and trapped ions .
 An open-system quantum simulator with trapped ions, J. T. Barreiro*, M. Müller*, P. Schindler, D. Nigg, T. Monz, M. Chwalla, M. Hennrich, C. F. Roos, P. Zoller, R. Blatt, Nature 470, 486, (2011).
 Quantum Simulation of Dynamical Maps with Trapped Ions, P. Schindler*, M. Müller*, D. Nigg, J. T. Barreiro, E. A. Martinez, M. Hennrich, T. Monz, S. Diehl, P. Zoller, R. Blatt, Nature Physics 9, 361 (2013).
 Symmetry-protected topological phases in lattice gauge theories: topological QED2, G. Magnifico, D. Vodola, E. Ercolessi, S. P. Kumar, M. Müller, A. Bermudez, Phys. Rev. D 99, 014503 (2019).
 Quantum sensors for the generating functional of interacting quantum field theories, A. Bermudez, G. Aarts, M. Müller, Physical Review X 7, 041012 (2017).
 Rydberg-atom quantum simulation and Chern-number characterization of a topological Mott insulator, A. Dauphin, M. Müller and M. A. Martin-Delgado, Phys. Rev. A 86, 053618 (2012).
 Engineered Open Systems and Quantum Simulations with Atoms and Ions, M. Müller*, S. Diehl*, G. Pupillo, P. Zoller, Advances in Atomic, Molecular, and Optical Physics 61, 1-80 (2012).