# Ebook: Quantum Simulators

**Description**

The last century has been characterized by the development of information theory and the consequent transformative impact of new technologies on societies around the world. It seems likely that the tremendous progress in nanoscience – the ability to manipulate microscopic systems at the level of a single atom – and the emergence of quantum information science, will be the key components of the next revolution; that of the new quantum technologies. Indeed, the ability to manipulate and control quantum systems has already found a variety of potential applications, ranging from the development of molecular nanoscale machines which exploit quantum coherence for their functioning, to metrological schemes where quantum effects are used to enhance the accuracy of measurement and detection systems to achieve higher statistical precision than is possible using purely classical approaches.

This book presents the proceedings of the Enrico Fermi Summer School on Quantum Simulators (Course 198) held in Varenna, Italy, 22-27 July 2016. Topics covered included: cold atoms in optical lattices; trapped ions; solid state implementations; quantum many-body physics; quantum photonics; hybrid quantum systems; and transport phenomena.

The book will be of interest to all those whose work is connected to the rapidly growing field of quantum technologies.

**Preface**

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

**Abstract**

What could the preparation of a coffee and epidemic spreading of a disease have in common? These two very different situations are examples of the phenomenon known as percolation. The mechanism of percolation consists in the evolution of clusters of sites that can be bound with certain probability. At a specific threshold probability, the system experiences a phase transition that separates the regime in which only small clusters of connected sites form to that in which single clusters that cross the entire system appear, and thus percolation happens. If the propagation in the network has a preferred direction, it is called directed percolation. The model of percolation is theoretically appealing and it has been widely studied due to its simplicity and large applicability to several systems in nature. In my brief presentation I will also explain how we can implement the mechanism of percolation with ultracold Rydberg atoms and how we investigate experimentally its phase transition.

**Abstract**

In nature there is clear preference for coupling between pairs of entities. When there are more than two entities participating in an interaction, things tend to become problematic and “relationships” become frustrated. There are many social examples of polyamor relations and many historical books realted to triad relationships (e.g. the Divine Comedy of Dante). Similarly, in physics interactions appear in couples and many-body terms (where “many” stands more than two) introducing difficulties that need to be handled with care. These conflicts can be resolved with the introduction of chaperones that can help to disentangle the complicated many-body interactions. This is also happening in quantum computing. Mapping fermionic Hamiltonians into the qubit realization introduces undesired k-body interaction terms that can be disentangled using auxiliary qubits, called ancillas. The solution of this problem requires compromises or, more scientifically, approximations. In this work I provide a general overview of the already existing perturbative techniques for the reduction of k-local terms to 2-local terms. In addition, I evaluate new algorithms based on linear algebra and optimization techniques under the scope of natural and problematic relationships of humans and/or fermions.

**Abstract**

The problem of ultracold interacting Bose gas was intensively studied since 1995, when the condensation of trapped alkali atoms was achieved for the first time. The usual framework in order to deal with these systems consists in replacing the true interaction with an effective pseudo-potential. In these proceedings we analyze two different situations: the well-known contact interaction approximation and the currently widely under investigation case of dipolar condensates where, in addition to the usual zero-range contribution, we have a long-range anisotropic component. In the former case we present a slightly different approach to study the dynamics of a non-condensed Bose gas, based on a strong formal analogy with plasma physics; for example, we will be able to show the occurring of Landau damping in a quasi-1D Bose gas. Concerning the case of dipolar atoms, recent experimental activities showed strikingly new dynamical features: properly quenching the ratio between the dipolar interaction strength and contact one, the system displays a clusterization in a pattern of smaller droplets, which reminds the Rosensweig instability typical of classical ferrofluids.

**Abstract**

The Bose-Hubbard model is explored in a flat-band lattice by following the story of a community of bosons, whose movement is enforced by the mayor through taxes and earnings to illustrate the role of the tunnelling and the onsite interaction parameters. This model is then illustrated for a 1-D diamond lattice and the effect on the community is examined in order to explain the phase-type and the phase transition point. The model is frustrated, which is illustrated by having thieves along certain routes, meaning bosons can only get past by travelling in pairs. Additionally, the minimisation of the mayor's expenditure provides a parallel for obtaining the ground state energy of the model.

**Abstract**

This chapter focuses on quantum simulators realized through two-electron atoms, with particular attention to quantum simulators based on Yb. Details of some relevant physical models which can be efficiently simulated through these atomic systems are given, focusing in particular on two recent experiments: quantum simulation of multi-spin Luttinger liquids, and realization of Hall-like chiral systems, where edge states are detected, along with their associated chiral currents.

**Abstract**

In these lectures, we discuss theoretical aspects of analogue quantum simulation, focussing on ultracold atoms in optical lattices as a specific example. We touch on the motivation for quantum simulation, before going into detail on the first-principles microscopic understanding of the system that allows us to use ultracold atoms as quantum simulators. This level of understanding and control crucially also exists for the dominant dissipative processes in experiments. We conclude by discussing the verification of quantum simulators, comparison with classical simulations, and the current role of many-body entanglement in this discussion.

**Abstract**

Quantum computers promise to unlock a computational power beyond the reach of classical platforms. In this context, photonic Boson Sampling is believed to provide a strong evidence of such quantum speedup already with forthcoming technologies. However, in the regime where Boson Sampling devices start surpassing classical computers, it becomes fundamental to develop efficient and reliable techniques to validate their quantum operation. In this work we describe in a unified framework the accomplishments achieved in the scope of Boson Sampling, with a particular focus on its implementation and validation on integrated photonic platforms. The results consolidate the general belief that such devices are indeed capable to provide reliable demonstrations of quantum supremacy.

**Abstract**

Yesterday you decided to go for a quick drink with friends. One drink turns into one (or several) too many and the next thing you know you've woken up on your floor surrounded by cold pizza and a phone full of texts you should never have sent. Most people have memories of nights like this, but not many people have memories of their walk home. I would like to speak about the quantum random walk: the quantum analogue of the drunk man's walk home. After enough beers it's often the case that we end up in some strange positions. Unfortunately though, it's never possible that we can be in two places at once. So, instead, imagine that some friends all leave the bar together, in one single group. How would the group, as a whole, travel? Would it remain as a tight, compact group or would it spread and be in multiple places at the same time? What happens if someone becomes more interested in the local kebab shop or phone signal is lost and they can no longer contact each other? How do all these different factors affect the walk? Physically, quantum random walks have applications within quantum computation, search algorithms and cryptography. In general, understanding the features of a quantum walk, and determining the control that we need to have on them, can offer a significant advantage over classical methods within computation and security.

**Abstract**

Occasionally, we all stumble upon a certain physics problem that can only be solved with a computer larger than the known universe. In the following, I will describe how to work around this slight inconvenience, assuming you have some friends with enough free time, a borderline irresponsible amount of laser power and atoms colder than fridges that anyone will ever sell you.

**Abstract**

This paper presents the basic features of quantum simulation based on continuous-variable systems. Far from attempting at being complete, the paper is a compendium of the limitations, possibilities, and perspectives in quantum simulations with such a class of systems. It includes a short summary on the phase-space description of basic continuous-variable states and operations and a gentle introduction to the most crucial results on the simulation of Hamiltonians for continuous-variable systems.

**Abstract**

Single photons provide unique advantages that make this architecture ideally suited for quantum computation and quantum simulation. In particular the low-noise properties and the ease of manipulation at the individual particle level allow for the simulation of particular interesting questions in quantum chemistry, quantum biology and solid-state physics. Moreover, the advantage of the photons' mobility makes optical quantum systems unprecedented for quantum random walks and thus for mimicking quantum transport phenomena. Here we give an overview of the state-of-the-art technology for the photon generation using spontaneous parametric down-conversion and for the high-efficient detection using superconducting single-photon nanowire detectors. In addition we review some recent photonic quantum simulations and explain novel quantum computation architectures that exploit the superposition of the orders of quantum gates.

**Abstract**

At a first glance, the material world around you might seem inanimate. If you zoomed in on it, you would not only be faced with potentially malicious bacteria waiting on the surface to make their way into your body but also — magnifying further and further — uncountable molecules formed by atoms, which again consist of a bulk of nucleons surrounded by a cloud of quickly moving electrons. As all these particles are more or less interacting with each other it is obviously highly challenging to understand their mutual interplay. To establish order in that chaos I use laser light. I first make the atoms cool down and, as a next step, separate and align them in a well-defined 2D lattice structure. Having caught the atoms and successfully attracted their attention it is possible to communicate with them, change their individual state and even control how strong they can interact with one another. This setup also reveals that the realm of atoms is a really odd world: As an example, the behavior of the atoms can only be understood through quantum mechanics. I invite you to a journey into the world of tiny quanta with its amazing characteristics and will present you how the lattice setup helps to get a better understanding of that place and make use of its properties.

**Abstract**

Beware: You are about to read an entertaining approach to the concept of stimulated Raman adiabatic passage (STIRAP) applied to ultracold atoms. Here, the dating agency alkaLOVE uses STIRAP (which they claim stands for Stimulated Transfer Into Really Awesome Partnerships) to bring lonely ultracold atoms into happy molecular relationships. This happens with record success rates and in very short time. Here some of the key secrets of alkaLOVE's quantum two-step process are explained and the advantages of molecular life are highlighted.

**Abstract**

We present a perspective on quantum simulations, a quantum technology aiming at the intentional reproduction of physical or unphysical models in controllable quantum systems. We provide pedagogical examples and exercises in different quantum platforms as trapped ions and superconducting circuits.

**Abstract**

Classical computation has been at the centre of human life in the last fifty years, contributing to the evolution of our life style, from basic calculations to games, phones, very complex artificial networks and so on and so forth. Unfortunately, besides the wonders, there are problems that computers cannot solve so easily, as factoring big numbers or simulate phenomena belonging to the microscopical realm. Here the idea of the quantum computer. Surely you have noticed that inserting the word quantum in any context is very cool. But why? Well, because exploiting quantum properties there is a speed-up in computation with respect its classical counterpart. Lucky us, it is very difficult to reach this goal, therefore, the real question is: how can I find real evidences of this quantum supremacy? A possible answer comes from the Boson Sampling problem. Injecting indistinguishable photons into a complex interferometer and measuring the outputs, we could show the real advantage of quantum computation and open the way to a new quantum technological era. Or it is just a pretext to invoke the word “quantum” in ordinary life.

**Abstract**

First of all you can cook with them! You can prepare tea, you can cook vegetables. But also ions! To build a quantum computer you have many possibilities. In our case we use as our unit of information a beryllium ion. We trap it above a small chip similar to the ones you can find in modern computers. To operate on this information unit researchers typically use lots of lasers, which are the analogue of the electrical current in present-day computers. In our experiments, we use a more simple and compact microwave circuit. To drive simple operations is quite easy but for more complex ones you need to create a microwave with really specific properties which allows you to make different ions interact between each other. In my talk I will explain why this system is really interesting!

**Abstract**

A short detour into the world of fancy scientific words, Schrödinger's cat and window glass in the lab.

**Abstract**

Quantum Error Correction (QEC) is an important ingredient towards fault-tolerant quantum computation. In this article, the building blocks for QEC with trapped ion is briefly reviewed.