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.

* 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...

Gauge theories are fundamental to our understanding of interactions between the elementary constituents of matter as mediated by gauge bosons [1,2]. However, computing the real-time dynamics in gauge theories is a notorious challenge for classical computational methods. In the spirit of Feynman's vision of a quantum simulator [3,4], this has recently stimulated theoretical effort to devise schemes for simulating such theories on engineered quantum-mechanical devices, with the difficulty that gauge invariance and the associated local conservation laws (Gauss laws) need to be implemented [5][6][7]. Here we report the first experimental demonstration of a digital quantum simulation of a lattice gauge theory, by realising 1+1-dimensional quantum electrodynamics (Schwinger model [8,9]) on a few-qubit trapped-ion quantum computer. We are interested in the real-time evolution of the Schwinger mechanism [10,11], describing the instability of the bare vacuum due to quantum fluctuations, which manifests itself in the spontaneous creation of electron-positron pairs. To make efficient use of our quantum resources, we map the original problem to a spin model by eliminating the gauge fields [12] in favour of exotic longrange interactions, which have a direct and efficient implementation on an ion trap architecture [13]. We explore the Schwinger mechanism of particle-antiparticle generation by monitoring the mass production and the vacuum persistence amplitude. Moreover, we track the real-time evolution of entanglement in the system, which illustrates how particle creation and entanglement generation are directly related. Our work represents a first step towards quantum simulating high-energy theories with atomic physics experiments, the long-term vision being the extension to real-time quantum simulations of non-Abelian lattice gauge theories.Small-scale quantum computers exist today in the laboratory as programmable quantum devices [14]. In particular, trapped-ion quantum computers [13] provide a platform allowing a few hundred coherent quantum gates on a few qubits, with a clear roadmap towards scaling up FIG. 1. (a)The instability of the vacuum due to quantum fluctuations is one of the most fundamental effects in gauge theories. We simulate the coherent real time dynamics of particle-antiparticle creation by realising the Schwinger model (one-dimensional quantum electrodynamics) on a lattice, as described in the main text. (b) The experimental setup for the simulation consists of a linear Paul trap, where a string of 40 Ca + ions is confined. The electronic states of each ion encode a spin |↑ or |↓ ; these can be manipulated using laser beams (see Methods for details).these devices [4,15]. This provides the tools for universal digital quantum simulation [16], where the time evolution of a quantum system is approximated as a stroboscopic sequence of quantum gates [17]. Here we show how this quantum technology can be used to simulate the real time dynamics of a minimal model of a lattice gauge theory, realising the Schwinge...

A digital quantum simulator is an envisioned quantum device that can be programmed to efficiently simulate any other local system. We demonstrate and investigate the digital approach to quantum simulation in a system of trapped ions. With sequences of up to 100 gates and 6 qubits, the full time dynamics of a range of spin systems are digitally simulated. Interactions beyond those naturally present in our simulator are accurately reproduced, and quantitative bounds are provided for the overall simulation quality. Our results demonstrate the key principles of digital quantum simulation and provide evidence that the level of control required for a full-scale device is within reach.

The construction of a quantum computer remains a fundamental scientific and technological challenge because of the influence of unavoidable noise. Quantum states and operations can be protected from errors through the use of protocols for quantum computing with faulty components. We present a quantum error-correcting code in which one qubit is encoded in entangled states distributed over seven trapped-ion qubits. The code can detect one bit flip error, one phase flip error, or a combined error of both, regardless on which of the qubits they occur. We applied sequences of gate operations on the encoded qubit to explore its computational capabilities. This seven-qubit code represents a fully functional instance of a topologically encoded qubit, or color code, and opens a route toward fault-tolerant quantum computing.