Superconducting quantum circuits based on Josephson junctions have made rapid progress in demonstrating quantum behavior and scalability. However, the future prospects ultimately depend upon the intrinsic coherence of Josephson junctions, and whether superconducting qubits can be adequately isolated from their environment. We introduce a new architecture for superconducting quantum circuits employing a three-dimensional resonator that suppresses qubit decoherence while maintaining sufficient coupling to the control signal. With the new architecture, we demonstrate that Josephson junction qubits are highly coherent, with T2 ∼ 10 to 20  μs without the use of spin echo, and highly stable, showing no evidence for 1/f critical current noise. These results suggest that the overall quality of Josephson junctions in these qubits will allow error rates of a few 10(-4), approaching the error correction threshold.

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.

In contrast to a single quantum bit, an oscillator can store multiple excitations and coherences provided one has the ability to generate and manipulate complex multiphoton states. We demonstrate multiphoton control by using a superconducting transmon qubit coupled to a waveguide cavity resonator with a highly ideal off-resonant coupling. This dispersive interaction is much greater than decoherence rates and higher-order nonlinearities to allow simultaneous manipulation of hundreds of photons. With a tool set of conditional qubit-photon logic, we mapped an arbitrary qubit state to a superposition of coherent states, known as a "cat state." We created cat states as large as 111 photons and extended this protocol to create superpositions of up to four coherent states. This control creates a powerful interface between discrete and continuous variable quantum computation and could enable applications in metrology and quantum information processing.

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.

Photons are ideal carriers for quantum information as they can have a long coherence time and can be transmitted over long distances. These properties are a consequence of their weak interactions within a nearly linear medium. To create and manipulate nonclassical states of light, however, one requires a strong, nonlinear interaction at the single photon level. One approach to generate suitable interactions is to couple photons to atoms, as in the strong coupling regime of cavity QED systems [1, 2]. In these systems, however, one only indirectly controls the quantum state of the light by manipulating the atoms [3]. A direct photon-photon interaction occurs in so-called Kerr media, which typically induce only weak nonlinearity at the cost of significant loss. So far, it has not been possible to reach the single-photon Kerr regime, where the interaction strength between individual photons exceeds the loss rate. Here, using a 3D circuit QED architecture [4], we engineer an artificial Kerr medium which enters this regime and allows the observation of new quantum effects. We realize a Gedankenexperiment proposed by Yurke and Stoler [5], in which the collapse and revival of a coherent state can be observed. This time evolution is a consequence of the quantization of the light field in the cavity and the nonlinear interaction between individual photons. During this evolution non-classical superpositions of coherent states, i.e. multi-component Schrödinger cat states, are formed. We visualize this evolution by measuring the Husimi Q-function and confirm the non-classical properties of these transient states by Wigner tomography. The ability to create and manipulate superpositions of coherent states in such a high quality factor photon mode opens perspectives for combining the physics of continuous variables [6] with superconducting circuits. The single-photon Kerr effect could be employed in QND measurement of photons [7], single photon generation [8], autonomous quantum feedback schemes [9] and quantum logic operations [10].A material whose refractive index depends on the intensity of the light field is called a Kerr medium. A light beam traveling through such a material acquires a phase shift φ Kerr = Kτ I [11] where I is the intensity of the beam, τ is the interaction time of the light field with the material, and K is the Kerr constant. The Kerr effect is a widely used phenomenon in nonlinear quantum optics and has been successfully employed to generate quadrature and amplitude squeezed states [12], parametrically convert frequencies [13], and create ultra-fast pulses [14]. In the field of quantum optics with microwave circuits, the direct analog of the Kerr effect is naturally created by the nonlinear inductance of a Josephson junction (specifically the4 term in the Taylor expansion of the cos φ of the Josephson energy relation) [15,16]. This effect has been used to create Josephson parametric amplifiers [17][18][19] and to generate squeezing of microwave fields [20]. However, in both the microwave and optical dom...

We experimentally demonstrate a quantum walk on a line in phase space using one and two trapped ions. A walk with up to 23 steps is realized by subjecting an ion to state-dependent displacement operations interleaved with quantum coin tossing operations. To analyze the ion's motional state after each step we apply a technique that directly maps the probability density distribution onto the ion's internal state. The measured probability distributions and the position's second moment clearly show the nonclassical character of the quantum walk. To further highlight the difference between the classical (random) and the quantum walk, we demonstrate the reversibility of the latter. Finally, we extend the quantum walk by using two ions, giving the walker the additional possibility to stay instead of taking a step.

Today, ion traps are among the most promising physical systems for constructing a quantum device harnessing the computing power inherent in the laws of quantum physics 1,2 . For the implementation of arbitrary operations, a quantum computer requires a universal set of quantum logic gates. As in classical models of computation, quantum error correction techniques 3,4 enable rectification of small imperfections in gate operations, thus enabling perfect computation in the presence of noise. For fault-tolerant computation 5 , it is believed that error thresholds ranging between 10 −4 and 10 −2 will be required-depending on the noise model and the computational overhead for realizing the quantum gates 6-8 -but so far all experimental implementations have fallen short of these requirements. Here, we report on a Mølmer-Sørensen-type gate operation 9,10 entangling ions with a fidelity of 99.3(1)%. The gate is carried out on a pair of qubits encoded in two trapped calcium ions using an amplitudemodulated laser beam interacting with both ions at the same time. A robust gate operation, mapping separable states onto maximally entangled states is achieved by adiabatically switching the laser-ion coupling on and off. We analyse the performance of a single gate and concatenations of up to 21 gate operations.For ion traps, all building blocks necessary for the construction of a universal quantum computer 1 have been demonstrated over the past decade. Currently, the most important challenges consist of scaling up the present systems to a higher number of qubits and raising the fidelity of gate operations up to the point where quantum error correction techniques can be successfully applied. Although single-qubit gates are easily carried out with high quality, the realization of high-fidelity entangling two-qubit gates 11-16 is much more demanding because the inter-ion distance is orders of magnitude bigger than the characteristic length scale of any state-dependent ion-ion interaction. Apart from quantum gates of the Cirac-Zoller type 2,12 , where a laser couples a single qubit with a vibrational mode of the ion string at a time, most other gate realizations entangling ions have relied on collective interactions of the qubits with the laser control fields 11,[13][14][15] . These gate operations entangle transiently the collective pseudospin of the qubits with the vibrational mode and produce either a conditional phase shift 17 or a collective spin flip 9,10,18 of the qubits. Whereas the highest fidelity F = 97% reported until now 13 has been achieved with a conditional phase gate acting on a pair of hyperfine qubits in 9 Be + , spin-flip gates have been limited so far to F ≈ 85% (refs 11,14). All of these experiments have used qubits encoded in hyperfine or Zeeman ground states and a Raman transition mediated by an electric-dipole transition for coupling the qubits. Whereas spontaneous scattering from the mediating short-lived levels degrades the gate fidelity owing to the limited amount of laser power available in current experi...

The question of whether quantum phenomena can be explained by classical models with hidden variables is the subject of a long lasting debate [1]. In 1964, Bell showed that certain types of classical models cannot explain the quantum mechanical predictions for specific states of distant particles [2]. Along this line, some types of hidden variable models have been experimentally ruled out [3,4,5,6,7,8,9]. An intuitive feature for classical models is non-contextuality: the property that any measurement has a value which is independent of other compatible measurements being carried out at the same time. However, the results of Kochen, Specker, and Bell[10,11,12] show that non-contextuality is in conflict with quantum mechanics. The conflict resides in the structure of the theory and is independent of the properties of special states. It has been debated whether the Kochen-Specker theorem could be experimentally tested at all [13,14]. Only recently, first tests of quantum contextuality have been proposed and undertaken with photons [15] and neutrons [16,17]. Yet these tests required the generation of special quantum states and left various loopholes open. Here, using trapped ions, we experimentally demonstrate a state-independent conflict with non-contextuality. The experiment is not subject to the detection loophole and we show that, despite imperfections and possible measurement disturbances, our results cannot be explained in non-contextual terms. PACS numbers:Hidden variable models assert that the result v(A) of measuring the observable A on an individual quantum system is predetermined by a hidden variable λ. Two observables A and B are mutually compatible, if the result of A does not depend on whether B is measured before, after, or simultaneously with A and vice versa. Non-contextuality is the property of a hidden variable model that the value v(A) is determined, regardless of which other compatible observable is measured jointly with A. As a consequence, for compatible observables the relation v(AB) = v(A)v(B) holds. Kochen and Specker showed that the assumption of noncontextuality cannot be reconciled with quantum mechanics. A considerable simplification of the original Kochen-Specker argument by Mermin and Peres [18,19] uses a 3 × 3 square of observables A ij with possible outcomes v(A ij ) = ±1, where the observables in each row or column are mutually compatible. Considering the products of rows, the total product would be k=1,2,3 R k C k = 1, since any v(A ij ) appears twice in the total product.In quantum mechanics, however, one can take a fourlevel quantum system, for instance two spin-1 2 -particles, * Electronic address: christian.roos@uibk.ac.at and the following array of observables,(1) Here, σ (k) i denotes the Pauli matrix acting on the k-th particle, and all the observables have the outcomes ±1. Moreover, in each of the rows or columns of (1), the observables are mutually commuting and can be measured simultaneously or in any order. In any row or column, their measurement product R k or C k equa...