arXiv:1911.00003 · The European Physical Journal D
Simulating lattice gauge theories within quantum technologies
Mari Carmen Bañuls, R. Blatt, Jacopo Catani, Alessio Celi, Juan Ignacio Cirac, Marcello Dalmonte, Leonardo Fallani, Karl Jansen, Maciej Lewenstein, Simone Montangero, Christine A. Muschik, Benni Reznik, Enrique Rico, Luca Tagliacozzo, Karel Van Acoleyen, Frank Verstraete, U.‐J. Wiese, Matthew Wingate, Jakub Zakrzewski, Peter Zoller
Abstract Lattice gauge theories, which originated from particle physics in the context of Quantum Chromodynamics (QCD), provide an important intellectual stimulus to further develop quantum information technologies. While one long-term goal is the reliable quantum simulation of currently intractable aspects of QCD itself, lattice gauge theories also play an important role in condensed matter physics and in quantum information science. In this way, lattice gauge theories provide both motivation and a framework for interdisciplinary research towards the development of special purpose digital and analog quantum simulators, and ultimately of scalable universal quantum computers. In this manuscript, recent results and new tools from a quantum science approach to study lattice gauge theories are reviewed. Two new complementary approaches are discussed: first, tensor network methods are presented – a classical simulation approach – applied to the study of lattice gauge theories together with some results on Abelian and non-Abelian lattice gauge theories. Then, recent proposals for the implementation of lattice gauge theory quantum simulators in different quantum hardware are reported, e.g., trapped ions, Rydberg atoms, and superconducting circuits. Finally, the first proof-of-principle trapped ions experimental quantum simulations of the Schwinger model are reviewed. Graphical abstract
Abstract Lattice gauge theories, which originated from particle physics in the context of Quantum Chromodynamics (QCD), provide an important intellectual stimulus to further develop quantum information technologies. While one long-term goal is the reliable quantum simulation of currently intractable aspects of QCD itself, lattice gauge theories also play an important role in condensed matter physics and in quantum information science. In this way, lattice gauge theories provide both motivation and a framework for interdisciplinary research towards the development of special purpose digital and analog quantum simulators, and ultimately of scalable universal quantum computers. In this manuscript, recent results and new tools from a quantum science approach to study lattice gauge theories are reviewed. Two new complementary approaches are discussed: first, tensor network methods are presented – a classical simulation approach – applied to the study of lattice gauge theories together with some results on Abelian and non-Abelian lattice gauge theories. Then, recent proposals for the implementation of lattice gauge theory quantum simulators in different quantum hardware are reported, e.g., trapped ions, Rydberg atoms, and superconducting circuits. Finally, the first proof-of-principle trapped ions experimental quantum simulations of the Schwinger model are reviewed. Graphical abstract
Abstract: Lattice gauge theories, which originated from particle physics in the context of Quantum Chromodynamics (QCD), provide an important intellectual stimulus to further develop quantum information technologies. While one long-term goal is the reliable quantum simulation of currently intractable aspects of QCD itself, lattice gauge theories also play an important role in condensed matter physics and in quantum information science. In this way, lattice gauge theories provide both motivation and a framework for interdisciplinary research towards the development of special purpose digital and analog quantum simulators, and ultimately of scalable universal quantum computers. In this manuscript, recent results and new tools from a quantum science approach to study lattice gauge theories are reviewed. Two new complementary approaches are discussed: first, tensor network methods are presented – a classical simulation approach – applied to the study of lattice gauge theories together with some results on Abelian and non-Abelian lattice gauge theories. Then, recent proposals for the implementation of lattice gauge theory quantum simulators in different quantum hardware are reported, e.g., trapped ions, Rydberg atoms, and superconducting circuits. Finally, the first proof-of-principle trapped ions experimental quantum simulations of the Schwinger model are reviewed. Graphical abstract:
Lattice gauge theories, which originated from particle physics in the context of Quantum Chromodynamics (QCD), provide an important intellectual stimulus to further develop quantum information technologies. While one long-term goal is the reliable quantum simulation of currently intractable aspects of QCD itself, lattice gauge theories also play an important role in condensed matter physics and in quantum information science. In this way, lattice gauge theories provide both motivation and a framework for interdisciplinary research towards the development of special purpose digital and analog quantum simulators, and ultimately of scalable universal quantum computers. In this manuscript, recent results and new tools from a quantum science approach to study lattice gauge theories are reviewed. Two new complementary approaches are discussed: first, tensor network methods are presented - a classical simulation approach -applied to the study of lattice gauge theories together with some results on Abelian and non-Abelian lattice gauge theories. Then, recent proposals for the implementation of lattice gauge theory quantum simulators in different quantum hardware are reported, e.g., trapped ions, Rydberg atoms, and superconducting circuits. Finally, the first proof-of-principle trapped ions experimental quantum simulations of the Schwinger model are reviewed.
Formulating gauge theories on a lattice offers a genuinely non-perturbative way of studying quantum field theories, and has led to impressive achievements. In particular, it significantly deepened our understanding of quantum chromodynamics. Yet, some very relevant problems remain inherently challenging, such as real time evolution, or the presence of a chemical potential, cases in which Monte Carlo simulations are hindered by a sign problem. In the last few years, a number of possible alternatives have been put forward, based on quantum information ideas, which could potentially open the access to areas of research that have so far eluded more standard methods. They include tensor network calculations, quantum simulations with different physical platforms and quantum computations, and constitute nowadays a vibrant research area. Experts from different fields, including experimental and theoretical high energy physics, condensed matter, and quantum information, are turning their attention to these interdisciplinary possibilities, and driving the progress of the field. The aim of this article is to review the status and perspectives of these new avenues for the exploration of lattice gauge theories.
Abelian and non‐Abelian gauge theories are of central importance in many areas of physics. In condensed matter physics, Abelian U (1) lattice gauge theories arise in the description of certain quantum spin liquids. In quantum information theory, Kitaev's toric code is a lattice gauge theory. In particle physics, Quantum Chromodynamics (QCD), the non‐Abelian gauge theory of the strong interactions between quarks and gluons, is non‐perturbatively regularized on a lattice. Quantum link models extend the concept of lattice gauge theories beyond the Wilson formulation, and are well suited for both digital and analog quantum simulation using ultracold atomic gases in optical lattices. Since quantum simulators do not suffer from the notorious sign problem, they open the door to studies of the real‐time evolution of strongly coupled quantum systems, which are impossible with classical simulation methods. A plethora of interesting lattice gauge theories suggests itself for quantum simulation, which should allow us to address very challenging problems, ranging from confinement and deconfinement, or chiral symmetry breaking and its restoration at finite baryon density, to color superconductivity and the real‐time evolution of heavy‐ion collisions, first in simpler model gauge theories and ultimately in QCD.
Abstract A hallmark of the computational campaign in nuclear and particle physics is the lattice-gauge-theory program. It continues to enable theoretical predictions for a range of phenomena in nature from the underlying Standard Model. The emergence of a new computational paradigm based on quantum computing, therefore, can introduce further advances in this program. In particular, it is believed that quantum computing will make possible first-principles studies of matter at extreme densities, and in and out of equilibrium, hence improving our theoretical description of the early universe, astrophysical environments, and high-energy particle collisions. Developing and advancing a quantum-computing-based lattice-gauge-theory program, therefore, have turned into a vibrant and fast-moving area of research in theoretical nuclear and particle physics. These lecture notes introduce the topic of quantum computing lattice gauge theories in a pedagogical manner, with an emphasis on theoretical and algorithmic aspects of the program, and on the most common approaches and practices, to keep the presentation focused and useful. The Hamiltonian formulation of lattice gauge theories is introduced within the Kogut–Susskind framework, the notion of Hilbert space and physical states is discussed, and some elementary numerical methods for performing Hamiltonian simulations are discussed. Quantum-simulation preliminaries and digital quantum-computing basics are presented, which set the stage for concrete examples of gauge-theory quantum-circuit design and resource analysis. A step-by-step analysis is provided for a simpler Abelian gauge theory, and an overview of our current understanding of the cost of quantum computing quantum chromodynamics is presented in the end. Examples and exercises supplement the material, and reinforce the concepts and methods introduced throughout.
In the last decade there has been an intense activity aimed at the quantum simulation of interacting many-body systems using cold atoms [1, 2]. The idea of quantum simulations traces back to Feynman [3], who argued that the ideal setting to study quantum systems would be a quantum experimental setup rather then a classical one - the latter one being fundamentally limited due to its hardware classical structure. This is a particularly important problem given the intrinsic complexity of interacting many-body problems, and the difficulties that arise when tackling them with numerical simulations - two paradigmatic examples being the sign(s) problem affecting Monte Carlo simulations of fermionic systems, and the real time dynamics in more than one spatial dimension. Ultracold atoms offer a very powerful setting for quantum simulations. Atoms can be trapped in tailored optical and magnetic potentials, also controlling their dimensionality. The inter-atomic interactions can be tuned by external knobs, such as Feshbach resonances. This gives a large freedom on model building and, with suitable mappings, they allow the implementation of desired target models. This allowed an impressive exploitation of quantum simulators on the context of condensed matter physics. The simulation of high-energy physics is an important line of research in this field and it is less direct. In particular it requires the implementation of symmetries like Lorentz and gauge invariance which are not immediately available in a cold atomic setting. Gauge fields are ubiquitous in physics ranging from condensed matter [4–6] and quantum computation [7, 8] to particle physics [9], an archetypical example being Quantum Chromodynamics (QCD) [10,11], the theory of strong nuclear forces. Currently open problems in QCD, providing a long-term goal of cold atomic simulations, include confinement/deconfinement and the structure of color superconducting phases at finite chemical potential [12]. Even though QCD is a very complicated theory (to simulate or study), it is possible to envision a path through implementation of simpler models. Furthermore, it is also expected that interesting physics is found on such “intermediate models” which may deserve attention irrespectively of the QCD study. A very relevant model in this regard is the Schwinger model (Quantum Electrodynamics in 1+1 dimensions) [13]. This theory exhibits features of QCD, such as confinement [14], and is at the very same time amenable to both theoretical studies and simpler experimental schemes. This model was the target of the first experimental realization of a gauge theory with a quantum simulator [15]. The work on this Thesis is, in part, motivated by the study of toy models which put in evidence certain aspects that can be found in QCD. Such toy models provide also intermediate steps in the path towards more complex simulations. The two main aspects of QCD which are addressed here are symmetry-locking and confinement. The other main motivation for this study is to develop a systematic framework, through dimensional mismatch, for theoretical understanding and quantum simulations of long-range theories using gauge theories. The model used to study symmetry-locking consists of a four-fermion mixture [16]. It has the basic ingredients to exhibit a non-Abelian symmetry-locked phase: the full Hamiltonian has an SU (2) × SU (2) (global) symmetry which can break to a smaller SU (2) group. Such phase is found in a extensive region of the phase diagram by using a mean-field approach and a strong coupling expansion. A possible realization of such system is provided by an Ytterbium mixture. Even without tuning interactions, it is shown that such mixture falls inside the the locked-symmetry phase pointing towards a possible realization in current day experiments. The models with dimensional mismatch investigated here have fermions in a lower dimensionality d + 1 and gauge fields in higher dimensionality D + 1. They serve two purposes: establish mappings to non-local theories by integration of fields [17] and the study of confinement [18]. In the particular case of d = 1 and D = 2 it is found that some general non-local terms can be obtained on the Lagrangian [17]. This is found in the form of power-law expansions of the Laplacian mediating either kinetic terms (for bosons) or interactions (for fermions). The fact that such expansions are not completely general is not surprising since constraints do exist, preventing unphysical features like breaking of unitarity. The non-local terms obtained are physically acceptable, in this regard, since they are derived from unitary theories. The above mapping is done exactly. In certain cases it is shown that it is possible to construct an effective long-range Hamiltonian in a perturbative expansion. In particular it is shown how this is done for non-relativistic fermions (in d = 1) and 3 + 1 gauge fields. These results are relevant in the context of state of the art experiments which implement models with long-range interactions and where theoretical results are less abundant than for the case of local theories. The above mappings establish a direct relation with local theories which allow theoretical insight onto these systems. Examples of this would consist on the application of Mermin- Wagner-Hohenberg theorem [19, 20] and Lieb-Robinson bounds [21], on the propagation of quantum correlations, to non-local models. In addition they can also provide a path towards implementation of tunable long-range interactions with cold atoms. Furthermore, in terms of quantum simulations, they are in between the full higher dimensional system and the full lower dimensional one. Such property is attractive from the point of view of a gradual increase of complexity for quantum simulations of gauge theories. Another interesting property of these models is that they allow the study of confinement beyond the simpler case of the Schwinger model. The extra dimensions are enough to attribute dynamics to the gauge field, which are no longer completely fixed by the Gauss law. The phases of the Schwinger model are shown to be robust under variation of the dimension of the gauge fields [18]. Both the screened phase, of the massless case, and the confined phase, of the massive one, are found for gauge fields in 2+1 and 3+1 dimensions. Such results are also obtained in the Schwinger- Thirring model. This shows that these phases are very robust and raises interesting questions about the nature of confinement. Robustness under Thirring interactions are relevant because it shows that errors on the experimental implementation will not spoil the phase. Even more interesting is the case of gauge fields in higher dimensions since confinement in the Schwinger model is intuitively atributed to the dimensionality of the gauge fields (creating linear potentials between particles). This Thesis is organized as follows. In Chapter 1, some essential background regarding quantum simulations of gauge theories is provided. It gives both a brief introduction to cold atomic physics and lattice gauge theories. In Chapter 2, it is presented an overview over proposals of quantum simulators of gauge potentials and gauge fields. At the end of this Chapter, in Section 2.3 it is briefly presented ongoing work on a realization of the Schwinger model that we term Half Link Schwinger model. There, it is argued, some of the generators of the gauge symmetry on the lattice can be neglected without comprimising gauge invariance. In Chapter 3, the results regarding the phase diagram of the fourfermion mixture, exhibiting symmetry-locking, is presented. In Chapter 4, the path towards controlling non-local kinetic terms and interactions is provided, after a general introduction to the formalism of dimensional mismatch. At the end the construction of effective Hamiltonians is described. Finally, Chapter 5 concerns the study of confinement and the robustness of it for 1+1 fermions. The first part regards the Schwinger-Thirring with the presence of a -term while, in the second part, models with dimensional mismatch are considered. The thesis ends with conclusions and perspectives of future work based on the results presented here.