## Final Report Summary - ATOMICGAUGESIMULATOR (Classical and Atomic Quantum Simulation of Gauge Theories in Particle and Condensed Matter Physics)

The ERC advanced grant project "AtomicGaugeSimulator" aims at the simulation of quantum systems with many strongly coupled degrees of freedom, both from condensed matter and from particle physics. The efficient simulation of such systems on classical computers is often prevented by so-called sign problems, which reflect the entanglement of the quantum degrees of freedom.

In the framework of the project, novel classical simulation methods have been developed that allow us to overcome the sign problem for a number of quantum systems of physical interest. In this way CP(N-1) models, which mimic several aspects of strongly interacting quarks and gluons in particle physics, have been simulated at non-zero particle density. This has led to the identification of a new ferromagnetic Bose-Einstein condensed phase in this system. New efficient classical simulation methods have also been developed for quantum link and quantum dimer models. Quantum dimer models play an important role in condensed matter physics in the context of high-temperature superconductivity. In our project, we were able to correctly identify its phase diagram. Newly developed cluster algorithms for quantum link models, an alternative formulation of gauge theories in particle physics, led to the discovery of new exotic "crystalline" confined phases with fractionalized strings connecting charge-anti-charge pairs.

Another important class of problems concerns the real-time evolution of large strongly correlated quantum systems, for which no efficient classical simulation methods exist. We have developed new classical simulation techniques for specific open quantum systems, whose dynamics is driven by dissipative processes. In this way, we have been able to simulate the dissipative cooling process that leads into an ultra-cold Bose-Einstein condensate, which in realized on a daily basis in cold-atom experiments world-wide.

In general, classical computers are not sufficiently powerful to simulate the real-time evolution of large strongly interacting quantum systems, such as quarks and gluons in particle physics or strongly correlated electrons in condensed matter physics. This provides a strong motivation to develop quantum simulation methods. A quantum simulator is a well-controlled quantum system that mimics another quantum system that one would like to understand. It thus acts as a special-purpose (digital or analog) quantum computer. E.g. ultra-cold atoms in an optical lattice have been used to quantum simulate condensed matter systems, such as quantum spin systems as undoped precursors of high-temperature superconductors. Due to the quantum nature of their hardware, unlike classical computers, quantum simulators are unaffected by the sign problem. Quantum link models provide an alternative formulation of gauge theories using quantum-spin-like degrees of freedom, which makes them very promising in the context of quantum simulation of gauge theories. In collaboration with Peter Zoller's theory group in Innsbruck, we have constructed quantum simulators for Abelian quantum link models using superconducting quantum circuits, and for non-Abelian quantum link and CP(N-1) models using ultra-cold alkaline-earth atoms in optical lattices. These are ready for experimental implementation. Other types of quantum simulators for Abelian gauge theories have been developed together with Markus Oberthaler's experimental group in Heidelberg.

The development of universal quantum computers promises to revolutionize our understanding of quantum systems. Protecting this envisioned type of computer from decoherence is a major challenge. Topological quantum computers store quantum information non-locally in topological modes, thus protecting it against decoherence. Chern-Simons theories naturally provide topological excitations, so-called anyons, whose non-Abelian statistics facilitates universal quantum computation. Motivated by these ideas, we have constructed novel Chern-Simons theories that indeed have anyons and provide an extension of Kitaev's toric code. As such, they are promising for storing quantum information. It still remains to be seen whether some of these new models are even capable of universal quantum computation.

In the framework of the project, novel classical simulation methods have been developed that allow us to overcome the sign problem for a number of quantum systems of physical interest. In this way CP(N-1) models, which mimic several aspects of strongly interacting quarks and gluons in particle physics, have been simulated at non-zero particle density. This has led to the identification of a new ferromagnetic Bose-Einstein condensed phase in this system. New efficient classical simulation methods have also been developed for quantum link and quantum dimer models. Quantum dimer models play an important role in condensed matter physics in the context of high-temperature superconductivity. In our project, we were able to correctly identify its phase diagram. Newly developed cluster algorithms for quantum link models, an alternative formulation of gauge theories in particle physics, led to the discovery of new exotic "crystalline" confined phases with fractionalized strings connecting charge-anti-charge pairs.

Another important class of problems concerns the real-time evolution of large strongly correlated quantum systems, for which no efficient classical simulation methods exist. We have developed new classical simulation techniques for specific open quantum systems, whose dynamics is driven by dissipative processes. In this way, we have been able to simulate the dissipative cooling process that leads into an ultra-cold Bose-Einstein condensate, which in realized on a daily basis in cold-atom experiments world-wide.

In general, classical computers are not sufficiently powerful to simulate the real-time evolution of large strongly interacting quantum systems, such as quarks and gluons in particle physics or strongly correlated electrons in condensed matter physics. This provides a strong motivation to develop quantum simulation methods. A quantum simulator is a well-controlled quantum system that mimics another quantum system that one would like to understand. It thus acts as a special-purpose (digital or analog) quantum computer. E.g. ultra-cold atoms in an optical lattice have been used to quantum simulate condensed matter systems, such as quantum spin systems as undoped precursors of high-temperature superconductors. Due to the quantum nature of their hardware, unlike classical computers, quantum simulators are unaffected by the sign problem. Quantum link models provide an alternative formulation of gauge theories using quantum-spin-like degrees of freedom, which makes them very promising in the context of quantum simulation of gauge theories. In collaboration with Peter Zoller's theory group in Innsbruck, we have constructed quantum simulators for Abelian quantum link models using superconducting quantum circuits, and for non-Abelian quantum link and CP(N-1) models using ultra-cold alkaline-earth atoms in optical lattices. These are ready for experimental implementation. Other types of quantum simulators for Abelian gauge theories have been developed together with Markus Oberthaler's experimental group in Heidelberg.

The development of universal quantum computers promises to revolutionize our understanding of quantum systems. Protecting this envisioned type of computer from decoherence is a major challenge. Topological quantum computers store quantum information non-locally in topological modes, thus protecting it against decoherence. Chern-Simons theories naturally provide topological excitations, so-called anyons, whose non-Abelian statistics facilitates universal quantum computation. Motivated by these ideas, we have constructed novel Chern-Simons theories that indeed have anyons and provide an extension of Kitaev's toric code. As such, they are promising for storing quantum information. It still remains to be seen whether some of these new models are even capable of universal quantum computation.