Author Correction: Self-verifying variational quantum simulation of lattice models.

An amendment to this paper has been published and can be accessed via a link at the top of the paper.

Self-verifying variational quantum simulation of lattice models.

Hybrid classical-quantum algorithms aim to variationally solve optimization problems using a feedback loop between a classical computer and a quantum co-processor, while benefiting from quantum resources. Here we present experiments that demonstrate self-verifying, hybrid, variational quantum simulation of lattice models in condensed matter and high-energy physics. In contrast to analogue quantum simulation, this approach forgoes the requirement of realizing the targeted Hamiltonian directly in the laboratory, thus enabling the study of a wide variety of previously intractable target models. We focus on the lattice Schwinger model, a gauge theory of one-dimensional quantum electrodynamics. Our quantum co-processor is a programmable, trapped-ion analogue quantum simulator with up to 20 qubits, capable of generating families of entangled trial states respecting the symmetries of the target Hamiltonian. We determine ground states, energy gaps and additionally, by measuring variances of the Schwinger Hamiltonian, we provide algorithmic errors for the energies, thus taking a step towards verifying quantum simulation.

Experimental Realization of Nonunitary Multiqubit Operations.

We demonstrate a novel experimental tool set that enables irreversible multiqubit operations on a quantum platform. To exemplify our approach, we realize two elementary nonunitary operations: the or and nor gates. The electronic states of two trapped ^{40}Ca^{+} ions encode the logical information, and a cotrapped ^{88}Sr^{+} ion provides the irreversibility of the gate by a dissipation channel through sideband cooling. We measure 87% and 81% success rates for the or and nor gates, respectively. The presented methods are a stepping stone toward other nonunitary operations such as in quantum error correction and quantum machine learning.

Measuring Ion Oscillations at the Quantum Level with Fluorescence Light.

We demonstrate an optical method for detecting the mechanical oscillations of an atom with single-phonon sensitivity. The measurement signal results from the interference between the light scattered by a trapped atomic ion and that of its mirror image. We detect the oscillations of the atom in the Doppler cooling limit and reconstruct average trajectories in phase space. We demonstrate single-phonon sensitivity near the ground state of motion after electronically induced transparency cooling. These results could be applied for motion detection of other light scatterers of fundamental interest, such as trapped nanoparticles.

Quasiparticle engineering and entanglement propagation in a quantum many-body system.

The key to explaining and controlling a range of quantum phenomena is to study how information propagates around many-body systems. Quantum dynamics can be described by particle-like carriers of information that emerge in the collective behaviour of the underlying system, the so-called quasiparticles. These elementary excitations are predicted to distribute quantum information in a fashion determined by the system's interactions. Here we report quasiparticle dynamics observed in a quantum many-body system of trapped atomic ions. First, we observe the entanglement distributed by quasiparticles as they trace out light-cone-like wavefronts. Second, using the ability to tune the interaction range in our system, we observe information propagation in an experimental regime where the effective-light-cone picture does not apply. Our results will enable experimental studies of a range of quantum phenomena, including transport, thermalization, localization and entanglement growth, and represent a first step towards a new quantum-optic regime of engineered quasiparticles with tunable nonlinear interactions.

Cluster State Generation with Spin-Orbit Coupled Fermionic Atoms in Optical Lattices.

Measurement-based quantum computation, an alternative paradigm for quantum information processing, uses simple measurements on qubits prepared in cluster states, a class of multiparty entangled states with useful properties. Here we propose and analyze a scheme that takes advantage of the interplay between spin-orbit coupling and superexchange interactions, in the presence of a coherent drive, to deterministically generate macroscopic arrays of cluster states in fermionic alkaline earth atoms trapped in three dimensional (3D) optical lattices. The scheme dynamically generates cluster states without the need of engineered transport, and is robust in the presence of holes, a typical imperfection in cold atom Mott insulators. The protocol is of particular relevance for the new generation of 3D optical lattice clocks with coherence times >10 s, 2 orders of magnitude larger than the cluster state generation time. We propose the use of collective measurements and time reversal of the Hamiltonian to benchmark the underlying Ising model dynamics and the generated many-body correlations.

Interference of Single Photons Emitted by Entangled Atoms in Free Space.

The generation and manipulation of entanglement between isolated particles has precipitated rapid progress in quantum information processing. Entanglement is also known to play an essential role in the optical properties of atomic ensembles, but fundamental effects in the controlled emission and absorption from small, well-defined numbers of entangled emitters in free space have remained unobserved. Here we present the control of the emission rate of a single photon from a pair of distant, entangled atoms into a free-space optical mode. Changing the length of the optical path connecting the atoms modulates the single-photon emission rate in the selected mode with a visibility V=0.27±0.03 determined by the degree of entanglement shared between the atoms, corresponding directly to the concurrence C_{ρ}=0.31±0.10 of the prepared state. This scheme, together with population measurements, provides a fully optical determination of the amount of entanglement. Furthermore, large sensitivity of the interference phase evolution points to applications of the presented scheme in high-precision gradient sensing.

Tunable ion-photon entanglement in an optical cavity.

Proposed quantum networks require both a quantum interface between light and matter and the coherent control of quantum states. A quantum interface can be realized by entangling the state of a single photon with the state of an atomic or solid-state quantum memory, as demonstrated in recent experiments with trapped ions, neutral atoms, atomic ensembles and nitrogen-vacancy spins. The entangling interaction couples an initial quantum memory state to two possible light-matter states, and the atomic level structure of the memory determines the available coupling paths. In previous work, the transition parameters of these paths determined the phase and amplitude of the final entangled state, unless the memory was initially prepared in a superposition state (a step that requires coherent control). Here we report fully tunable entanglement between a single (40)Ca(+) ion and the polarization state of a single photon within an optical resonator. Our method, based on a bichromatic, cavity-mediated Raman transition, allows us to select two coupling paths and adjust their relative phase and amplitude. The cavity setting enables intrinsically deterministic, high-fidelity generation of any two-qubit entangled state. This approach is applicable to a broad range of candidate systems and thus is a promising method for distributing information within quantum networks.

Direct Observation of Dynamical Quantum Phase Transitions in an Interacting Many-Body System.

The theory of phase transitions represents a central concept for the characterization of equilibrium matter. In this work we study experimentally an extension of this theory to the nonequilibrium dynamical regime termed dynamical quantum phase transitions (DQPTs). We investigate and measure DQPTs in a string of ions simulating interacting transverse-field Ising models. During the nonequilibrium dynamics induced by a quantum quench we show for strings of up to 10 ions the direct detection of DQPTs by revealing nonanalytic behavior in time. Moreover, we provide a link between DQPTs and the dynamics of other quantities such as the magnetization, and we establish a connection between DQPTs and entanglement production.

Trapped-ion antennae for the transmission of quantum information.

More than 100 years ago, Hertz succeeded in transmitting signals over a few metres to a receiving antenna using an electromagnetic oscillator, thus proving the electromagnetic theory developed by Maxwell. Since this seminal work, technology has developed, and various oscillators are now available at the quantum mechanical level. For quantized electromagnetic oscillations, atoms in cavities can be used to couple electric fields. However, a quantum mechanical link between two mechanical oscillators (such as cantilevers or the vibrational modes of trapped atoms or ions) has been rarely demonstrated and has been achieved only indirectly. Examples include the mechanical transport of atoms carrying quantum information or the use of spontaneously emitted photons. Here we achieve direct coupling between the motional dipoles of separately trapped ions over a distance of 54 micrometres, using the dipole-dipole interaction as a quantum mechanical transmission line. This interaction is small between single trapped ions, but the coupling is amplified by using additional trapped ions as antennae. With three ions in each well, the interaction is increased by a factor of seven compared to the single-ion case. This enhancement facilitates bridging of larger distances and relaxes the constraints on the miniaturization of trap electrodes. The system provides a building block for quantum computers and opportunities for coupling different types of quantum systems.

Quantum simulation of the Dirac equation.

The Dirac equation successfully merges quantum mechanics with special relativity. It provides a natural description of the electron spin, predicts the existence of antimatter and is able to reproduce accurately the spectrum of the hydrogen atom. The realm of the Dirac equation-relativistic quantum mechanics-is considered to be the natural transition to quantum field theory. However, the Dirac equation also predicts some peculiar effects, such as Klein's paradox and 'Zitterbewegung', an unexpected quivering motion of a free relativistic quantum particle. These and other predicted phenomena are key fundamental examples for understanding relativistic quantum effects, but are difficult to observe in real particles. In recent years, there has been increased interest in simulations of relativistic quantum effects using different physical set-ups, in which parameter tunability allows access to different physical regimes. Here we perform a proof-of-principle quantum simulation of the one-dimensional Dirac equation using a single trapped ion set to behave as a free relativistic quantum particle. We measure the particle position as a function of time and study Zitterbewegung for different initial superpositions of positive- and negative-energy spinor states, as well as the crossover from relativistic to non-relativistic dynamics. The high level of control of trapped-ion experimental parameters makes it possible to simulate textbook examples of relativistic quantum physics.

Enhanced quantum interface with collective ion-cavity coupling.

We prepare a maximally entangled state of two ions and couple both ions to the mode of an optical cavity. The phase of the entangled state determines the collective interaction of the ions with the cavity mode, that is, whether the emission of a single photon into the cavity is suppressed or enhanced. By adjusting this phase, we tune the ion-cavity system from sub- to superradiance. We then encode a single qubit in the two-ion superradiant state and show that this encoding enhances the transfer of quantum information onto a photon.

Spectroscopy of Interacting Quasiparticles in Trapped Ions.

The static and dynamic properties of many-body quantum systems are often well described by collective excitations, known as quasiparticles. Engineered quantum systems offer the opportunity to study such emergent phenomena in a precisely controlled and otherwise inaccessible way. We present a spectroscopic technique to study artificial quantum matter and use it for characterizing quasiparticles in a many-body system of trapped atomic ions. Our approach is to excite combinations of the system's fundamental quasiparticle eigenmodes, given by delocalized spin waves. By observing the dynamical response to superpositions of such eigenmodes, we extract the system dispersion relation, magnetic order, and even detect signatures of quasiparticle interactions. Our technique is not limited to trapped ions, and it is suitable for verifying quantum simulators by tuning them into regimes where the collective excitations have a simple form.

Experimental violation of multipartite Bell inequalities with trapped ions.

We report on the experimental violation of multipartite Bell inequalities by entangled states of trapped ions. First, we consider resource states for measurement-based quantum computation of between 3 and 7 ions and show that all strongly violate a Bell-type inequality for graph states, where the criterion for violation is a sufficiently high fidelity. Second, we analyze Greenberger-Horne-Zeilinger states of up to 14 ions generated in a previous experiment using stronger Mermin-Klyshko inequalities, and show that in this case the violation of local realism increases exponentially with system size. These experiments represent a violation of multipartite Bell-type inequalities of deterministically prepared entangled states. In addition, the detection loophole is closed.

Atom-atom entanglement by single-photon detection.

A scheme for entangling distant atoms is realized, as proposed in the seminal paper by [C. Cabrillo et al., Phys. Rev. A 59, 1025 (1999)]. The protocol is based on quantum interference and detection of a single photon scattered from two effectively one meter distant laser cooled and trapped atomic ions. The detection of a single photon heralds entanglement of two internal states of the trapped ions with high rate and with a fidelity limited mostly by atomic motion. Control of the entangled state phase is demonstrated by changing the path length of the single-photon interferometer.

Experimental generation of quantum discord via noisy processes.

Quantum systems in mixed states can be unentangled and yet still nonclassically correlated. These correlations can be quantified by the quantum discord and might provide a resource for quantum information processing tasks. By precisely controlling the interaction of two ionic qubits with their environment, we investigate the capability of noise to generate discord. Firstly, we show that noise acting on only one quantum system can generate discord between two. States generated in this way are restricted in terms of the rank of their correlation matrix. Secondly, we show that classically correlated noise processes are capable of generating a much broader range of discordant states with correlation matrices of any rank. Our results show that noise processes prevalent in many physical systems can automatically generate nonclassical correlations and highlight fundamental differences between discord and entanglement.

Heralded entanglement of two ions in an optical cavity.

We demonstrate precise control of the coupling of each of two trapped ions to the mode of an optical resonator. When both ions are coupled with near-maximum strength, we generate ion-ion entanglement heralded by the detection of two orthogonally polarized cavity photons. The entanglement fidelity with respect to the Bell state Ψ+ reaches F≥(91.9±2.5)%. This result represents an important step toward distributed quantum computing with cavities linking remote atom-based registers.

Shot-noise-limited monitoring and phase locking of the motion of a single trapped ion.

We perform a high-resolution real-time readout of the motion of a single trapped and laser-cooled Ba+ ion. By using an interferometric setup, we demonstrate a shot-noise-limited measurement of thermal oscillations with a resolution of 4 times the standard quantum limit. We apply the real-time monitoring for phase control of the ion motion through a feedback loop, suppressing the photon recoil-induced phase diffusion. Because of the spectral narrowing in the phase-locked mode, the coherent ion oscillation is measured with a resolution of about 0.3 times the standard quantum limit.

Measurement-based quantum computation with trapped ions.

Measurement-based quantum computation represents a powerful and flexible framework for quantum information processing, based on the notion of entangled quantum states as computational resources. The most prominent application is the one-way quantum computer, with the cluster state as its universal resource. Here we demonstrate the principles of measurement-based quantum computation using deterministically generated cluster states, in a system of trapped calcium ions. First we implement a universal set of operations for quantum computing. Second we demonstrate a family of measurement-based quantum error correction codes and show their improved performance as the code length is increased. The methods presented can be directly scaled up to generate graph states of several tens of qubits.

'Designer atoms' for quantum metrology.

Entanglement is recognized as a key resource for quantum computation and quantum cryptography. For quantum metrology, the use of entangled states has been discussed and demonstrated as a means of improving the signal-to-noise ratio. In addition, entangled states have been used in experiments for efficient quantum state detection and for the measurement of scattering lengths. In quantum information processing, manipulation of individual quantum bits allows for the tailored design of specific states that are insensitive to the detrimental influences of an environment. Such 'decoherence-free subspaces' (ref. 10) protect quantum information and yield significantly enhanced coherence times. Here we use a decoherence-free subspace with specifically designed entangled states to demonstrate precision spectroscopy of a pair of trapped Ca+ ions; we obtain the electric quadrupole moment, which is of use for frequency standard applications. We find that entangled states are not only useful for enhancing the signal-to-noise ratio in frequency measurements--a suitably designed pair of atoms also allows clock measurements in the presence of strong technical noise. Our technique makes explicit use of non-locality as an entanglement property and provides an approach for 'designed' quantum metrology.