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...

We report high-fidelity laser-beam-induced quantum logic gates on magnetic-field-insensitive qubits comprised of hyperfine states in 9 Be + ions with a memory coherence time of more than 1 s. We demonstrate single-qubit gates with error per gate of 3.8(1) × 10 −5 . By creating a Bell state with a deterministic two-qubit gate, we deduce a gate error of 8(4) × 10 −4 . We characterize the errors in our implementation and discuss methods to further reduce imperfections towards values that are compatible with fault-tolerant processing at realistic overhead.Quantum computers can solve certain problems that are thought to be intractable on conventional computers. An important general goal is to realize universal quantum information processing (QIP), which could be used for algorithms having a quantum advantage over processing with conventional bits as well as to simulate other quantum systems of interest [1][2][3]. For large problems, it is generally agreed that individual logic gate errors must be reduced below a certain threshold, often taken to be around 10 −4 [4][5][6], to achieve fault tolerance without excessive overhead in the number of physical qubits required to implement a logical qubit. This level has been achieved in some experiments for all elementary operations including state preparation and readout, with the exception of two-qubit gates, emphasizing the importance of improving multi-qubit gate fidelities. [7,8]. As various ions differ in mass, electronic, and hyperfine structure, they each have technical advantages and disadvantages. For example, 9 Be + is the lightest ion currently considered for QIP, and as such, has several potential advantages. The relatively light mass yields deeper traps and higher motional frequencies for given applied potentials, and facilitates fast ion transport [9,10]. Light mass also yields stronger laser-induced effective spin-spin coupling (inversely proportional to the mass), which can yield less spontaneous emission error for a given laser intensity [11]. However, a disadvantage of 9 Be + ion qubits compared to some heavier ions such as 40 Ca + and 43 Ca + [12, 13] has been the difficulty of producing and controlling the ultraviolet (313 nm) light required to drive 9 Be + stimulated-Raman transitions. In the work reported here, we use an ion trap array designed for scalable QIP [14] and take advantage of recent technological developments with lasers and optical fibers that improve beam quality and pointing stability. We also implement active control of laser pulse intensities to re- duce errors. We demonstrate laser-induced single-qubit computational gate errors of 3.8(1) × 10 −5 and realize a deterministic two-qubit gate to ideally produce the Bell state |Φ + = 1 √ 2 (|↑↑ + |↓↓ ). By characterizing the effects of known error sources with numerical simulations and calibration measurements, we deduce an entangling gate infidelity or error of = 8(4) × 10 −4 , where = 1 -F, and F is the fidelity. Along with Ref.[13]; these appear to be the highest two-qubit gate fidelitie...

We demonstrate spectroscopy and thermometry of individual motional modes in a mesoscopic 2D ion array using entanglement-induced decoherence as a method of transduction. Our system is a ∼400 µm-diameter planar crystal of several hundred 9 Be + ions exhibiting complex drumhead modes in the confining potential of a Penning trap. Exploiting precise control over the 9 Be + valence electron spins, we apply a homogeneous spin-dependent optical dipole force to excite arbitrary transverse modes with an effective wavelength approaching the interparticle spacing (∼20 µm). Center-of-mass displacements below 1 nm are detected via entanglement of spin and motional degrees of freedom. Studies of quantum physics at the interface of microscopic and mesoscopic regimes have recently focused on the observation of quantum coherent phenomena in optomechanical systems [1][2][3]. The realization of quantum coherence in mechanical oscillations involving many particles behaving approximately as a continuum provides exciting insights into the quantum-classical transition. Previous work has shown that crystals of cold, trapped ions behave as atomic-scale nanomechanical oscillators [4][5][6], with the benefits of in-situ tunable motional modes and exploitable single-particle quantum degrees of freedom (e.g. valence electron spin). Our system of hundreds of crystallized ions in a Penning trap provides a bottom-up approach to studying mesoscopic quantum coherence. In this context, the relevant particle numbers are sufficiently small to permit excellent quantum control without sacrificing continuum mechanical features. Beyond these capabilites, trapped ions have long provided a laboratory platform for studying diverse physical phenomena including: strongly-coupled one-component plasmas (OCPs) [7,8]; quantum computation [9,10] and simulation [11][12][13][14][15]; dynamical decoupling [16]; and atomic clocks and precision measurement [17].In this Letter, we present an experimental and theoretical study of motional drumhead modes in a 2D crystal of 9 Be + ions confined within a Penning trap. We excite inhomogeneous modes of arbitrary wavelength (see Fig. 1(a)) through application of a homogeneous, spinstate-dependent optical dipole force (ODF) to a largescale spin superposition. Distinct drumhead modes are entangled with the 9 Be + valence electron spins by tuning a beat frequency (µ R ) between two ODF lasers near a mode resonance. This spin-motion entanglement is de-

Publisher's Note: Phonon-mediated quantum spin simulator employing a planar ionic crystal in a Penning trap [Phys. Rev. A 87, 013422 (2013)]