These lecture notes were created for a graduate-level course on quantum simulation taught at Leibniz University Hannover in 2013. The first part of the course discusses various state of the art methods for the numerical description of many-body quantum systems. In particular, I explain successful applications and inherent limitations due to the exponential complexity of the many-body problem. In the second part of the course, I show how using highly controllable quantum system such as ultracold atoms will provide a way to overcome the limitations of classical simulation methods. I discuss several theoretical and experimental achievements and outline the road for future developments.
These lecture notes were created for a graduate-level course on quantum simulation taught at Leibniz University Hannover in 2013. The first part of the course discusses various state of the art methods for the numerical description of many-body quantum systems. In particular, I explain successful applications and inherent limitations due to the exponential complexity of the many-body problem. In the second part of the course, I show how using highly controllable quantum system such as ultracold atoms will provide a way to overcome the limitations of classical simulation methods. I discuss several theoretical and experimental achievements and outline the road for future developments.
The study of quantum computations that can be simulated efficiently classically is of interest for numerous reasons. On a fundamental level, such an investigation sheds light on the intrinsic computational power that is harnessed in quantum mechanics as compared to classical physics. More practically, understanding which quantum computations do not offer any speed-ups over classical computation provides insights in where (not) to look for novel quantum algorithmic primitives. On the other hand, classical simulation of many-body systems is a challenging task, as the dimension of the Hilbert space scales with the number of particles. Therefore, to understand the properties of the systems, suitable approximation methods need to be employed. The lectures will be divided into two parts. In the first part we discuss classical simulation of quantum computation from several perspectives. We review a number of well-known examples of classically simulatable quantum computations, such as the Gottesman-Knill theorem, matchgate simulation and tensor contraction methods. We further discuss simulation methods that are centred on classical sampling methods (‘weak simulation’), and illustrate how these techniques outperform methods that rely on the exact computation of measurement probabilities (‘strong simulation’). The second part focuses on "Entanglement and variational wavefunctions in quantum many body physics". We review the idea of entanglement in quantum many-body systems and how it helps us to understand the success of numerical renormalization group methods. In particular we will discuss a few variational wave-function based methods for simulating strongly correlated quantum systems, which include (1) matrix product states (2) multiscale entanglement renormalization ansatz (3) projected entangled pair states and (4) continuous matrix product states for quantum field theories.
The study of quantum computations that can be simulated efficiently classically is of interest for numerous reasons. On a fundamental level, such an investigation sheds light on the intrinsic computational power that is harnessed in quantum mechanics as compared to classical physics. More practically, understanding which quantum computations do not offer any speed-ups over classical computation provides insights in where (not) to look for novel quantum algorithmic primitives. On the other hand, classical simulation of many-body systems is a challenging task, as the dimension of the Hilbert space scales with the number of particles. Therefore, to understand the properties of the systems, suitable approximation methods need to be employed. The lectures will be divided into two parts. In the first part we discuss classical simulation of quantum computation from several perspectives. We review a number of well-known examples of classically simulatable quantum computations, such as the Gottesman-Knill theorem, matchgate simulation and tensor contraction methods. We further discuss simulation methods that are centred on classical sampling methods (‘weak simulation’), and illustrate how these techniques outperform methods that rely on the exact computation of measurement probabilities (‘strong simulation’). The second part focuses on "Entanglement and variational wavefunctions in quantum many body physics". We review the idea of entanglement in quantum many-body systems and how it helps us to understand the success of numerical renormalization group methods. In particular we will discuss a few variational wave-function based methods for simulating strongly correlated quantum systems, which include (1) matrix product states (2) multiscale entanglement renormalization ansatz (3) projected entangled pair states and (4) continuous matrix product states for quantum field theories.
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 ultra-cold 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.
This practical guidebook provides a comprehensive set of exercises which illustrate the most relevant concepts in a first course on quantum many-body theory either as part of advanced undergraduate or graduate courses. The problems come with detailed solutions which can easily be followed either by both professors and students. Quantum many-body theory is relevant in a number of fields, from condensed matter to astrophysics. The problems proposed in this book cover this variety of topics and are illustrated whenever possible with state-of-the-art-examples. Key Features: Provides a problems-based approach to quantum many-body theory, unlike existing textbooks. In-depth solutions to problems are presented, with the aim to maximize understanding and improve the teaching experience of the subject. The multidisciplinary nature of quantum many-body theory is explored in problems that deal with nuclear physics to ultracold atoms and astrophysics.
Quantum computers, although not yet available on the market, will revolutionise the future of information processing. Already now, quantum computers of special purpose, i.e., quantum simulators, are within reach. The physics of ultracold atoms, ions, and molecules offers unprecedented possibilities of control of quantum many systems, and novel possibilities of applications for quantum information and quantum metrology. Particularly fascinating is the possibility of using ultracold atoms in lattices to simulate condensed matter or even high energy physics. This book provides a comprehensive overview of ultracold lattice gases as quantum simulators, an interdisciplinary field involving atomic, molecular, and optical physics; quantum optics; quantum information; and condensed matter and high energy physics. It includes some introductory chapters on basic concepts and methods, and focuses on the physics of spinor, dipolar, disordered, and frustrated lattice gases, before reviewing in detail artificial lattice gauge fields with ultracold gases. The last part of the book moves onto a discussion of possible implementations universal quantum computers with ultracold atoms. After a crash course in quantum information theory, several models of quantum computation with ultracold gases are presented, as well as the current understanding of condensed matter from a quantum information perspective. The book ends with the general discussion of various detection methods that are unique for ultracold atoms.
These are the lecture notes for a course that I am teaching at Zhiyuan College of Shanghai Jiao Tong University (available at https://www.youtube.com/derekkorg), though the first draft was created for a previous course I taught at the University of Erlangen-Nuremberg in Germany. It has been designed for students who have only had basic training on quantum mechanics, and hence, the course is suited for people at all levels. The notes are a work in progress, meaning that some proofs and many figures are still missing. However, I've tried my best to write everything in such a way that a reader can follow naturally all arguments and derivations even with these missing bits. Quantum optics treats the interaction between light and matter. We may think of light as the optical part of the electromagnetic spectrum, and matter as atoms. However, modern quantum optics covers a wild variety of systems, including superconducting circuits, confined electrons, excitons in semiconductors, defects in solid state, or the center-of-mass motion of micro-, meso-, and macroscopic systems. Moreover, quantum optics is at the heart of the field of quantum information. The ideas and experiments developed in quantum optics have also allowed us to take a fresh look at many-body problems and even high-energy physics. In addition, quantum optics holds the promise of testing foundational problems in quantum mechanics as well as physics beyond the standard model in table-sized experiments. Quantum optics is therefore a topic that no future researcher in quantum physics should miss.