Verifiable Blind Quantum Computing with Trapped Ions and Single Photons.

We report the first hybrid matter-photon implementation of verifiable blind quantum computing. We use a trapped-ion quantum server and a client-side photonic detection system networked via a fiber-optic quantum link. The availability of memory qubits and deterministic entangling gates enables interactive protocols without postselection-key requirements for any scalable blind server, which previous realizations could not provide. We quantify the privacy at ≲0.03 leaked classical bits per qubit. This experiment demonstrates a path to fully verified quantum computing in the cloud.

Breaking the Entangling Gate Speed Limit for Trapped-Ion Qubits Using a Phase-Stable Standing Wave.

All laser-driven entangling operations for trapped-ion qubits have hitherto been performed without control of the optical phase of the light field, which precludes independent tuning of the carrier and motional coupling. By placing ^{88}Sr^{+} ions in a λ=674 nm standing wave, whose relative position is controlled to ≈λ/100, we suppress the carrier coupling by a factor of 18, while coherently enhancing the spin-motion coupling. We experimentally demonstrate that the off-resonant carrier coupling imposes a speed limit for conventional traveling-wave Mølmer-Sørensen gates; we use the standing wave to surpass this limit and achieve a gate duration of 15 µs, restricted by the available laser power.

Coherent Control of Trapped-Ion Qubits with Localized Electric Fields.

We present a new method for coherent control of trapped ion qubits in separate interaction regions of a multizone trap by simultaneously applying an electric field and a spin-dependent gradient. Both the phase and amplitude of the effective single-qubit rotation depend on the electric field, which can be localized to each zone. We demonstrate this interaction on a single ion using both laser-based and magnetic-field gradients in a surface-electrode ion trap, and measure the localization of the electric field.

Robust Quantum Memory in a Trapped-Ion Quantum Network Node.

We integrate a long-lived memory qubit into a mixed-species trapped-ion quantum network node. Ion-photon entanglement first generated with a network qubit in ^{88}Sr^{+} is transferred to ^{43}Ca^{+} with 0.977(7) fidelity, and mapped to a robust memory qubit. We then entangle the network qubit with a second photon, without affecting the memory qubit. We perform quantum state tomography to show that the fidelity of ion-photon entanglement decays ∼70 times slower on the memory qubit. Dynamical decoupling further extends the storage duration; we measure an ion-photon entanglement fidelity of 0.81(4) after 10 s.

An elementary quantum network of entangled optical atomic clocks.

Optical atomic clocks are our most precise tools to measure time and frequency1-3. Precision frequency comparisons between clocks in separate locations enable one to probe the space-time variation of fundamental constants4,5 and the properties of dark matter6,7, to perform geodesy8-10 and to evaluate systematic clock shifts. Measurements on independent systems are limited by the standard quantum limit; measurements on entangled systems can surpass the standard quantum limit to reach the ultimate precision allowed by quantum theory-the Heisenberg limit. Although local entangling operations have demonstrated this enhancement at microscopic distances11-16, comparisons between remote atomic clocks require the rapid generation of high-fidelity entanglement between systems that have no intrinsic interactions. Here we report the use of a photonic link17,18 to entangle two 88Sr+ ions separated by a macroscopic distance19 (approximately 2 m) to demonstrate an elementary quantum network of entangled optical clocks. For frequency comparisons between the ions, we find that entanglement reduces the measurement uncertainty by nearly [Formula: see text], the value predicted for the Heisenberg limit. Today's optical clocks are typically limited by dephasing of the probe laser20; in this regime, we find that entanglement yields a factor of 2 reduction in the measurement uncertainty compared with conventional correlation spectroscopy techniques20-22. We demonstrate this enhancement for the measurement of a frequency shift applied to one of the clocks. This two-node network could be extended to additional nodes23, to other species of trapped particles or-through local operations-to larger entangled systems.

Experimental quantum key distribution certified by Bell's theorem.

Cryptographic key exchange protocols traditionally rely on computational conjectures such as the hardness of prime factorization1 to provide security against eavesdropping attacks. Remarkably, quantum key distribution protocols such as the Bennett-Brassard scheme2 provide information-theoretic security against such attacks, a much stronger form of security unreachable by classical means. However, quantum protocols realized so far are subject to a new class of attacks exploiting a mismatch between the quantum states or measurements implemented and their theoretical modelling, as demonstrated in numerous experiments3-6. Here we present the experimental realization of a complete quantum key distribution protocol immune to these vulnerabilities, following Ekert's pioneering proposal7 to use entanglement to bound an adversary's information from Bell's theorem8. By combining theoretical developments with an improved optical fibre link generating entanglement between two trapped-ion qubits, we obtain 95,628 key bits with device-independent security9-12 from 1.5 million Bell pairs created during eight hours of run time. We take steps to ensure that information on the measurement results is inaccessible to an eavesdropper. These measurements are performed without space-like separation. Our result shows that provably secure cryptography under general assumptions is possible with real-world devices, and paves the way for further quantum information applications based on the device-independence principle.

Benchmarking a High-Fidelity Mixed-Species Entangling Gate.

We implement a two-qubit logic gate between a ^{43}Ca^{+} hyperfine qubit and a ^{88}Sr^{+} Zeeman qubit. For this pair of ion species, the S-P optical transitions are close enough that a single laser of wavelength 402 nm can be used to drive the gate but sufficiently well separated to give good spectral isolation and low photon scattering errors. We characterize the gate by full randomized benchmarking, gate set tomography, and Bell state analysis. The latter method gives a fidelity of 99.8(1)%, comparable to that of the best same-species gates and consistent with known sources of error.

High-Rate, High-Fidelity Entanglement of Qubits Across an Elementary Quantum Network.

We demonstrate remote entanglement of trapped-ion qubits via a quantum-optical fiber link with fidelity and rate approaching those of local operations. Two ^{88}Sr^{+} qubits are entangled via the polarization degree of freedom of two spontaneously emitted 422 nm photons which are coupled by high-numerical-aperture lenses into single-mode optical fibers and interfere on a beam splitter. A novel geometry allows high-efficiency photon collection while maintaining unit fidelity for ion-photon entanglement. We generate heralded Bell pairs with fidelity 94% at an average rate 182 s^{-1} (success probability 2.18×10^{-4}).

Probing Qubit Memory Errors at the Part-per-Million Level.

Robust qubit memory is essential for quantum computing, both for near-term devices operating without error correction, and for the long-term goal of a fault-tolerant processor. We directly measure the memory error ε_{m} for a ^{43}Ca^{+} trapped-ion qubit in the small-error regime and find ε_{m}<10^{-4} for storage times t≲50 ms. This exceeds gate or measurement times by three orders of magnitude. Using randomized benchmarking, at t=1 ms we measure ε_{m}=1.2(7)×10^{-6}, around ten times smaller than that extrapolated from the T_{2}^{*} time, and limited by instability of the atomic clock reference used to benchmark the qubit.

Magnetic field stabilization system for atomic physics experiments.

Atomic physics experiments commonly use millitesla-scale magnetic fields to provide a quantization axis. As atomic transition frequencies depend on the magnitude of this field, many experiments require a stable absolute field. Most setups use electromagnets, which require a power supply stability not usually met by commercially available units. We demonstrate the stabilization of a field of 14.6 mT to 4.3 nT rms noise (0.29 ppm), compared to noise of >100 nT without any stabilization. The rms noise is measured using a field-dependent hyperfine transition in a single 43Ca+ ion held in a Paul trap at the center of the magnetic field coils. For the 43Ca+ "atomic clock" qubit transition at 14.6 mT, which depends on the field only in second order, this would yield a projected coherence time of many hours. Our system consists of a feedback loop and a feedforward circuit that control the current through the field coils and could easily be adapted to other field amplitudes, making it suitable for other applications such as neutral atom traps.

Recommended electron-impact excitation and ionization cross sections for Be I.

Analytic fits to the recommended electron-impact excitation and ionization cross sections for Be I are presented. The lowest 19 terms of configurations 2snl (n ≤ 4) and 2p 2 terms below the first ionization limit are considered. The fits are based on the accurate calculations with the convergent close coupling (CCC) method as well as the B-spline R-matrix (BSR) approach. The fitted cross sections provide rate coefficients that are believed to approximate the original data within 10% with very few exceptions. The oscillator strengths for the dipole-allowed transitions between all the considered states are calculated with the relativistic multi-configuration Dirac-Hartree-Fock (MCDHF) approach and compared with the CCC and BSR results. This comparison shows a very good agreement except for a handful of cases with likely strong cancellations.

A short response time atomic source for trapped ion experiments.

Ion traps are often loaded from atomic beams produced by resistively heated ovens. We demonstrate an atomic oven which has been designed for fast control of the atomic flux density and reproducible construction. We study the limiting time constants of the system and, in tests with 40Ca, show that we can reach the desired level of flux in 12 s, with no overshoot. Our results indicate that it may be possible to achieve an even faster response by applying an appropriate one-off heat treatment to the oven before it is used.

Fast quantum logic gates with trapped-ion qubits.

Quantum bits (qubits) based on individual trapped atomic ions are a promising technology for building a quantum computer. The elementary operations necessary to do so have been achieved with the required precision for some error-correction schemes. However, the essential two-qubit logic gate that is used to generate quantum entanglement has hitherto always been performed in an adiabatic regime (in which the gate is slow compared with the characteristic motional frequencies of the ions in the trap), resulting in logic speeds of the order of 10 kilohertz. There have been numerous proposals of methods for performing gates faster than this natural 'speed limit' of the trap. Here we implement one such method, which uses amplitude-shaped laser pulses to drive the motion of the ions along trajectories designed so that the gate operation is insensitive to the optical phase of the pulses. This enables fast (megahertz-rate) quantum logic that is robust to fluctuations in the optical phase, which would otherwise be an important source of experimental error. We demonstrate entanglement generation for gate times as short as 480 nanoseconds-less than a single oscillation period of an ion in the trap and eight orders of magnitude shorter than the memory coherence time measured in similar calcium-43 hyperfine qubits. The power of the method is most evident at intermediate timescales, at which it yields a gate error more than ten times lower than can be attained using conventional techniques; for example, we achieve a 1.6-microsecond-duration gate with a fidelity of 99.8 per cent. Faster and higher-fidelity gates are possible at the cost of greater laser intensity. The method requires only a single amplitude-shaped pulse and one pair of beams derived from a continuous-wave laser. It offers the prospect of combining the unrivalled coherence properties, operation fidelities and optical connectivity of trapped-ion qubits with the submicrosecond logic speeds that are usually associated with solid-state devices.

High-Fidelity Trapped-Ion Quantum Logic Using Near-Field Microwaves.

We demonstrate a two-qubit logic gate driven by near-field microwaves in a room-temperature microfabricated surface ion trap. We introduce a dynamically decoupled gate method, which stabilizes the qubits against fluctuating energy shifts and avoids the need to null the microwave field. We use the gate to produce a Bell state with fidelity 99.7(1)%, after accounting for state preparation and measurement errors. The gate is applied directly to ^{43}Ca^{+} hyperfine "atomic clock" qubits (coherence time T_{2}^{*}≈50 s) using the oscillating magnetic field gradient produced by an integrated microwave electrode.

High-Fidelity Quantum Logic Gates Using Trapped-Ion Hyperfine Qubits.

We demonstrate laser-driven two-qubit and single-qubit logic gates with respective fidelities 99.9(1)% and 99.9934(3)%, significantly above the ≈99% minimum threshold level required for fault-tolerant quantum computation, using qubits stored in hyperfine ground states of calcium-43 ions held in a room-temperature trap. We study the speed-fidelity trade-off for the two-qubit gate, for gate times between 3.8 µs and 520 µs, and develop a theoretical error model which is consistent with the data and which allows us to identify the principal technical sources of infidelity.

Hybrid quantum logic and a test of Bell's inequality using two different atomic isotopes.

Entanglement is one of the most fundamental properties of quantum mechanics, and is the key resource for quantum information processing (QIP). Bipartite entangled states of identical particles have been generated and studied in several experiments, and post-selected or heralded entangled states involving pairs of photons, single photons and single atoms, or different nuclei in the solid state, have also been produced. Here we use a deterministic quantum logic gate to generate a 'hybrid' entangled state of two trapped-ion qubits held in different isotopes of calcium, perform full tomography of the state produced, and make a test of Bell's inequality with non-identical atoms. We use a laser-driven two-qubit gate, whose mechanism is insensitive to the qubits' energy splittings, to produce a maximally entangled state of one (40)Ca(+) qubit and one (43)Ca(+) qubit, held 3.5 micrometres apart in the same ion trap, with 99.8 ± 0.6 per cent fidelity. We test the CHSH (Clauser-Horne-Shimony-Holt) version of Bell's inequality for this novel entangled state and find that it is violated by 15 standard deviations; in this test, we close the detection loophole but not the locality loophole. Mixed-species quantum logic is a powerful technique for the construction of a quantum computer based on trapped ions, as it allows protection of memory qubits while other qubits undergo logic operations or are used as photonic interfaces to other processing units. The entangling gate mechanism used here can also be applied to qubits stored in different atomic elements; this would allow both memory and logic gate errors caused by photon scattering to be reduced below the levels required for fault-tolerant quantum error correction, which is an essential prerequisite for general-purpose quantum computing.

Optical injection and spectral filtering of high-power ultraviolet laser diodes.

We demonstrate injection locking of high-power laser diodes operating at 397 nm. We achieve stable operation with an injection power of ∼100 µW and a slave laser output power of up to 110 mW. We investigate the spectral purity of the slave laser light via photon scattering experiments on a single trapped (40)Ca(+) ion. We show that it is possible to achieve a scattering rate indistinguishable from that of monochromatic light by filtering the laser light with a diffraction grating to remove amplified spontaneous emission.

High-Fidelity Preparation, Gates, Memory, and Readout of a Trapped-Ion Quantum Bit.

We implement all single-qubit operations with fidelities significantly above the minimum threshold required for fault-tolerant quantum computing, using a trapped-ion qubit stored in hyperfine "atomic clock" states of ^{43}Ca^{+}. We measure a combined qubit state preparation and single-shot readout fidelity of 99.93%, a memory coherence time of T_{2}^{*}=50 sec, and an average single-qubit gate fidelity of 99.9999%. These results are achieved in a room-temperature microfabricated surface trap, without the use of magnetic field shielding or dynamic decoupling techniques to overcome technical noise.

Injection locking of two frequency-doubled lasers with 3.2 GHz offset for driving Raman transitions with low photon scattering in 43Ca+.

We describe the injection locking of two infrared (794 nm) laser diodes that are each part of a frequency-doubled laser system. An acousto-optic modulator in the injection path gives an offset of 1.6 GHz between the lasers for driving Raman transitions between states in the hyperfine split (by 3.2 GHz) ground level of 43Ca+. The offset can be disabled for use in 40Ca+. We measure the relative linewidth of the frequency-doubled beams to be 42 mHz in an optical heterodyne measurement. The use of both injection locking and frequency doubling combines spectral purity with high optical power. Our scheme is applicable for providing Raman beams across other ion species and neutral atoms where coherent optical manipulation is required.

Contribution of near threshold states to recombination in plasmas.

Resonance states in atoms or ions at low energies can control the rates of important plasma processes (e.g., dielectronic recombination). We examine the role of states at negative energies just below the ionization threshold of the recombined system and find that they can contribute as much, or more, to recombination as positive energy states. In plasmas, negative energy states can be populated by three body recombination, photorecombination, or continuum lowering. Properly including these negative energy states in a theoretical treatment of plasma processes can change the thermally averaged rate coefficients and, in some cases, removes much of the sensitivity to the energy of a state.