Single trapped ions represent elementary quantum systems that are well isolated from the environment. They can be brought nearly to rest by laser cooling, and both their internal electronic states and external motion can be coupled to and manipulated by light fields. This makes them ideally suited for quantum-optical and quantum-dynamical studies under well-controlled conditions. Theoretical and experimental work on these topics is reviewed in the paper, with a focus on ions trapped in radio-frequency (Paul) traps.

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We report the creation of Greenberger-Horne-Zeilinger states with up to 14 qubits. By investigating the coherence of up to 8 ions over time, we observe a decay proportional to the square of the number of qubits. The observed decay agrees with a theoretical model which assumes a system affected by correlated, Gaussian phase noise. This model holds for the majority of current experimental systems developed towards quantum computation and quantum metrology.

Entanglement, its generation, manipulation and fundamental understanding is at the very heart of quantum mechanics. The phrase entanglement was coined by Erwin Schrödinger in 1935 for particles that are described by a common wave function where individual particles are not independent of each other but where their quantum properties are inextricably interwoven 1 . Entanglement properties of two and three particles have been studied extensively and are very well understood. Entanglement of four 2 and five 3 particles was demonstrated experimentally. However, both creation and characterization of entanglement become exceedingly difficult for multi-particle systems. Thus the availability of such multiparticle entangled states together with the full information on these states in form of their 1

The key to explaining a wide range of quantum phenomena is understanding how entanglement propagates around many-body systems. Furthermore, the controlled distribution of entanglement is of fundamental importance for quantum communication and computation. In many situations, quasiparticles are the carriers of information around a quantum system and are expected to distribute entanglement in a fashion determined by the system interactions [1]. Here we report on the observation of magnon quasiparticle dynamics in a one-dimensional many-body quantum system of trapped ions representing an Ising spin model [2,3]. Using the ability to tune the effective interaction range [4][5][6], and to prepare and measure the quantum state at the individual particle level, we observe new quasiparticle phenomena. For the first time, we reveal the entanglement distributed by quasiparticles around a many-body system. Second, for long-range interactions we observe the divergence of quasiparticle velocity and breakdown of the light-cone picture [7][8][9][10] that is valid for short-range interactions. Our results will allow experimental studies of a wide range of phenomena, such as quantum transport [11,12], thermalisation [13], localisation [14] and entanglement growth [15], and represent a first step towards a new quantum-optical regime with on-demand quasiparticles with tunable non-linear interactions.Quasiparticles, such as magnons, phonons, and anyons, are elementary excitations in the collective behaviour of an underlying many-body quantum system. While precise control is already possible in the laboratory for systems of individual atoms, ions, or photons, it remains a challenge to extend this to quasiparticles. In systems with nearest-neighbour interactions, quasiparticles are expected to distribute entanglement within light-like-cones defined by a strict quantum information speed limit, enforced not by relativity but by the finite interaction range itself [7,16,17]. These results, known as Lieb-Robinson bounds, have allowed various important theorems to be proven about systems with nearest-neighbour interactions, including restrictions on ground-state correlations [18,19] and the time to create states for topological quantum computation [16]. Recently, wavefronts of correlations have been observed in bosonic atoms in optical lattices with nearest-neighbour interactions [20,21], and an outstanding challenge is to observe the entanglement dynamics.Extending these results to systems with long-range interactions is of great interest: the interactions in many natural systems fall into this class, exhibiting a power-law dependence (1/r α ), such as van-der-Waals (α=6), dipole-dipole (α=3), or Coulomb interactions (α=1). In each case, a new set of quasiparticles are predicted with unique properties. If the interactions fall off sufficiently fast, one can still formulate generalized Lieb-Robinson bounds [8][9][10]. However, the notion of a speed of information propagation becomes invalid. For even longer-range interactions, these bounds...

* These authors contributed equally to this work.The control of quantum systems is of fundamental scientific interest and promises powerful applications and technologies. Impressive progress has been achieved in isolating the systems from the environment and coherently controlling their dynamics, as demonstrated by the creation and manipulation of entanglement in various physical systems. However, for open quantum systems, engineering the dynamics of many particles by a controlled coupling to an environment remains largely unexplored. Here we report the first realization of a toolbox for simulating an open quantum system with up to five qubits. Using a quantum computing architecture with trapped ions, we combine multi-qubit gates with optical pumping to implement coherent operations and dissipative processes. We illustrate this engineering by the dissipative preparation of entangled states, the simulation of coherent many-body spin interactions and the quantum non-demolition measurement of multi-qubit observables. By adding controlled dissipation to coherent operations, this work offers novel prospects for open-system quantum simulation and computation.Every quantum system is inevitably coupled to its surrounding environment. Significant progress has been made in isolating systems from their enviroment and coherently controlling the dynamics of several qubits [1][2][3][4]. These achievements have enabled the realization of highfidelity quantum gates, the implementation of small-scale quantum computing and communication devices as well as the measurement-based probabilistic preparation of entangled states, in atomic [5, 6], photonic [7] and solidstate setups [8][9][10]. In particular, successful demonstrations of quantum simulators [11, 12], which allow one to mimic and study the dynamics of complex quantum systems, have been reported [13].In contrast, controlling the more general dynamics of open systems amounts to engineering both the Hamiltonian time evolution of the system as well as the coupling to the environment. Although open-system dynamics in a many-body or multi-qubit system are typically associated with decoherence [14][15][16], the ability to design dissipation can be a useful resource. For example, controlled dissipation allows the preparation of a desired entangled state from an arbitrary state [17][18][19] or an enhanced sensitivity for precision measurements [20]. In a broader context, by combining suitably chosen coherent and dissipative time steps, one can realize the most general nonunitary open-system evolution of a many-particle system. This engineering of the system-environment coupling generalizes the concept of Hamiltonian quantum simulation to open quantum systems. In addition, this engineering enables the dissipative preparation and manipulation of many-body states and quantum phases [21], and also quantum computation based on dissipation [22].Here we provide the first experimental demonstration of a complete toolbox, through coherent and dissipative manipulations of a multi-qubit syst...

To process information using quantum-mechanical principles, the states of individual particles need to be entangled and manipulated. One way to do this is to use trapped, laser-cooled atomic ions. Attaining a general-purpose quantum computer is, however, a distant goal, but recent experiments show that just a few entangled trapped ions can be used to improve the precision of measurements. If the entanglement in such systems can be scaled up to larger numbers of ions, simulations that are intractable on a classical computer might become possible.

Quantum computers hold the promise to solve certain computational task much more efficiently than classical computers. We review the recent experimental advancements towards a quantum computer with trapped ions. In particular, various implementations of qubits, quantum gates and some key experiments are discussed. Furthermore, we review some implementations of quantum algorithms such as a deterministic teleportation of quantum information and an error correction scheme.Comment: Review article, accepted for publication in Physics Reports, 99pages, 38 Figures, ~2 MByt