# AtomicGaugeSimulator Report Summary

Project ID:
339220

Funded under:
FP7-IDEAS-ERC

Country:
Switzerland

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

The ERC Advanced Grant project "Classical and Atomic Quantum Simulation of Gauge Theories in Particle and Condensed Matter Physics" is concerned with the calculation of the behavior of quantum systems with a large number of strongly coupled degrees of freedom. In particle physics such systems arise in the context of Quantum Chromodynamics (QCD) - the fundamental theory of the strong interaction between quarks and gluons - as well as in other strongly coupled quantum field theories. In order to access its nonperturbative physics, QCD can be considered on a lattice that regularizes the space-time continuum. In several

cases of physical interest, lattice QCD can then be treated with numerical Monte Carlo simulation techniques. In other cases, including the real-time evolution or the finite-density behavior of quark-gluon systems, the Monte Carlo method fails due to sign problems, which make it exponentially hard to solve these systems with classical computers. In condensed matter physics it is a grand challenge to understand systems of strongly correlated electrons, such as high-temperature superconductors or some quantum antiferromagnets. Again, while Monte Carlo simulation is a very powerful tool in some cases, in other cases it again fails due to severe sign problems, thus preventing, for example, a deeper understanding of the mechanisms responsible for high-temperature superconductivity. Some strongly correlated condensed matter systems are described by quantum dimer models. These fall in the class of quantum link models, which - like QCD - are gauge theories. Indeed quantum link models provide an alternative regularization of QCD, in which gluons arise as collective excitations of discrete quantum link variables. In the quantum link formulation, QCD is thus closely related to strongly correlated condensed matter systems.

The ERC project is divided into four major tasks, each comprised of several subprojects, of which about one half have already been completed successfully. The subject of task 1 is to "address severe sign problems" by constructing new powerful numerical simulation algorithms. In the context of quantum link and quantum dimer models, after an analytic dualization procedure, we have constructed a numerical method that allowed us to conclusively settle a long-term debate about the phase structure of the square lattice quantum dimer model, which turns out to be dominated by the so-called columnar phase. In addition, we have developed new numerical methods to solve the sign problem in the real-time evolution of dissipation-driven quantum systems. In this way, we were able, for the first time, to investigate in detail the dissipative cooling process that leads into an ultracold Bose-Einstein condensate.

Task 2 is concerned with the "simulation of strongly coupled gauge theories". Here we have constructed a new method that drastically reduces the cut-off artifacts associated with a finite space-time lattice spacing in lattice QCD calculations.

Task 3 aims at the "construction of quantum simulators for gauge theories". Quantum simulators are highly controllable quantum devices that mimic the physics of other much less well controllable quantum systems. In this way, certain strongly coupled condensed matter systems, for which the sign problem could not be solved with classical computers, can be addressed, for example, with ultracold atoms in an optical lattice formed by counter-propagating laser beams. It is a grand challenge to construct quantum simulators for gauge theories in order to address those particle physics problems for which the sign problem could again not be solved using classical computers. Due to its finite-dimensional link Hilbert space, the quantum link formulation of gauge theories is ideally suited for implementation in ultra-cold matter. In an interdisciplinary collaboration with Peter Zoller's group in Innsbruck, we have constructed a quantum simulator for a simple so-called CP(2) toy model of QCD, using ultracold alkaline-earth atoms in an optical lattice. Furthermore, again in close collaboration with the Zoller group, we have constructed a quantum simulator for a 2-dimensional quantum link model using ultracold superconducting quantum circuits. Finally, we are also collaborating with the experimental and theoretical physics groups of Markus Oberthaler and Juergen Berges in Heidelberg in order to realize a lattice gauge quantum simulator in cold atom experiments.

Task 4 concerns the realization of "topological quantum computation". In contrast to special purpose quantum simulators, a universal quantum computer is capable of addressing much more general problems. Realizing a quantum computer is a grand challenge, because omnipresent decoherence endangers the entanglement in the state of the quantum computer, which carries the quantum information. A topological quantum computer aims at protecting quantum information from decoherence by storing it nonlocally in topological modes of the quantum device. The quantum information is then manipulated by braiding the world-lines of anyon particles, which is possible in the framework of Chern-Simons gauge theories. Also in this case, quantum systems with a finite-dimensional Hilbert space are best suited for implementation in ultracold matter. While it was originally proposed to realize Chern-Simons theories with quantum links, it turned out that a new type of lattice gauge theory with discrete gauge group is better suited for this purpose. In this way we have realized novel Chern-Simons lattice gauge theories, which are partly related to Kitaev's toric code - a prototype for a topological quantum information store device.

Altogether, until now 25 scientific papers have been published in the framework of this ERC project, of which only a few have been highlighted here. The second phase of the project has just started and we anticipate many other exciting new developments.

cases of physical interest, lattice QCD can then be treated with numerical Monte Carlo simulation techniques. In other cases, including the real-time evolution or the finite-density behavior of quark-gluon systems, the Monte Carlo method fails due to sign problems, which make it exponentially hard to solve these systems with classical computers. In condensed matter physics it is a grand challenge to understand systems of strongly correlated electrons, such as high-temperature superconductors or some quantum antiferromagnets. Again, while Monte Carlo simulation is a very powerful tool in some cases, in other cases it again fails due to severe sign problems, thus preventing, for example, a deeper understanding of the mechanisms responsible for high-temperature superconductivity. Some strongly correlated condensed matter systems are described by quantum dimer models. These fall in the class of quantum link models, which - like QCD - are gauge theories. Indeed quantum link models provide an alternative regularization of QCD, in which gluons arise as collective excitations of discrete quantum link variables. In the quantum link formulation, QCD is thus closely related to strongly correlated condensed matter systems.

The ERC project is divided into four major tasks, each comprised of several subprojects, of which about one half have already been completed successfully. The subject of task 1 is to "address severe sign problems" by constructing new powerful numerical simulation algorithms. In the context of quantum link and quantum dimer models, after an analytic dualization procedure, we have constructed a numerical method that allowed us to conclusively settle a long-term debate about the phase structure of the square lattice quantum dimer model, which turns out to be dominated by the so-called columnar phase. In addition, we have developed new numerical methods to solve the sign problem in the real-time evolution of dissipation-driven quantum systems. In this way, we were able, for the first time, to investigate in detail the dissipative cooling process that leads into an ultracold Bose-Einstein condensate.

Task 2 is concerned with the "simulation of strongly coupled gauge theories". Here we have constructed a new method that drastically reduces the cut-off artifacts associated with a finite space-time lattice spacing in lattice QCD calculations.

Task 3 aims at the "construction of quantum simulators for gauge theories". Quantum simulators are highly controllable quantum devices that mimic the physics of other much less well controllable quantum systems. In this way, certain strongly coupled condensed matter systems, for which the sign problem could not be solved with classical computers, can be addressed, for example, with ultracold atoms in an optical lattice formed by counter-propagating laser beams. It is a grand challenge to construct quantum simulators for gauge theories in order to address those particle physics problems for which the sign problem could again not be solved using classical computers. Due to its finite-dimensional link Hilbert space, the quantum link formulation of gauge theories is ideally suited for implementation in ultra-cold matter. In an interdisciplinary collaboration with Peter Zoller's group in Innsbruck, we have constructed a quantum simulator for a simple so-called CP(2) toy model of QCD, using ultracold alkaline-earth atoms in an optical lattice. Furthermore, again in close collaboration with the Zoller group, we have constructed a quantum simulator for a 2-dimensional quantum link model using ultracold superconducting quantum circuits. Finally, we are also collaborating with the experimental and theoretical physics groups of Markus Oberthaler and Juergen Berges in Heidelberg in order to realize a lattice gauge quantum simulator in cold atom experiments.

Task 4 concerns the realization of "topological quantum computation". In contrast to special purpose quantum simulators, a universal quantum computer is capable of addressing much more general problems. Realizing a quantum computer is a grand challenge, because omnipresent decoherence endangers the entanglement in the state of the quantum computer, which carries the quantum information. A topological quantum computer aims at protecting quantum information from decoherence by storing it nonlocally in topological modes of the quantum device. The quantum information is then manipulated by braiding the world-lines of anyon particles, which is possible in the framework of Chern-Simons gauge theories. Also in this case, quantum systems with a finite-dimensional Hilbert space are best suited for implementation in ultracold matter. While it was originally proposed to realize Chern-Simons theories with quantum links, it turned out that a new type of lattice gauge theory with discrete gauge group is better suited for this purpose. In this way we have realized novel Chern-Simons lattice gauge theories, which are partly related to Kitaev's toric code - a prototype for a topological quantum information store device.

Altogether, until now 25 scientific papers have been published in the framework of this ERC project, of which only a few have been highlighted here. The second phase of the project has just started and we anticipate many other exciting new developments.