C. J. Ballance

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.The generation of entanglement is a fundamental resource for quantum technology, and trapped ions are one of the most promising systems for storage and manipulation of quantum information. Here we study the speed/fidelity trade-off for a two-qubit phase gate implemented in Ca hyperfine trapped-ion qubits. We characterize various error sources contributing to the measured fidelity, allowing us to account for errors due to single-qubit state preparation, rotation and measurement (each at the ∼ 0.1% level), and to identify the leading sources of error in the two-qubit entangling operation. We achieve gate fidelities ranging between 97.1(2)% (for a gate time tg = 3.8μs) and 99.9(1)% (for tg = 100μs), representing respectively the fastest and lowest-error two-qubit gates reported between trapped-ion qubits by nearly an order of magnitude in each case. We perform a two-qubit geometric phase gate in the σz basis [1], where the qubits are stored in the S 4,+4 1/2 and S 1/2 states of the ground hyperfine manifold of Ca. The two-qubit gate operation is implemented by a pair of Raman laser beams at a detuning ∆ from the 4S1/2 ↔ 4P1/2 transition. To vary tg we adjust ∆ while holding the Raman beam intensity constant (at 5 mW per beam in a spot size of w = 27μm); smaller ∆ enables a faster gate, at the cost of increased error due to photon scattering [2]. The Raman difference frequency is δ = νz + δg where δg = 2/tg and the axial trap frequency is νz = 1.95 MHz. The Raman beams propagate at 45◦ to the trap z-axis, such that their wave-vector difference is along z. We cool both axial modes of the ions close to the ground state of motion by Raman sideband cooling; the centre-of-mass mode, rather than the stretch mode, is used to implement the gate to avoid coupling to the (uncooled) radial modes of the trap [3].

Reduction of heating rate in a microfabricated ion trap by pulsed-laser cleaning

Laser cleaning of the electrodes in a planar micro-fabricated ion trap has been attempted using ns pulses from a tripled Nd:YAG laser at 355?nm. The effect of the laser pulses at several energy density levels has been tested by measuring the heating rate of a single 40Ca+ trapped ion as a function of its secular frequency ?z. A reduction of the electric-field noise spectral density by ?50% has been observed and a change in the frequency dependence also noticed. This is the first reported experiment where the ?anomalous heating? phenomenon has been reduced by removing the source as opposed to reducing its thermal driving by cryogenic cooling. This technique may open up the way to better control of the electrode surface quality in ion microtraps.

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 40Ca+ qubit and one 43Ca+ 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.

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.

Microwave control electrodes for scalable, parallel, single-qubit operations in a surface-electrode ion trap

We propose a surface ion trap design incorporating microwave control electrodes for near-field single-qubit control. The electrodes are arranged so as to provide arbitrary frequency, amplitude and polarization control of the microwave field in one trap zone, whilst a similar set of electrodes is used to null the residual microwave field in a neighbouring zone. The geometry is chosen to reduce the residual field to the 0.5 % level without nulling fields; with nulling, the crosstalk may be kept close to the 0.01 % level for realistic microwave amplitude and phase drift. Using standard photolithography and electroplating techniques, we have fabricated a proof-of-principle electrode array with two trapping zones. We discuss requirements for the microwave drive system and prospects for scalability to a large 2-D trap array.

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.

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.

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.

Experimental Implementations of Two-Qubit Gates

In this chapter we present the results of our two-qubit gate experiments. We start by discussing how we define gate fidelity and the experimental signals we measure. We present preliminary experiments of a light-shift gate on a pair of \(^{40}\mathrm{Ca}^{+}\) qubits.

Trapped-Ion Qubits

In this chapter we give an overview of how one can use a calcium ion as a qubit. We then review the operation of a Paul trap, and discuss the quantised behaviour of a ‘crystal’ of trapped ions near their motional ground state. The shared motional degrees of freedom of such a crystal allow the implementation of quantum ‘logic gates’, i.e. multi-qubit entangling operations.