Abstract
Today, ion traps are among the most promising physical systems for constructing a quantum device harnessing the computing power inherent in the laws of quantum physics^{1,2}. For the implementation of arbitrary operations, a quantum computer requires a universal set of quantum logic gates. As in classical models of computation, quantum error correction techniques^{3,4} enable rectification of small imperfections in gate operations, thus enabling perfect computation in the presence of noise. For faulttolerant computation^{5}, it is believed that error thresholds ranging between 10^{−4} and 10^{−2} will be required—depending on the noise model and the computational overhead for realizing the quantum gates^{6,7,8}—but so far all experimental implementations have fallen short of these requirements. Here, we report on a Mølmer–Sørensentype gate operation^{9,10} entangling ions with a fidelity of 99.3(1)%. The gate is carried out on a pair of qubits encoded in two trapped calcium ions using an amplitudemodulated laser beam interacting with both ions at the same time. A robust gate operation, mapping separable states onto maximally entangled states is achieved by adiabatically switching the laser–ion coupling on and off. We analyse the performance of a single gate and concatenations of up to 21 gate operations.
Main
For ion traps, all building blocks necessary for the construction of a universal quantum computer^{1} have been demonstrated over the past decade. Currently, the most important challenges consist of scaling up the present systems to a higher number of qubits and raising the fidelity of gate operations up to the point where quantum error correction techniques can be successfully applied. Although singlequbit gates are easily carried out with high quality, the realization of highfidelity entangling twoqubit gates^{11,12,13,14,15,16} is much more demanding because the interion distance is orders of magnitude bigger than the characteristic length scale of any statedependent ion–ion interaction. Apart from quantum gates of the Cirac–Zoller type^{2,12}, where a laser couples a single qubit with a vibrational mode of the ion string at a time, most other gate realizations entangling ions have relied on collective interactions of the qubits with the laser control fields^{11,13,14,15}. These gate operations entangle transiently the collective pseudospin of the qubits with the vibrational mode and produce either a conditional phase shift^{17} or a collective spin flip^{9,10,18} of the qubits. Whereas the highest fidelity F=97% reported until now^{13} has been achieved with a conditional phase gate acting on a pair of hyperfine qubits in ^{9}Be^{+}, spinflip gates have been limited so far to F≈85% (refs 11, 14). All of these experiments have used qubits encoded in hyperfine or Zeeman ground states and a Raman transition mediated by an electricdipole transition for coupling the qubits. Whereas spontaneous scattering from the mediating shortlived levels degrades the gate fidelity owing to the limited amount of laser power available in current experiments^{19}, this source of decoherence does not occur for optical qubits, that is, qubits encoded in a ground state and a metastable electronic state of an ion. In the experiment presented here, where the qubit comprises the states S〉≡S_{1/2}(m=1/2) and D〉≡D_{5/2}(m=3/2) of the isotope ^{40}Ca^{+}, spontaneous decay of the metastable state reduces the gate fidelity by less than 5×10^{−5}.
A Mølmer–Sørensen gate inducing collective spin flips is achieved with a bichromatic laser field with frequencies ω_{±}=ω_{0}±δ, with ω_{0} being the qubit transition frequency and δ close to the vibrational mode frequency ν (Fig. 1). For optical qubits, the bichromatic field can be a pair of copropagating lasers, which is equivalent to a single laser beam resonant with the qubit transition and amplitudemodulated with frequency δ. For a gate mediated by the axial centreofmass (COM) mode, the hamiltonian describing the laser–qubit interaction is given by $H=\hslash \Omega {\text{e}}^{i\phi}S+({\text{e}}^{i(\delta t+\zeta )}+{\text{e}}^{i(\delta t+\zeta )}){\text{e}}^{i\eta (a{\text{e}}^{ivt+{a}^{\u2020}{\text{e}}^{ivt}})}+\text{h.c.}$ Here, S_{j}=σ_{j}^{(1)}+σ_{j}^{(2)},j∈{+,−,x,y,z}, denotes a collective atomic operator constructed from Pauli spin operators σ_{j}^{(i)} acting on ion i, and σ_{+}^{(i)}S〉_{i}=D〉_{i}. The operators a, a^{†} annihilate and create phonons of the COM mode with Lamb–Dicke factor η. The optical phase of the laser field (with coupling strength Ω) is labelled φ, and the phase ζ accounts for a time difference between the start of the gate operation and the maximum of the amplitude modulation of the laser beam. In the Lamb–Dicke regime, and for φ=0, the gate operation is very well described by the propagator^{20}
Here, the operator to the right describes collective spin flips induced by the operator S_{y,ψ}=S_{y}cosψ+S_{z}sinψ, ψ=(4Ω/δ)cos ζ, and λ≈η^{2}Ω^{2}/(ν−δ), χ≈η^{2}Ω^{2}/(ν−δ)^{2}. With α(t)=α_{0}(e^{i(ν−δ)t}−1), the displacement operator accounting for the transient entanglement between the qubits and the harmonic oscillator becomes equal to the identity after the gate time τ_{gate}=2π/ν−δ. The operator e^{−iF(t)Sx} with F(t)=(2Ω/δ)(sin(δ t+φ)−sinφ) describes fast nonresonant excitations of the carrier transition that occur in the limit of short gates when Ω≪δ no longer strictly holds. Nonresonant excitations are suppressed by intensityshaping the laser pulse so that the Rabi frequency Ω(t) is switched on and off smoothly. Moreover, adiabatic switching makes the collective spinflip operator independent of ζ as S_{y,ψ}→S_{y} for Ω→0. To achieve adiabatic following, it turns out to be sufficient to switch on the laser within 2.5 oscillation periods of the ions’ axial COM mode. When the laser is switched on adiabatically, equation (1) can be simplified by dropping the factor e^{−iF(t)Sx} and replacing S_{y,ψ} by S_{y}. To realize an entangling gate of duration τ_{gate} described by the unitary operator U_{gate}=exp(−i(π/8)S_{y}^{2}), the laser intensity needs to be set such that η Ω≈δ−ν/4.
Two ^{40}Ca^{+} ions are confined in a linear trap^{21} with axial and radial COM mode frequencies of ν_{axial}/2π=1.23 MHz and ν_{radial}/2π=4 MHz, respectively. After Doppler cooling and frequencyresolved optical pumping^{22} in a magnetic field of 4 G, the two axial modes are cooled close to the motional ground state (, ). Both ions are now initialized to S S〉 with a probability of more than 99.8%. Then, the gate operation is carried out, followed by an optional carrier pulse for analysis. Finally, we measure the probability p_{k} of finding k ions in the S〉 state by detecting light scattered on the dipole transition with a photomultiplier for 3 ms. The error in state detection due to spontaneous decay from the D state is estimated to be less than 0.15%. Each experimental cycle is synchronized with the frequency of the a.c.power line and repeated 50–200 times. The laser beam carrying out the entangling operation is controlled by a doublepass acoustooptic modulator, which enables setting the frequency ω_{L} and phase φ of the beam. By means of a variable gain amplifier, we control the radiofrequency input power and hence the intensity profile of each laser pulse. To generate a bichromatic light field, the beam is passed through another acoustooptic modulator in singlepass configuration that is driven simultaneously by two radiofrequency signals with difference frequency δ/π (see the first paragraph of the Methods section). Phase coherence of the laser frequencies is maintained by phaselocking all radiofrequency sources to an ultrastable quartz oscillator. We use 1.8 mW average light power focused down to a spot size of 14 μm gaussian beam waist illuminating both ions from an angle of 45^{∘} with equal intensity to achieve the Rabi frequencies Ω/(2π)≈110 kHz required for carrying out a gate operation with (ν−δ)/(2π)=20 kHz and η=0.044. To make the bichromatic laser pulses independent of the phase ζ, the pulse is switched on and off by using Blackmanshaped pulse slopes of duration τ_{r}=2 μs.
Multiple application of the bichromatic pulse of duration τ_{gate} ideally maps the state S S〉 to
up to global phases. Maximally entangled states occur at instances τ_{m}=m·τ_{gate} (m=1,3,…). A similar mapping of product states onto Bell states and vice versa also occurs when starting from state S D〉. To assess the fidelity of the gate operation, we adapt the strategy first applied in refs 11, 13 consisting of measuring the fidelity of Bell states created by a single application of the gate to the state S S〉 (Fig. 2a). The fidelity F=〈Ψ_{1}ρ^{exp}Ψ_{1}〉=(ρ_{S S,S S}^{exp}+ρ_{D D,D D}^{exp})/2+Imρ_{D D,S S}^{exp}, with the density matrix ρ^{exp} describing the experimentally produced qubits’ state, is inferred from measurements on a set of 42,400 Bell states continuously produced within a measurement time of 35 min. Fluorescence measurements on 13,000 Bell states reveal that ρ_{S S,S S}^{exp}+ρ_{D D,D D}^{exp}=p_{2}+p_{0}=0.9965(4). The offdiagonal element ρ_{D D,S S}^{exp} is determined by measuring P(φ)=〈σ_{φ}^{(1)}σ_{φ}^{(2)}〉 for different values of φ, where σ_{φ}=σ_{x}cosφ+σ_{y}sinφ, by applying (π/2)_{φ} pulses to the remaining 29,400 states and measuring p_{0}+p_{2}−p_{1} to obtain the parity 〈σ_{z}^{(1)}σ_{z}^{(2)}〉. The resulting parity oscillation P(φ) shown in Fig. 2b is fitted with a function P_{fit}(φ)=Asin(2φ+φ_{0}) that yields A=2ρ_{D D,S S}^{exp}=0.990(1). Combining the two measurements, we obtain the fidelity F=99.3(1)% for the Bell state Ψ_{1}.
A wealth of further information is obtained by studying the state dynamics under the action of the gate hamiltonian (see equation (2)). Starting from state S S〉, Fig. 3 shows the time evolution of the state populations for pulse lengths equivalent to up to 17 gate times. The ions are entangled and disentangled consecutively up to nine times, the populations closely following the predicted unitary evolution of the propagator (1) for ζ=0 shown in Fig. 3 as solid lines.
To study sources of gate imperfections we measured the fidelity of Bell states obtained after a pulse length τ_{m} for up to m=21 gate operations. The sum of the populations p_{0}(t)+p_{2}(t) does not return perfectly to one at times τ_{m} as shown in Fig. 4 but decreases by about 0.0022(1) per gate. This linear decrease could be explained by resonant spinflip processes caused by spectral components of the qubit laser that are far outside the laser’s linewidth of 20 Hz (ref. 21) (see the Methods section). The figure also shows the amplitude of parity fringe pattern scans at odd integer multiples of τ_{gate} similar to the one in Fig. 2b. The gaussian shape of the amplitude decay is consistent with variations in the coupling strength Ω that occur from one experiment to the next (see the Methods section).
The observed Bellstate infidelity of 7×10^{−3} indicates that the gate operation has an infidelity below the error threshold required by some models of faulttolerant quantum computation^{6,7,8} (an indication to be confirmed by full quantum gate tomography^{16} in future experiments). However, further experimental advances will be needed before faulttolerant computation will become a reality as the overhead implied by these models is considerable. Nevertheless, in addition to making the implementation of quantum algorithms with tens of entangling operations look realistic, the gate presented here also opens interesting perspectives for generating multiparticle entanglement^{23} by a single laser interacting with more than two qubits at once. For the generation of Nqubit Greenberger–Horne–Zeilinger states, there exist no constraints on the positioning of ions in the bichromatic beam that otherwise made generation of Greenberger–Horne–Zeilinger states beyond N=6 difficult in the experiment with hyperfine qubits described in ref. 24. Although the bichromatic force lacks a strong spatial modulation that would enable tailoring of the gate interaction by choosing particular ion spacings^{25,26}, more complex multiqubit interactions could be engineered by interleaving entangling laser pulses addressing all qubits with a focused laser inducing phase shifts in single qubits. Akin to nuclear magnetic resonance techniques, this method should enable refocusing of unwanted qubit–qubit interactions^{27} and open the door to a wide variety of entangling multiqubit interactions.
Methods
a.c.Starkshift compensation
The red and the bluedetuned frequency components ω_{±} of the bichromatic light field cause dynamic (a.c.) Stark shifts by nonresonant excitation on the carrier and the firstorder sidebands that exactly cancel each other if the corresponding laser intensities I_{±} are equal. The remaining a.c.Stark shift due to other Zeeman transitions and fardetuned dipole transitions amounts to 7 kHz for a gate time τ_{gate}=50 μs. These shifts could be compensated by using an extra fardetuned light field^{28} or by properly setting the intensity ratio I_{+}/I_{−}. We use the latter technique, which makes the coupling strengths , slightly unequal. However, the error is insignificant as in our experiments.
Sources of gate infidelity
A bichromatic force with timedependent Ω(t) acting on ions prepared in an eigenstate of S_{y} creates coherent states α(t) following trajectories in phase space that generally do not close^{20,29}. For the short rise times used in our experiments, this effect can be made negligibly (<10^{−4}) small by slightly increasing the gate time.
Spin flips induced by incoherent offresonant light of the bichromatic laser field reduce the gate fidelity. A beat frequency measurement between the gate laser and a similar independent laser system that was spectrally filtered indicates that a fraction γ of about 2×10^{−7} of the total laser power is contained in a 20 kHz bandwidth B around the carrier transition when the laser is tuned close to a motional sideband. A simple model predicts spin flips to cause a gate error with probability p_{flip}=(πγν−δ)/(2η^{2}B). This would correspond to a probability p_{flip}=8×10^{−4}, whereas the measured state populations shown in Fig. 4 would be consistent with p_{flip}=2×10^{−3}. Spinflip errors could be further reduced by two orders of magnitude by spectrally filtering the laser light and increasing the trap frequency ν/(2π) to above 2 MHz where noise caused by the laser frequency stabilization is much reduced.
Imperfections due to lowfrequency noise randomly shifting the laser frequency ω_{L} with respect to the atomic transition frequency ω_{0} were estimated from Ramsey measurements on a single ion showing that an average frequency deviation σ_{(ωL−ω0)}/(2π)=160 Hz occurred. From numerical simulations, we infer that for a single gate operation this frequency uncertainty gives rise to a fidelity loss of 0.25% (an infidelity of 10^{−4} would require σ_{(ωL−ω0)}/(2π)=30 Hz). In our parity oscillation experiments shown in Figs 2b and 4, however, this loss is not directly observable because a small error in the frequency of the bichromatic laser beam carrying out the gate operation is correlated with a similar frequency error of the carrier pulse probing the entanglement produced by the gate so that the phase φ of the analysing pulse with respect to the qubit state remains well defined.
Variations in the coupling strength δ Ω induced by lowfrequency laser intensity noise and thermally occupied radial modes were inferred from an independent measurement by recording the amplitude decay of carrier oscillations. Assuming a gaussian decay, we find a relative variation of δ Ω/Ω=1.4(1)×10^{−2}. For m entangling gate operations, the loss of fidelity is approximately given by 1−F=(πm/2)^{2}(δ Ω/Ω)^{2} and contributes with 5×10^{−4} to the error of a single gate operation. For the multiple gate operations shown in Fig. 4, this source of noise explains the gaussian decay of the parity fringe amplitude, whereas laser frequency noise reduces the fringe amplitude by less than 1% even for 21 gate operations. In combination with error estimates for state preparation, detection and laser noise, the analysis of multiple gates provides us with a good understanding of the most important sources of gate infidelity.
References
 1
Nielsen, M. A. & Chuang, I. L. Quantum Computation and Quantum Information (Cambridge Univ. Press, Cambridge, 2000).
 2
Cirac, J. I. & Zoller, P. Quantum computations with cold trapped ions. Phys. Rev. Lett. 74, 4091–4094 (1995).
 3
Shor, P. W. Scheme for reducing decoherence in quantum computer memory. Phys. Rev. A 52, R2493–R2496 (1995).
 4
Steane, A. M. Error correcting codes in quantum theory. Phys. Rev. Lett. 77, 793–797 (1996).
 5
Shor, P. W. 37th Symposium on Foundations of Computing 56–65 (IEEE Computer Society Press, Washington DC, 1996).
 6
Knill, E. Quantum computing with realistically noisy devices. Nature 434, 39–44 (2005).
 7
Raussendorf, R. & Harrington, J. Faulttolerant quantum computation with high threshold in two dimensions. Phys. Rev. Lett. 98, 190504 (2007).
 8
Reichardt, B. W. Improved ancilla preparation scheme increases faulttolerant threshold. Preprint at <http://arxiv.org/abs/quantph/0406025v1> (2004).
 9
Sørensen, A. & Mølmer, K. Quantum computation with ions in thermal motion. Phys. Rev. Lett. 82, 1971–1974 (1999).
 10
Sørensen, A. & Mølmer, K. Entanglement and quantum computation with ions in thermal motion. Phys. Rev. A 62, 022311 (2000).
 11
Sackett, C. A. et al. Experimental entanglement of four particles. Nature 404, 256–259 (2000).
 12
SchmidtKaler, F. et al. Realization of the Cirac–Zoller controlledNOT quantum gate. Nature 422, 408–411 (2003).
 13
Leibfried, D. et al. Experimental demonstration of a robust, highfidelity geometric two ionqubit phase gate. Nature 422, 412–415 (2003).
 14
Haljan, P. C. et al. Entanglement of trappedion clock states. Phys. Rev. A 72, 062316 (2005).
 15
Home, J. P. et al. Deterministic entanglement and tomography of ion spin qubits. New J. Phys. 8, 188 (2006).
 16
Riebe, M. et al. Process tomography of ion trap quantum gates. Phys. Rev. Lett. 97, 220407 (2006).
 17
Milburn, G. J., Schneider, S. & James, D. F. V. Ion trap quantum computing with warm ions. Fortschr. Phys. 48, 801–810 (2000).
 18
Solano, E., de Matos Filho, R. L. & Zagury, N. Deterministic Bell states and measurement of the motional state of two trapped ions. Phys. Rev. A 59, R2539–R2543 (1999).
 19
Ozeri, R. et al. Errors in trappedion quantum gates due to spontaneous photon scattering. Phys. Rev. A 75, 042329 (2007).
 20
Roos, C. F. Ion trap quantum gates with amplitudemodulated laser beams. New J. Phys. 10, 013002 (2008).
 21
Benhelm, J. et al. Measurement of the hyperfine structure of the S1/2–D5/2 transition in ^{43}Ca^{+}. Phys. Rev. A 75, 032506 (2007).
 22
Roos, C. F., Chwalla, M., Kim, K., Riebe, M. & Blatt, R. ‘Designer atoms’ for quantum metrology. Nature 443, 316–319 (2006).
 23
Mølmer, K. & Sørensen, A. Multiparticle entanglement of hot trapped ions. Phys. Rev. Lett. 82, 1835–1838 (1999).
 24
Leibfried, D. et al. Creation of a sixatom ‘Schrödinger cat’ state. Nature 438, 639–642 (2005).
 25
Chiaverini, J. et al. Realization of quantum error correction. Nature 432, 602–605 (2004).
 26
Reichle, R. et al. Experimental purification of twoatom entanglement. Nature 443, 838–841 (2006).
 27
Vandersypen, L. M. K. & Chuang, I. L. NMR techniques for quantum control and computation. Rev. Mod. Phys. 76, 1037–1069 (2004).
 28
Häffner, H. et al. Precision measurement and compensation of optical Stark shifts for an iontrap quantum processor. Phys. Rev. Lett. 90, 143602 (2003).
 29
Leibfried, D., Knill, E., Ospelkaus, C. & Wineland, D. J. Transport quantum logic gates for trapped ions. Phys. Rev. A 76, 032324 (2007).
Acknowledgements
We gratefully acknowledge the support of the European network SCALA and the Disruptive Technology Office and the Institut für Quanteninformation GmbH. We thank R. Gerritsma and F. Zähringer for help with the experiments.
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Benhelm, J., Kirchmair, G., Roos, C. et al. Towards faulttolerant quantum computing with trapped ions. Nature Phys 4, 463–466 (2008). https://doi.org/10.1038/nphys961
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