Quantum computers can solve certain problems more efficiently than any possible conventional computer. Small quantum algorithms have been demonstrated on multiple quantum computing platforms, many specifically tailored in hardware to implement a particular algorithm or execute a limited number of computational paths. Here we demonstrate a five-qubit trapped-ion quantum computer that can be programmed in software to implement arbitrary quantum algorithms by executing any sequence of universal quantum logic gates. We compile algorithms into a fully connected set of gate operations that are native to the hardware and have a mean fidelity of 98 per cent. Reconfiguring these gate sequences provides the flexibility to implement a variety of algorithms without altering the hardware. As examples, we implement the Deutsch-Jozsa and Bernstein-Vazirani algorithms with average success rates of 95 and 90 per cent, respectively. We also perform a coherent quantum Fourier transform on five trapped-ion qubits for phase estimation and period finding with average fidelities of 62 and 84 per cent, respectively. This small quantum computer can be scaled to larger numbers of qubits within a single register, and can be further expanded by connecting several such modules through ion shuttling or photonic quantum channels.

The field of quantum computing has grown from concept to demonstration devices over the past 20 years. Universal quantum computing offers efficiency in approaching problems of scientific and commercial interest, such as factoring large numbers, searching databases, simulating intractable models from quantum physics, and optimizing complex cost functions. Here, we present an 11-qubit fully-connected, programmable quantum computer in a trapped ion system composed of 13 171Yb+ ions. We demonstrate average single-qubit gate fidelities of 99.5, average two-qubit-gate fidelities of 97.5, and SPAM errors of 0.7. To illustrate the capabilities of this universal platform and provide a basis for comparison with similarly-sized devices, we compile the Bernstein-Vazirani and Hidden Shift algorithms into our native gates and execute them on the hardware with average success rates of 78 and 35, respectively. These algorithms serve as excellent benchmarks for any type of quantum hardware, and show that our system outperforms all other currently available hardware.

We run a selection of algorithms on two state-of-the-art 5-qubit quantum computers that are based on different technology platforms. One is a publicly accessible superconducting transmon device (www.research.ibm.com/ibm-q) with limited connectivity, and the other is a fully connected trapped-ion system. Even though the two systems have different native quantum interactions, both can be programed in a way that is blind to the underlying hardware, thus allowing a comparison of identical quantum algorithms between different physical systems. We show that quantum algorithms and circuits that use more connectivity clearly benefit from a better-connected system of qubits. Although the quantum systems here are not yet large enough to eclipse classical computers, this experiment exposes critical factors of scaling quantum computers, such as qubit connectivity and gate expressivity. In addition, the results suggest that codesigning particular quantum applications with the hardware itself will be paramount in successfully using quantum computers in the future.quantum computing | quantum information | quantum information science | quantum physics | quantum computing architecture I nspired by the vast computing power a universal quantum computer could offer, several candidate systems are being explored. They have allowed experimental demonstrations of quantum gates, operations, and algorithms of ever-increasing sophistication. Recently, two architectures, superconducting transmon qubits (1-5) and trapped ions (6, 7), have reached a new level of maturity. They have become fully programmable multiqubit machines that provide the user with the flexibility to implement arbitrary quantum circuits from a high-level interface. This makes it possible for the first time to test quantum computers irrespective of their particular physical implementation.Whereas the quantum computers considered here are still small scale and their capabilities do not currently reach beyond small demonstration algorithms, this line of inquiry can still provide useful insights into the performance of existing systems and the role of architecture in quantum computer design. These will be crucial for the realization of more advanced future incarnations of the present technologies.The standard abstract model of quantum computation assumes that interactions between arbitrary pairs of qubits are available. However, physical architectures will in general have certain constraints on qubit connectivity, such as nearest-neighbor couplings only. These restrictions do not in principle limit the ability to perform arbitrary computations, because swap operations may be used to effect gates between arbitrary qubits using the connections available. For a general circuit, reducing a fully connected system to the more sparse star-shaped or linear nearestneighbor connectivity requires an increase in the number of gates of O(n), where n is the number of qubits (8). How much overhead is incurred in practice depends on the connections used in a particular circuit and how effi...

PACS numbers:Quantum entanglement is the central resource behind applications in quantum information science, from quantum computers [1] and simulators of complex quantum systems [2] to metrology [3] and secure communication [1]. All of these applications require the quantum control of large networks of quantum bits (qubits) to realize gains and speedups over conventional devices. However, propagating quantum entanglement generally becomes difficult or impossible as the system grows in size, owing to the inevitable decoherence from the complexity of connections between the qubits and increased couplings to the environment. Here, we demonstrate the first step in a modular approach [4] to scaling entanglement by utilizing a hierarchy of quantum buses on a collection of three atomic ion qubits stored in two remote ion trap modules. Entanglement within a module is achieved with deterministic near-field interactions through phonons [5], and remote entanglement between modules is achieved through a probabilistic interaction through photons [6]. This minimal system allows us to address generic issues in synchronization and scalability of entanglement with multiple buses, while pointing the way toward a modular large-scale quantum computer architecture that promises less spectral crowding and less decoherence. We generate this modular entanglement faster than the observed qubit decoherence rate, thus the system can be scaled to much larger dimensions by adding more modules.Small modules of qubits have been entangled through native local interactions in many physical platforms, such as trapped atomic ions through their Coulomb interaction [5], Rydberg atoms through their electric dipoles [7,8], nitrogen-vacancy centers in diamond through their magnetic dipoles [9], and superconducting Josephson junctions through capacitive or inductive couplings [10,11]. However, each of these systems is confronted with practical limits to the number of qubits that can be reliably controlled, stemming from inhomogeneities, the complexity and density of the interactions between the qubits, or quantum decoherence. Scaling beyond these limits can be achieved by invoking a second type of interaction that can extend the entanglement to other similar qubit modules. Such an architecture should therefore exploit both the local interactions within the qubit modules, and also remote interactions between modules (an example architecture is shown in Fig 1). Optical interfaces provide ideal buses for this purpose [12,13], as optical photons can propagate over macroscopic distances with negligible loss. Several qubit systems have been entangled through remote optical buses, such as atomic ions [14], neutral atoms [15], and nitrogen-vacancy centers in diamond [16].In the experiment reported here, we juxtapose local and remote entanglement buses utilizing trapped atomic ion qubits, balancing the requirements of each interface within the same qubit system. The observed entanglement rate within and between modules is faster than the observed entangled qub...

Quantum computing leverages the quantum resources of superposition and entanglement to efficiently solve computational problems considered intractable for classical computers. Examples include calculating molecular and nuclear structure, simulating strongly-interacting electron systems, and modeling aspects of material function. While substantial theoretical advances have been made in mapping these problems to quantum algorithms, there remains a large gap between the resource requirements for solving such problems and the capabilities of currently available quantum hardware. Bridging this gap will require a co-design approach, where the expression of algorithms is developed in conjunction with the hardware itself to optimize execution. Here, we describe a scalable co-design framework for solving chemistry problems on a trapped ion quantum computer, and apply it to compute the ground-state energy of the water molecule. The robust operation of the trapped ion quantum computer yields energy estimates with errors approaching the chemical accuracy, which is the target threshold necessary for predicting the rates of chemical reaction dynamics.

The Grover quantum search algorithm is a hallmark application of a quantum computer with a well-known speedup over classical searches of an unsorted database. Here, we report results for a complete three-qubit Grover search algorithm using the scalable quantum computing technology of trapped atomic ions, with better-than-classical performance. Two methods of state marking are used for the oracles: a phase-flip method employed by other experimental demonstrations, and a Boolean method requiring an ancilla qubit that is directly equivalent to the state marking scheme required to perform a classical search. We also report the deterministic implementation of a Toffoli-4 gate, which is used along with Toffoli-3 gates to construct the algorithms; these gates have process fidelities of 70.5% and 89.6%, respectively.

We show the fault-tolerant encoding, measurement, and operation of a logical qubit via quantum error detection.

We demonstrate high fidelity entangling quantum gates within a chain of five trapped ion qubits by optimally shaping optical fields that couple to multiple collective modes of motion. We individually address qubits with segmented optical pulses to construct multipartite entangled states in a programmable way. This approach enables both high fidelity and fast quantum gates that can be scaled to larger qubit registers for quantum computation and simulation.Trapped atomic ion crystals are the leading architecture for quantum information processing, with their unsurpassed level of qubit coherence and near perfect initialization and detection efficiency [1,2]. Moreover, trapped ion qubits can be controllably entangled through their Coulomb-coupled motion by applying external fields that provide a qubit state-dependent force [3][4][5][6]. However, scaling to large numbers of ions N within a single crystal is complicated by the many collective modes of motion, which can cause gate errors from mode crosstalk. Such errors can be mitigated by coupling to a single motional mode, at a cost of significantly slowing the gate operation. The gate time τ g must generally be much longer than the inverse of the frequency splitting of the motional modes, which for axial motion in a linear chain implies τ g 1/ω z > N 0.86 /ω x , where ω z and ω x are the center-of-mass axial and transverse mode frequencies [7]. For gates using transverse motion in a linear chainIn either case, the slowdown with qubit number N can severely limit the practical size of trapped ion qubit crystals.In this letter, we circumvent this scaling problem by applying qubit state-dependent optical forces that simultaneously couple to multiple modes of motion. We address subsets of ions immersed in a five-ion linear crystal and engineer laser pulse shapes to entangle pairs of ions with high fidelity while suppressing mode crosstalk and maintaining short gate times [8,9]. The pre-calculated pulse shapes optimize target gate fidelity, achieving unity for sufficiently complex pulses. In the experiment, we concatenate these shaped gates to entangle multiple pairs of qubits and directly measure multi-qubit entanglement in the crystal. Extensions of this approach can be scaled to larger ion chains and also incorporate higher levels of pulse shaping to reduce sensitivity to particular experimental errors and drifts [10][11][12].In the experiment, five 171 Yb + ions are confined in a three-layer linear rf trap [13] with transverse centerof-mass (CM) frequency ranging from ω x /2π = 2.5 − 4.5 MHz and axial CM frequency ω z /2π = 310−550 kHz, with a minimal ion separation of ∼ 5 µm. Each qubit is represented by the 2 S 1/2 hyperfine "clock" states within 171 Yb + , denoted by |0 and |1 and having a splitting of ω 0 /2π = 12.642821 GHz [14]. We initialize each qubit by optically pumping to state |0 using laser light resonant with the 2 S 1/2 ↔ 2 P 1/2 transition near 369.5 nm. We then coherently manipulate the qubits with a mode-locked laser at 355 nm whose frequency comb be...