The groups participating in the ExQM School focus on seven primary research areas between the broader field of quantum optics, numerical tensor networks methods and theoretical quantum information. The individual focus areas are:

Focus Area I: Ultracold Atoms in Optical Lattices

Ultracold Atoms in Optical Lattices have proven to be a versatile model system in which strong correlation effects in a many-body quantum system can be studied. The tunability and control over almost all underlying system parameters make them rather unique and allows for switching between different ground states of the underlying many-body quantum system. One of the most prominent effects in this respect is the unique quantum transition from a superfluid to a Mott-insulator phase that started the field of strong-correlation physics with ultracold gases.

Another major advantage of cold atoms for elucidating many-body phenomena is the possibility to change parameters characterising the relative strength of kinetic and interaction energy dynamically. It paves the way to study real-time dynamics of strongly correlated systems in a controlled way, a simple example being the quantum quench of the system from the superfluid to the Mott-insulator regime. One of the future challenges amounts to cool strongly interacting bosonic and fermionic spin mixtures below the super-exchange energy scale for observing magnetically ordered quantum phases, for which new cooling schemes have been proposed.

*Primary Working Groups and PhD Students: * Research Group Bloch: Michael Lohse

Focus Area II: Scalable Networks of Solid-State Quantum Circuits

Scalable Networks of Solid-State Quantum Circuits are becoming increasingly attractive for quantum simulations. For example, networks of nonlinear superconducting transmission line resonators or optical nanocavities can be used as scalable quantum simulators for the Bose-Hubbard Hamiltonian. The resonators are made nonlinear by a controllable coupling to superconducting or semiconductor quantum bits, thereby forming harmonic oscillators with tunable Kerr nonlinearity.

Networks of these entities would be particularly well suited for accessing the strongly correlated regime and for investigating quantum many-body dynamics of interacting particles under the influence of driving and dissipation. Solid state quantum circuits with multiple drives are another attracting system. E.g., superconducting quantum bits strongly coupled to a resonator field mode and subjected to multiple classical drives can be used for quantum simulations of relativistic quantum physics (e.g. dynamics of the Dirac equations, Klein paradox). The key advantage is the controllability of the relevant physical parameters via the strength of the longitudinal and two transverse drives. Moreover, quantum bits with two-tone multiple drives can be used for quantum simulation of strong and ultra-strong coupling dynamics.

*Primary Working Groups and PhD Students:* Research Group Gross: Michael Fischer

Focus Area IIIa: Density-Matrix Renormalisation Group Methods (DMRG)

Density-Matrix Renormalisation Group Methods (DMRG) have firmly established themselves as (currently) the most powerful method for calculating static and dynamic properties of lower-energy eigenstates of strongly correlated Hamiltonians in 1D quantum lattices as well as their time evolution far from equilibrium. For the latter, it is the only method of importance. It can be applied to analyse 2D classical and 1D quantum systems in a highly precise fashion.

The extension to 2D quantum systems is more difficult and a key challenge of current computational physics in spite of the impressive results already obtained. DMRG methods have been applied to spin systems, bosonic and fermionic systems with equal success. This has become particularly obvious in the simulation of non-equilibrium properties of 1D quantum systems in scenarios with both pure and mixed states as well as in the presence or absence of dissipation.

The Cirac group devised a new variational algorithm to find a matrix product operator (MPO) description for the steady states of dissipative one dimensional systems described by a master equation [PRL 114, 220601 (2015)], which allows immediate application to explore dissipative phase transitions. – Moreover, numerical applications of matrix product states to lattice gauge theories have produced results of unprecedented numerical accuracy, which the Cirac group extended to the real time evolution of a non-Abelian theory. This allows to include the study of the string breaking phenomenon in the presence of fully dynamical fermions, beyond the reach of the most traditional lattice methods.

*Primary Working Groups and PhD Students:* Research Group Cirac; Research Group Schollwöck: Claudius Hubig

Focus Area IIIb: Extending Matrix-Product State Approaches

DMRG can be considered a variational method over the class of Matrix Product States (MPS) with the variational criterion based on a Matrix Product Operator (MPO) representation of the Hamiltonian. Using this approach, it is possible to find the ground states and simulate time evolutions of arbitrary finite systems, as they commonly occur during the central solution steps of embedding methods such as Dynamical Mean Field Theory (DMFT) and Density Matrix Embedding Theory (DMET). With MPS-DMRG replacing exact diagonalisation methods as the central solver, it becomes possible to consider larger embedded patches; in turn leading to more accurate results including physical interactions that are impossible to model on smaller patches.

Furthermore, it is feasible to study both one- and two-dimensional finite systems, the latter with near-linear scaling of computational cost in the larger of the two dimensions. By numerically including more symmetries of the system in question, that cost can additionally be reduced further and with new parallelisation techniques, large-scale calculations become more and more viable. Lastly, as a generalisation of the MPS structure to two or more dimensions, Projected Entangled Pair States (PEPS) are a straightforward generalisation of MPS with great potential for improvements in the future.

The three approaches — embedding techniques, large-scale MPS-DMRG calculations and improvements for PEPS-based methods — each offer the possibility to study quantum states at high numerical accuracy and sufficiently-large system sizes to allow for scaling to the thermodynamic limit. Together, they combine into an arsenal of numerical methods with the potential to unravel many standing problems in condensed-matter physics like the nature of high temperature superconductivity or solving the doped Fermi-Hubbard model.

The Schollwöck group has developed very powerful impurity solvers based on matrix product states for simulations of correlated materials both in and out of equilibrium in combination with the dynamical mean-field theory (DMFT). By a complete revision of its MPS codes (large scale parallelization and asynchronous memory management, improved convergence schemes and quantum-information driven rephrasing of the Hamiltonians) it is now also able to access very large 2D cylinders of Hubbard or frustrated Heisenberg models.

*Primary Working Groups and PhD Students:* Research Group Huckle: Moritz August; Research Group Schollwöck: Claudius Hubig

Focus Area IIIc: Entanglement Structure of Quantum Fields

Static and dynamic properties of strongly interacting quantum fields determine the behaviour of a variety of interacting many-body systems, a key example being the fractional quantum Hall effect. Contrary to conventional wisdom, recent work has shown that these systems are amenable to variational 2methods. This opens up a fertile playground for tensor network methods, the objective being the development and application of certifiable finite-dimensional approximations and simulations of quantum field theories. While critical 1D spin chains provide a natural testbed, the study of higher-dimensional systems is most relevant as it encompasses systems of experimental interest. Advances in corresponding tensor network techniques provide new quantitative insights into the nature of quantum correlations of corresponding continuous degrees of freedom. This in turn connects to modern information-theoretic and geometric approaches to strongly interacting quantum fields, including the holographic principle.

*Primary Working Groups and PhD Students: *Research Group König

Focus Area IIId: Efficient Computations with Tensors

Efficient Computations with Tensors become increasingly important in high dimensional problems. Also in computer science, finding an accurate approximation to the smallest eigenvalue of a matrix larger than one can store on a powerful computer is a challenge in its own right. While in quantum physics techniques like MPS or PEPS were developed, in mathematics, besides the Tucker decomposition and the canonical decomposition, analogous concepts like Tensor Trains (TT) were introduced, thus expressing a common interest in powerful numerical methods specifically designed for coping with high-dimensional tensor networks. — Unifying variational approaches to ground-state calculations in a common framework of tensor approximations will be highly useful, in particular in view of optimizing numerical algorithms by exploiting sparsity, symmetry and geometry of MPS-based Riemannian manifolds of quantum-state dynamics.

The field has developed very dynamically now providing links to important areas including high-dimensional big data, (quantum) machine learning, and neural networks. ExQM will extend its research activities to encompass functions with matrix and tensor arguments for numerical applications in quantum dynamics and quantum simulation. Recent developments provided new insight to study the interrelation of tensor networks and neural networks in view of possible applications in machine learning and quantum simulation.

*Primary Working Groups and PhD Students:* Research Group Huckle: Moritz August

Focus Area IIIe: Variational Wavefunctions and Tensor Networks

Variational wavefunctions form a powerful way to probe the physics of strongly interacting quantum systems, such as in the fractional quantum Hall effect. Tensor networks, and in particular Projected Entangled Pair States (PEPS), form a versatile framework for the modelling of strongly correlated quantum systems based on their entanglement structure. They allow for engineering variational families by encoding the physics locally into the tensor, which in turn has, e.g., allowed for studying the physics of topological spin liquids such as Resonating Valence Bond states with unprecedented accuracy. Future applications include the use of PEPS models to design variational families which allow to study the nature of phase transitions between different topologically ordered phases, including non-abelian ones, in great detail, and the investigation of models with chiral topological order.

*Primary Working Groups and PhD Students:* Research Group Schuch

Focus Area IV: Interfacing Light and Matter

Single photons are well suited as quantum carriers for information exchange over long distances. However, if one wants to store or process quantum information then matter, such as atoms or solids, are much better suited. Long term goals, such as a quantum computer or a quantum internet require—among other things—the development of efficient and versatile interfaces between photons and quantum matter, a vibrant research field that currently evolves at a rapid pace.

We plan to generate a single photon from a quantum dot and subsequently store it in an atomic system. The development of such technologies is highly demanding since the spectral properties must be tailored to match each other. Connecting several such systems leads to a few-body system. As another example of a few-body system, schemes for the generation of entangled states of three or more photons will be conceived and implemented. Furthermore, the interaction between Rydberg atoms in a BEC, a genuine form of many-body quantum matter, can be exploited for processing quantum information. Combination of this interaction with an atom-light interface offers a way to process photonic quantum information.

*Primary Working Groups and PhD Students:*Research Group Rempe: Stephan Welte

Focus Area V: All-Optical Quantum Simulation

New approaches to quantum simulation can be based on complex photonic wave-guide systems. Novel fabrication processes enable the design of 3D optical structures allowing the simulation of high-dimensional systems and band- structures in solids. With these fabrication processes, waveguide-based photonic quantum technology will reach a high degree of maturity and reliability to enable the realization of versatile quantum simulators and to test some basic quantum functionalities that will be at the heart of new QIC devices based on condensed-matter quantum systems.

These technologically relevant systems rely on quantum operations such as controlling electronic charge or spin degrees of freedom in coupled quantum dots, manipulating the dynamics of Josephson junctions, and using spin chains as quantum communication channels. Laser-written waveguides offer— for the first time—the potential to design 3D photonic structures, which in turn enable an entire new regime in all-optical quantum simulation.

*Primary Working Groups and PhD Students:* Research Group Weinfurter: Lukas Knips

Focus Area VI: Exploring Open Quantum Systems

The evolution of open quantum systems is described by quantum channels. As mathematical objects, they do not only describe time evolution, but also play a central role in the analysis of Matrix-Product-States, which in turn form the basis for the DMRG-type algorithms above. From the perspective of information processing or the coherent simulation of quantum systems, decoherence and dissipation were long regarded as adversarial. In recent years, however, it became more and more clear that interactions with an environment also lead to opportunities that can be exploited in an advantageous way. This point of view was pioneered by a part of our consortium, and dissipative state preparation, for instance, has now become a well-established, very successful direction of research promising further significant impact on the design of quantum memories and noise modulation as well as fixed-point and environment engineering.

*Primary Working Groups and PhD Students:* Research Group Wolf: Anna-Lena Hashagen, and Research Group Schulte-Herbrüggen

Focus Area VII: Polariton Condensation in Emergent Nanomaterials

Sub-wavelength scale semi-conductor micro-cavities can enter new regimes of strong light-matter interactions that produce optically active bosonic quasi-particles – exciton polaritons – that facilitate new regimes of lasing and even optically induced Bose–Einstein condensation (BEC). In both organic and inorganic semiconductor micro cavities, recent discoveries have shown that these polaritons differ completely from excitons by virtue of giant optical nonlinearities and polariton lasing – the spontaneous emission of coherent light by condensates of exciton-polaritons. However, such phenomena typically exist only at cryogenic temperatures, limited by the small energy splitting of polaritons below the excitons. Most recently, group-III Nitrides and organic crystals have been found to exhibit polariton BECs at room temperature, by virtue of their large exciton binding energies (>100meV). In ExQM, we will explore new materials that promise the realization of polaritonic condensates in emergent “graphene-like” layered nano-materials, such as 2D transition metal chalcogenides (MoS 2 , WS 2 and GaS) in which polaritonic effects can be expected at room temperature. By structuring the 2D films over sub 20nm length scales, interactions between polariton condensates in structured matter will be explored.

*Primary Working Groups and PhD Students:* Research Group Finley: Jakob Wierzbowski