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Quantum computing with trapped ions

H. Haeffner, C. F. Roos, R. Blatt

arXiv:0809.4368v1quant-ph

TL;DR

Quantum computing requires controllable qubits and operations that can scale beyond classical capabilities, motivating trapped-ion implementations. This review surveys ion-trap qubits, gates, experiments, algorithms, and outstanding challenges, reporting high-fidelity demonstrations alongside practical limits on tomography, scalability, and speed.

  • Problem

    Quantum computing needs physical systems that support fault-tolerant operations, but limited operation fidelity remains a major obstacle to repeated high-fidelity quantum error correction.

  • Method

    The paper reviews experimental trapped-ion quantum computing, covering qubits, gates, key experiments, quantum algorithms, tomography, and future challenges.

  • Results

    The review reports a 0.9994 detection efficiency and 99.3(1)% Bell-state fidelity in trapped-ion experiments.

  • Takeaways & Limitations

    Trapped ions have demonstrated core quantum-computing capabilities and milestones including teleportation, error correction, and high-fidelity entanglement.

  • Takeaways & Limitations

    Full quantum process tomography becomes impractical for large systems because its measurement requirements scale exponentially with qubit number.

Abstract

from arXiv · show

Quantum computers hold the promise to solve certain computational task much more efficiently than classical computers. We review the recent experimental advancements towards a quantum computer with trapped ions. In particular, various implementations of qubits, quantum gates and some key experiments are discussed. Furthermore, we review some implementations of quantum algorithms such as a deterministic teleportation of quantum information and an error correction scheme.

1 Introduction

The article reviews experimental ion-trap quantum computing, from physical requirements and competing approaches to milestones including teleportation and error correction.

  • Scope: The review presents a coherent overview of important experimental issues rather than covering every facet of the rapidly developing field.It refers readers to original publications for detailed treatments and only briefly touches on segmented-trap shuttling.
  • Motivation: Quantum computers use qubits and quantum mechanics to address mathematical and many-body problems that can challenge classical computers.A forty-qubit state requires 2^40 complex numbers to describe.
  • Ion-trap approach: Trapped ions encode quantum information in two-level systems manipulated and read out with focused laser beams.Cirac and Zoller proposed the platform, and a controlled bit flip on one ion was demonstrated within a year.
  • Milestones: Quantum error-correction protocols allow arbitrary-length quantum algorithms without perfect control.The article identifies teleportation of quantum states and quantum error correction as milestones in ion-trap computing.
  • Requirements: A universal quantum computer requires scalable, characterizable qubits, initialization, long coherence, universal gates, and qubit-specific measurement.The criteria also include interconversion and faithful transmission of stationary and flying qubits for quantum networks.

2 Ion trap quantum computers

By 2008, trapped-ion quantum computing had become one of the most advanced approaches, with demonstrations spanning gates, algorithms, entanglement, teleportation, error correction, and integrated traps.

  • Field development: More than 25 groups pursued experimental ion-trap quantum computing in 2008, compared with about six groups around 2000.The authors use this growth as evidence of rapid field development.
  • Core capabilities: Four of DiVincenzo’s five core criteria had been demonstrated in trapped ions before experimental quantum computation began.These included initialization, read-out, long coherence times, and laser-cooled ion crystals serving as registers.
  • Gate experiments: Experiments demonstrated conditioned phase shifts, multiple two-qubit gate candidates, entanglement of up to four ions, a decoherence-free subspace, and a nonlinear beam-splitter simulation.These milestones were achieved primarily by the NIST ion-storage group.
  • Algorithms and registers: The Innsbruck group demonstrated the Deutsch-Josza algorithm, universal gates on two ions, entangled states, and partial read-out of an entangled register.The demonstrations used a single Ca+ ion and a two-ion string.
  • Advanced experiments: Further milestones included quantum teleportation, error correction, entanglement of six and eight particles, entanglement purification, and ion-photon entanglement between separate traps.These experiments were reported by the NIST, Innsbruck, and Ann-Arbor groups.
  • Trap technology: Miniaturization and integration of segmented ion traps were rapidly progressing to address single-ion addressing and scalability challenges.Virtually all ion-trap groups had begun developing segmented-trap technologies.

2.1 Principles of ion-trap quantum computers

Ion-trap quantum computers use long-lived internal states as qubits, laser-driven gates, and collective motion as a quantum bus, while networking can map ionic states to photons.

  • Architecture: The review organizes ion-trap architectures around the DiVincenzo criteria for universal quantum computation.It evaluates scalability, initialization, coherence, gates, measurement, and networking capabilities.
  • Qubits and scalability: Long-lived internal ion levels form qubits in linear strings, while distributing ions among multiple traps can mitigate large-string complications.The approach is described as scalable in principle.
  • Initialization: Optical pumping initializes ions, with typical fidelities of 0.99 and methods for higher fidelities discussed.Initialization is the preparation of a well-defined electronic state.
  • Coherence: Coherence times in current experiments are typically a few milliseconds, about one to two orders of magnitude longer than quantum-operation timescales.More than 10 s was demonstrated with Raman transitions between magnetic-field-insensitive transitions, and more than 10 minutes with microwave drive.
  • Quantum gates: Single-qubit gates use laser-driven Rabi oscillations and phase-controlled Bloch-sphere rotations, while two-qubit gates commonly use Coulomb-mediated interactions and shared motion.The Cirac–Zoller scheme swaps information to collective motion, applies a conditioned operation, and swaps it back.
  • Networking: Ionic states can be mapped onto photons through a cavity and transmitted through optical fibers to connect distant ion traps.A second cavity can couple the photon to a target ion.

2.2 The basic Hamiltonian

The trapped-ion Hamiltonian describes a two-level ion coupled to a quantized motional mode, with carrier and sideband transitions selected by laser detuning.

  • Hamiltonian model: The model treats a trapped two-level ion interacting with near-resonant laser light and one vibrational mode along the trap axis.The formulation provides the starting point for analyzing laser-driven ion motion.
  • Parameters: The Lamb-Dicke parameter η = k_z z_0 characterizes coupling between the laser field and motional displacement.Here k_z is the laser-wavevector projection and z_0 is the ground-state wavefunction’s spatial extension.
  • Approximations: The rotating-wave approximation assumes laser detuning and Rabi frequency are much smaller than optical frequencies, with a second approximation selecting one transition at a time.The treatment also applies to Raman qubits by eliminating the virtual coupling level.
  • Carrier transition: At Δ = 0, the carrier transition changes only the ion’s electronic state without changing its motional state.The carrier is one of the three detunings emphasized in the model.
  • Sideband transitions: At Δ = ω_t, the blue sideband simultaneously excites the electronic state and creates a phonon; at Δ = −ω_t, the red sideband destroys one.The blue-sideband Rabi frequency Ω+ describes flopping between |g, 0⟩ and |e, 1⟩.
  • Speed limitations: For small η, strong fields used to accelerate sidebands induce AC-Stark shifts and off-resonant carrier excitation that can reduce gate fidelity.These effects can be partially canceled, but sideband flops much faster than ηω_t remain difficult.

2.3 Choice of qubit ions

Trapped-ion qubit choices balance long coherence relative to gate times, laser feasibility, accessible transitions, and susceptibility to magnetic-field fluctuations. Optical and radio-frequency encodings offer different trade-offs in coherence and technical demands.

  • A suitable trapped-ion qubit requires coherence times longer than manipulation times and technically feasible lasers.Typical gate operation times are 0.1−500µs.
  • Optical qubits use superpositions of electronic ground and metastable excited states, whereas radio-frequency qubits use superpositions within the electronic ground state.The Ca+ D5/2 state has a lifetime exceeding one second, while radio-frequency qubits can achieve even larger coherence times.
  • Optical qubits require laser linewidths below 200 mHz, while radio-frequency qubits generally operate below 10 GHz and need a more accessible stable phase reference.The optical requirement corresponds to fractional stability of about 10−15.
  • Radio-frequency qubits can be driven directly by microwaves or through Raman transitions, where the laser-frequency difference supplies the population-transfer energy.Only the frequency difference between the Raman laser fields is important.
  • Magnetic-field fluctuations often limit coherence because the qubit basis states have different magnetic moments and acquire additional phase evolution.Strategies to avoid these decoherence sources are discussed elsewhere in the paper.
  • Calcium, strontium, and ytterbium are attractive ions because of their relatively large wavelength transitions, while beryllium and magnesium offer large Lamb-Dicke factors due to lower atomic mass.Some NIST experiments report η ≈0.3 for beryllium and magnesium.

2.4 Initialization and read-out

Trapped-ion algorithms begin with optical pumping into a defined state and end with state-dependent fluorescence read-out. Frequency-selective pumping and auxiliary-ion or adaptive detection methods improve performance toward fault-tolerant requirements.

  • Optical pumping initializes atomic qubits by driving population into a state on which the drive no longer acts.The target state is typically occupied in less than 1 µs with probability greater than 0.99.
  • Fault-tolerant quantum computing appears to require initialization fidelities exceeding 0.9999, beyond what standard polarization-based pumping clearly achieves.Frequency selection offers an alternative when pure circular polarization along the quantization axis is unavailable.
  • In 40Ca+, frequency-selective pumping couples one S1/2 state through D5/2 and P3/2 while leaving the target S1/2 state untouched.The scheme uses a narrow-band laser and a broad-band 854 nm laser.
  • 10 µs pumping time constants were demonstrated in Innsbruck for the 40Ca+ frequency-selection procedure.
  • Within 1 ms, the authors estimate an initialization fidelity of 0.9999; Innsbruck experiments had already observed fidelities exceeding 0.999.The estimate assumes Rabi frequencies below 2π × 10 kHz to keep unwanted off-resonant excitations below 10−4.
  • Electron shelving reads out the register by coupling radiation to only one qubit level, producing fluorescence for one state but not the other.For 40Ca+, approximately 30 fluorescence photons per millisecond can be detected under typical conditions.
  • Auxiliary-qubit mapping and repeated adaptive detection improve state-detection fidelity when ancilla preparation and gate operations are sufficiently reliable.Myerson et al. achieved about 0.9999 efficiency in an average detection time of 145 µs for a single 40Ca+ qubit.

2.5 Single-qubit gates

Single-qubit gates in trapped ions are implemented through laser-driven Rabi oscillations, with pulse area and laser phase controlling Bloch-sphere rotations. High fidelities are routine, while addressing imperfections, motional effects, control electronics, and nearby transitions constrain speed and accuracy.

  • Gate implementation: Single-qubit operations use Rabi oscillations between qubit levels, represented mathematically as rotations RC(θ, ϕ) of the state vector.The pulse area θ = Ωτ and laser phase ϕ are the relevant control parameters.
  • Bloch-sphere control: The laser phase ϕ selects the rotation axis in the Bloch sphere’s equatorial plane, while θ specifies the rotation angle.Z-axis rotations can be synthesized from x- and y-axis rotations or produced through phase shifts and AC-Stark shifts.
  • Performance and limitations: Single-qubit manipulations in ion traps routinely achieve fidelities exceeding 0.99.Reported limitations include laser-intensity fluctuations, finite crystal temperature for optical qubits, and spontaneous emission for Raman transitions.
  • Individual addressing: A 2 µm FWHM laser waist distinguishes ions spaced by 5 µm, reducing adjacent-ion intensity by 1000-fold and yielding an unwanted Rabi frequency of 0.03 relative to the addressed ion.The beam angle is chosen as a compromise between spatial resolution and coupling to axial ion motion.
  • Performance and limitations: Increasing ion number tends to reduce the Lamb-Dicke factor, slowing two-qubit operations and potentially lowering fidelity through a more complex normal-mode spectrum.Single-qubit rotation speed is also limited by modulator chirps, available laser power, and unwanted nearby transitions.
  • Individual addressing: Composite pulses reduce addressing errors by splitting a single-qubit operation into several pulses whose effects cancel on unaddressed ions.For known imperfections, spatially inhomogeneous AC-Stark pulses can produce a π phase difference and tolerate addressing errors close to unity.

2.6 Two-qubit gates

Two-qubit gates use the ions’ Coulomb-coupled motion as a quantum bus to entangle qubits. The review contrasts gate requirements and reports experimental demonstrations of high-fidelity entanglement and multi-ion operations.

  • Two-qubit operations entangle ions and, together with single-qubit operations, enable arbitrary unitary operations.
  • The shared motional modes of an ion string mediate interactions between ionic qubits.For three ions, the axial modes include center-of-mass, breathing, and a third mode.
  • Ground-state cooling of the relevant bus mode is usually required for coherent sideband operations because sideband Rabi frequencies depend strongly on vibrational quantum number.
  • The Mølmer-Sørensen gate does not require individual addressing and does not fail completely without ground-state cooling.It was used to entangle up to four 9Be+ ions, create all four Bell states, and implement Grover’s search algorithm.
  • 99.3(1)% Bell-state fidelity was achieved experimentally with a Mølmer-Sørensen gate on optical qubits.

2.7 Apparative requirements

Trapped-ion quantum computing requires stable confinement, laser control, qubit preparation and readout, and protection from decoherence. The reviewed apparatus combines trap engineering, optical control, magnetic-field management, and pulse optimization, while pulse-area trade-offs remain.

  • Linear Paul traps confine ion chains with radio-frequency fields perpendicular to the axis and positive end-cap potentials along the axis.
  • Stable laser frequency and intensity control is provided with acousto-optical modulators and references to ultra-stable cavities or molecular transitions.Ultraviolet lasers are often required because ion energy splittings are relatively large.
  • The NIST apparatus encodes qubits in the 9Be+ ground-state hyperfine manifold and uses optical pumping, resolved-sideband cooling, Raman control, and state-dependent fluorescence readout.
  • Segmented microstructured traps support scaling by providing multiple trapping zones, easing addressing and enabling separate loading zones.
  • The Innsbruck setup uses optical qubits in 40Ca+ and normally cools the center-of-mass mode to the motional ground state.
  • Composite pulses and optimal control: Composite pulses can reduce control-parameter sensitivity and execution time, but their larger total pulse area can increase decoherence from spontaneous Raman-transition emission.

3 Decoherence in ion trap quantum computers

Decoherence in trapped-ion quantum computers arises from bit flips, phase fluctuations, and motional effects, with magnetic-field noise a major coherence constraint. Experiments mitigate these mechanisms through shielding, insensitive transitions, pulse sequences, and improved trap fabrication.

  • Phase coherence: Ramsey experiments use two π/2 pulses separated by a waiting time T to measure phase evolution through the resulting excitation probability and Ramsey contrast.Synchronizing experiments to the mains phase and optimizing implementations can reduce magnetic-field and control-related imperfections.
  • Magnetic-field fluctuations limit the coherence time of most qubits to a few milliseconds.
  • Shielding and active cancellation suppress magnetic-field fluctuations, while magnetic-field-insensitive transitions reduce dephasing at the qubit-design level.
  • Magnetic-field-independent hyperfine transitions have demonstrated coherence times τ > 10 s and memory error rates of approximately 10^-5.
  • Motional coherence: Heating from Johnson noise is expected to take τ ∼200 s/quantum for D = 100 µm, ReZ = 1 Ω, and room temperature.Measured heating rates depend strongly on electrode size, fabrication annealing, temperature, materials, and fabrication methods.

4 Key experiments

The review highlights trapped-ion demonstrations of quantum gates, deterministic entanglement, long-lived coherence, state characterization, and conditional operations. These experiments established important capabilities while revealing fidelity and scalability challenges.

  • Cirac-Zoller-type gates: CNOT fidelity increased from 0.73(2) to 0.91.0(6) after improvements to coherence time, addressing error, and computer control.The initial experiment used τ ∼800 µs and ǫ ∼0.05; later conditions reached τ ∼2 ms and ǫ ∼0.03.
  • Entangled states with trapped ions: Trapped ions enabled deterministic entanglement generation, producing states on demand without destroying them during creation.This distinguishes the experiments from photon-based generation and permits subsequent experiments with the entangled states.
  • Entangled states with trapped ions: GHZ states reached fidelities up to 0.89 and enabled phase estimation 1.45 times more accurately than with three uncorrelated particles.The improved phase estimation was obtained from generalized Ramsey fringes.
  • Entangled states with trapped ions: W states were created with four to eight ions, while a proposed faster scheme avoided individual addressing during entanglement.The original scheme’s pulse area grows logarithmically with ion number, so generation time grows sublinearly.
  • Decoherence free subspaces: Decoherence-free encodings achieved coherence times of 34 s and 7 s, with entanglement lasting up to 20 seconds between ions separated by 5 µm.Magnetic-field-gradient fluctuations were identified as the likely decoherence source in these experiments.
  • Process tomography: Full process tomography becomes impractical for large systems because measurement requirements scale exponentially with qubit number.A four-qubit process required 20736 settings and about 24 hours under the reported experimental parameters.

5 Algorithms with trapped ions

The review presents trapped-ion implementations of quantum algorithms, including Deutsch–Jozsa, teleportation, error correction, and quantum simulations. These experiments demonstrate algorithmic operations, state transfer, and protection against applied errors.

  • Deutsch–Jozsa algorithm: The single-ion Deutsch–Jozsa experiment encoded the input qubit in the internal state of 40Ca+ and the work qubit in axial motion.Ground-state cooling initialized the required joint state.
  • Deutsch–Jozsa algorithm: The Deutsch–Jozsa algorithm classified single-bit functions using one measurement of qubit |a⟩.Function-classification fidelity exceeded 0.97 for three functions and remained above 0.9 for the fourth.
  • Quantum teleportation: Teleportation transfers an unknown qubit state using a shared entangled pair, a Bell-state measurement, two classical bits, and a conditional receiver rotation.The protocol does not require an open quantum channel during transfer once the entangled link has been established.
  • Quantum teleportation: Trapped-ion teleportation experiments reached fidelities of about 0.75, 0.83, and 0.78 across Innsbruck and NIST implementations.A classical resource permits a maximum fidelity of 0.667, while fidelity above 0.87 is noted as necessary to rule out hidden-variable theories.
  • Quantum error correction: Quantum error correction kept fidelity near 0.8 for |↓⟩ under error angles θ_e < π/2, whereas the uncorrected implementation dropped to 0.5.The encoded implementation also improved fidelity for two superposition states.
  • Quantum simulations: A nonlinear interferometer became more sensitive to trap-frequency changes for larger motional quantum number n.The phase accumulated between two π/2 sideband pulses is proportional to the energy separation.

6 Shuttling and sympathetic cooling of ions

Ion shuttling, splitting, junction transport, and sympathetic cooling provide components for distributed trapped-ion architectures, but their integration while preserving quantum information remains incomplete.

  • Transport: Single 9Be+ ions were transported 1.2 mm within tens of microseconds without affecting hyperfine-qubit coherence.The measured contrast was 95.8%±0.8%, limited by magnetic-field fluctuations.
  • Transport: Ions were transported between traps within 54 µs with hardly observable motional heating, measured at ≲0.01 quanta per transport.The axial trap frequency was 2.9 MHz.
  • Splitting: Early ion-string splitting achieved 95% success but required about 10 ms and caused heating of 140 ± 70 quanta.The electrode structure was not optimized for splitting, and the separation frequency became small.
  • Junctions: Transport through junctions was demonstrated for charged nanoparticles and Cd+ ions in planar segmented traps.These experiments addressed four-way crossings and T-junctions, respectively.
  • Sympathetic cooling: Sympathetic cooling uses Coulomb coupling to cool part of an ion string while preserving other ions’ coherence.Be+–Mg+ experiments reached the motional ground state and showed that Mg+ cooling light did not affect a 9Be+ hyperfine qubit over 30 ms.
  • Remaining integration challenge: The combined operation of shuttling, splitting, and re-cooling while preserving quantum information had not yet been accomplished in one experiment.The individual prerequisites had been demonstrated separately.

7 New trap developments

Microfabricated and segmented traps are being developed to support more flexible, integrated ion-trap processors. Planar traps use asymmetric electrode arrangements to shape the trapping fields.

  • Microfabricated traps: Microfabrication enables complicated, precise electrode structures for medium-sized trapped-ion quantum computers.Trap sizes range from about 200 µm down to a few tens of µm, measured between RF electrodes.
  • Planar traps: Planar traps use asymmetric electrode arrangements whose three-dimensional and side views show the resulting electric-field lines.The configuration is presented as an electrode layout for planar trapping.

8 Future challenges and prospects for ion trap quantum computing

Trapped-ion platforms have demonstrated core requirements for quantum computing, but fault tolerance, scalability, and processing speed remain major challenges. The review identifies segmented-trap architectures and improved gate performance as central development directions.

  • Demonstrated capabilities: Initialization and single-qubit readout reach accuracies near 0.999, while ion strings can be shuttled, split, and merged with high fidelity and small decoherence.These capabilities address essential requirements for a general-purpose trapped-ion device.
  • Fault tolerance: Two-qubit gates remain the main limiting factor, although some fidelities are high enough in principle for fault-tolerant computation under Knill’s scheme.All requirements must be met simultaneously, including ancilla preparation, readout, coherence, layout, and parallelization.
  • Fault tolerance: Operational fidelities around 0.9999 per operation may suffice for fault tolerance when additional criteria such as error propagation and overhead are satisfied.The threshold depends on the error types and the amount of possible parallelization.
  • Scalability: Deterministic teleportation using purified entangled Bell states could transfer quantum information between locations without fault-tolerant state transfer.This provides a route for connecting separated ion-trap processors.
  • Processing speed: Typical basic operations take a few hundred microseconds, making large-scale factoring too slow even with substantial parallelism.Readout and ion-string separation may remain additional speed bottlenecks.
  • Scalability: Large ion strings become impractical because maintaining strong radial confinement and suitable axial trapping frequencies grows more difficult with system size.The review therefore highlights merging and shuttling small ion strings in segmented traps as a practicable scaling approach.
  • Prospects: The review concludes that core requirements have been demonstrated, while systems with many thousands of qubits remain extremely challenging.Control over a reasonable number of qubits is described as feasible in the next decade.
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