Universal logic gates for two quantum bits (qubits) form an essential ingredient of quantum computation. Dynamical gates have been proposed in the context of trapped ions; however, geometric phase gates (which change only the phase of the physical qubits) offer potential practical advantages because they have higher intrinsic resistance to certain small errors and might enable faster gate implementation. Here we demonstrate a universal geometric pi-phase gate between two beryllium ion-qubits, based on coherent displacements induced by an optical dipole force. The displacements depend on the internal atomic states; the motional state of the ions is unimportant provided that they remain in the regime in which the force can be considered constant over the extent of each ion's wave packet. By combining the gate with single-qubit rotations, we have prepared ions in an entangled Bell state with 97% fidelity-about six times better than in a previous experiment demonstrating a universal gate between two ion-qubits. The particular properties of the gate make it attractive for a multiplexed trap architecture that would enable scaling to large numbers of ion-qubits.

The presence of long-range quantum spin correlations underlies a variety of physical phenomena in condensed matter systems, potentially including high-temperature superconductivity [1,2]. However, many properties of exotic strongly correlated spin systems (e.g., spin liquids) have proved difficult to study, in part because calculations involving N-body entanglement become intractable for as few as N ∼ 30 particles [3]. Feynman divined that a quantum simulator -a special-purpose "analog" processor built using quantum particles (qubits) -would be inherently adept at such problems [4,5]. In the context of quantum magnetism, a number of experiments have demonstrated the feasibility of this approach [6][7][8][9][10][11][12][13][14]. However, simulations of quantum magnetism allowing controlled, tunable interactions between spins localized on 2D and 3D lattices of more than a few 10's of qubits have yet to be demonstrated, owing in part to the technical challenge of realizing large-scale qubit arrays. Here we demonstrate a variable-range Ising-type spin-spin interaction J i, j on a naturally occurring 2D triangular crystal lattice of hundreds of spin-1/2 particles ( 9 Be + ions stored in a Penning trap), a computationally relevant scale more than an order of magnitude larger than existing experiments. We show that a spindependent optical dipole force can produce an antiferromagnetic interaction J i, j ∝ d −a i, j , where a is tunable over 0 < a < 3; d i, j is the distance between spin pairs. These power-laws correspond physically to infinite-range (a = 0), Coulomb-like (a = 1), monopole-dipole (a = 2) and dipole-dipole (a = 3) couplings. Experimentally, we demonstrate excellent agreement with theory for 0.05 a 1.4. This demonstration coupled with the high spin-count, excellent quantum control and low technical complexity of the Penning trap brings within reach simulation of interesting and otherwise computationally intractable problems in quantum magnetism.A challenge in condensed matter physics is the fact that many quantum-magnetic interactions cannot currently be modeled in a meaningful way. A canonical example is the spin-liquid postulated by Anderson [1]. He suggested this exotic state would arise in a collection of spin-1/2 particles residing on a triangular lattice and coupled by a nearest-neighbor antiferromagnetic Heisenberg interaction. The spin-liquid's Figure 1. The Penning trap confines hundreds of spin-1/2 particles (qubits) on a two-dimensional (2D) triangular lattice. Each qubit is the valence electron spin of a 9 Be + ion. (lower) A Penning trap confines ions by use of a combination of static electric and magnetic fields. The trap parameters are configured so that laser-cooled ions form a triangular 2D crystal. A general spin-spin interactionĤ I is generated by a spin-dependent excitation of the transverse (alongẑ) motional modes of the ion crystal. This coupling is implemented with an optical dipole force (ODF) due to a pair of off-resonance laser beams (left side) with angular separation θ R and dif...

A key requirement for scalable quantum computing is that elementary quantum gates can be implemented with sufficiently low error. One method for determining the error behavior of a gate implementation is to perform process tomography. However, standard process tomography is limited by errors in state preparation, measurement and one-qubit gates. It suffers from inefficient scaling with number of qubits and does not detect adverse error-compounding when gates are composed in long sequences. An additional problem is due to the fact that desirable error probabilities for scalable quantum computing are of the order of 0.0001 or lower. Experimentally proving such low errors is challenging. We describe a randomized benchmarking method that yields estimates of the computationally relevant errors without relying on accurate state preparation and measurement. Since it involves long sequences of randomly chosen gates, it also verifies that error behavior is stable when used in long computations. We implemented randomized benchmarking on trapped atomic ion qubits, establishing a one-qubit error probability per randomized / 2 pulse of 0.00482͑17͒ in a particular experiment. We expect this error probability to be readily improved with straightforward technical modifications.

Among the classes of highly entangled states of multiple quantum systems, the so-called 'Schrödinger cat' states are particularly useful. Cat states are equal superpositions of two maximally different quantum states. They are a fundamental resource in fault-tolerant quantum computing and quantum communication, where they can enable protocols such as open-destination teleportation and secret sharing. They play a role in fundamental tests of quantum mechanics and enable improved signal-to-noise ratios in interferometry. Cat states are very sensitive to decoherence, and as a result their preparation is challenging and can serve as a demonstration of good quantum control. Here we report the creation of cat states of up to six atomic qubits. Each qubit's state space is defined by two hyperfine ground states of a beryllium ion; the cat state corresponds to an entangled equal superposition of all the atoms in one hyperfine state and all atoms in the other hyperfine state. In our experiments, the cat states are prepared in a three-step process, irrespective of the number of entangled atoms. Together with entangled states of a different class created in Innsbruck, this work represents the current state-of-the-art for large entangled states in any qubit system.

The precision in spectroscopy of any quantum system is fundamentally limited by the Heisenberg uncertainty relation for energy and time. For N systems, this limit requires that they be in a quantum-mechanically entangled state. We describe a scalable method of spectroscopy that can potentially take full advantage of entanglement to reach the Heisenberg limit and has the practical advantage that the spectroscopic information is transferred to states with optimal protection against readout noise. We demonstrate our method experimentally with three beryllium ions. The spectroscopic sensitivity attained is 1.45(2) times as high as that of a perfect experiment with three non-entangled particles.