The last century has been characterized by the development of information theory and its impact on technology that has influenced society and transformed the whole world. The tremendous progress in nano-science, the ability to manipulate microscopic systems at the level of a single atom and the emergence of quantum information science are the key ingredients for a new revolution: that of the new Quantum Technologies.

Numerous examples are provided in which a judicious control of the laws of quantum mechanics would allow to realize goals that are currently unattainable. The impact and advantages of quantum information protocols are impressive. In cryptography quantum dynamics guarantees secure protocols, in quantum computation factorization of large numbers, intractable with classical algorithms, can be solved enormously faster with a quantum computer. When these systems are realised, we will have at our disposal a computational power that is unthinkable as compared with the supercomputers we use at present. Meanwhile the ability to manipulate and control quantum systems has already found a variety of potential applications ranging from the developing of molecular nanoscale machines exploiting quantum coherence as a resource for their functioning, to the development of quantum metrological schemes where quantum effects are used to enhance the accuracy of measurement and detection schemes to achieve higher statistical precision than purely classical approaches.

In addition, quantum information is having a beneficial impact on other fields as sensing or in the design of new strategies to simulate complex systems. Despite the enormous progress in simulating complex quantum systems, this is still a formidable problem. Solving it would have many important consequences from the design of new molecules with prescribed functionalities, to the understanding of superconductivity at high (possibly room) temperature, or to the realization of magnetic devices just to mention some of them. In this field quantum technologies will play an instrumental role in the realization of the so-called quantum simulators, controlled quantum systems capable to, in their evolution, simulate a given complex system of interest.

As first envisaged by Feynman, nothing can beat a quantum system in simulating another quantum system. This apparently tautological observation turned out to have a profound implication when, in the last decade, nanofabricated quantum structures and trapped cold atomic species where realized in the laboratory. Nowadays quantum simulators are getting to the level of real devices, constituted by a quantum system that can be controlled in its preparation, evolution and measurement and whose dynamics can implement that of the target quantum system we want to simulate. Research efforts from many different theoretical and experimental groups have recently led to a variety of spectacular results with cold atoms, ion traps, solid state devices and quantum optical systems (just to mention the most promising implementations at present).

While rapidly growing, the field is already mature to be taught at a graduate school. The present proceedings bring together both the contributions given at the 2016 Varenna summer school on Quantum Simulators by some of the leading world experts and those of the graduate students and postdocs attending the school. The collection does not focus on a specific implementation but rather covers various directions that are presently being investigated more intensively in the field.

T. Calarco, R. Fazio and P. Mataloni