Abstract
We demonstrate the experimental realization of a twoqubit Mølmer–Sørensen gate on a magnetic fieldinsensitive hyperfine transition in ^{9}Be^{+} ions using microwave nearfields emitted by a single microwave conductor embedded in a surfaceelectrode ion trap. The design of the conductor was optimized to produce a high oscillating magnetic field gradient at the ion position. The measured gate fidelity is determined to be 98.2 ± 1.2% and is limited by technical imperfections, as is confirmed by a comprehensive numerical error analysis. The conductor design can potentially simplify the implementation of multiqubit gates and represents a selfcontained, scalable module for entangling gates within the quantum CCD architecture for an iontrap quantum computer.
Introduction
Following the proposal by Cirac et al.,^{1} trapped atomic ions have shown to be a promising and pioneering platform for implementing elements of quantum information processing (QIP).^{2,3} Qubits are encoded in the internal states of individual ions, and shared motional modes are used as a “quantum bus” for multiqubit operations. Toward a largescale universal quantum processor based on trappedion qubits, the “Quantum ChargeCoupled Device” (QCCD)^{4,5} is considered as a possible scalable hardware implementation. It relies on microfabricated multizone iontrap arrays, in which quantum information is processed in dedicated zones interconnected via ion transport. While some key requirements such as highfidelity ion transport^{6} and faulttolerant singlequbit gates^{7,8} have already been demonstrated in multiple setups, highfidelity multiqubit gates^{9,10} below the faulttolerant threshold still remain challenging. In this context, entangling gates driven by microwave fields^{11,12} represent a technically less demanding alternative to laserinduced gates, as microwave signals can typically be controlled more easily than optical fields from highly specialized laser systems. The microwave approach avoids spontaneous scattering as a fundamental source of infidelities^{13} and experimental fidelities^{14,15,16,17} are approaching the fidelities of the best laserdriven gates.^{9,10} Here, we focus on the nearfield microwave^{12} gate approach, where the leading sources of infidelity in implementations so far comprise the spatiotemporal stability of the microwave nearfield pattern^{14,18} or fluctuating AC Zeeman shifts.^{17,19} We note that in the latter work, the error contribution arising from fluctuating AC Zeeman shifts has been reduced to ≲0.1% through the use of a dynamical decoupling scheme.^{17}
In this letter, we realize a twoqubit gate using a tailored microwave conductor embedded in a surfaceelectrode trap optimized to produce high oscillating magnetic nearfield gradients and low residual fields at the ion position, thus directly addressing the main sources of error in previous nearfield gates. The gate is realized on a fieldindependent hyperfine qubit in ^{9}Be^{+} ions, a promising ion species for scalable QIP,^{20,21,22} following the Mølmer–Sørensen (MS)^{23,24,25,26} protocol. The implementation is based on an optimized singleconductor design, which can be thought of as the prototype of a scalable multiqubit gate module for an iontrap quantum computer based on surfaceelectrode trap arrays. The measured gate fidelity of 98.2 ± 1.2% is purely limited by technical imperfections, in agreement with a numerical analysis.
Results and discussion
The surfaceelectrode trap was fabricated at the PTB cleanroom facility employing the singlelayer method as detailed in ref. ^{27} on an AlN substrate (for the present trap, we chose the singlelayer process in order to quickly test improvements that were made to the setup and trap orientation compared with ref. ^{28}). Gold electrodes are about 10 μm thick and separated by 5 μm gaps. Aiming to remove potential organic residuals on top of the electrode surfaces, the trap was cleaned in an ex situ dryetching process before being installed in a UHV vacuum chamber at room temperature. Electrical connectivity is provided by wire bonding to a printedcircuit board for DC signal filtering and signal routing.
Figure 1 shows a top view of the trap center. Besides the microwave conductor MWM, which produces a magnetic field quadrupole^{29} suitable for driving multiqubit gates, the electrode layout includes two microwave conductors labeled MWC to generate an oscillating magnetic field for global spinstate manipulation. The ions are radially and axially confined at an iontoelectrode distance of about 70 μm using one radio frequency (RF) and 10 DC control electrodes, respectively. With an applied RF voltage of 150V_{pp} at Ω_{RF} ≃ 2π × 88.2 MHz, the motionalmode frequencies of the radial rocking modes of a twoion ^{9}Be^{+} crystal were measured to be (ω_{r1}, ω_{r2}) ≃ 2π × (6.275, 6.318) MHz.
The static magnetic field B_{0} defining the quantization axis at an angle of 45° with respect to the trap axis is produced by a hybrid setup consisting of two permanent magnet assemblies and a pair of compensation coils.^{30} At the ion position, this setup generates a magnetic field of B_{0} = 22.3 mT forming a firstorder magnetic fieldinsensitive qubit^{31} on the hyperfine levels ^{2}S_{1/2}F = 1, m_{F} = 1〉 ≡ ↑〉 and ^{2}S_{1/2}F = 2, m_{F} = 1〉 ≡ ↓〉 with an unperturbed transition frequency of ω_{0} ≃ 2π × 1082.55 MHz, cf. Fig. 2. Here, F and m_{F} represent the quantum numbers for the ion’s total angular momentum and its projection on the quantization axis, respectively.
Ions are loaded using laser ablation combined with photoionization.^{29} State initialization is done via optical pumping to the 2, 2〉 state (also referred to as the bright state) and subsequent Doppler cooling on the closedcycle transition ^{2}S_{1/2}2, 2〉 ↔ ^{2}P_{3/2}\(\left {m_{\mathrm{J}} = \frac{3}{2},m_{\mathrm{I}} = \frac{3}{2}} \right\rangle\) (where m_{J} and m_{I} are the projections of the total electronic and nuclear angular momenta onto the quantization axis). Resolved sideband cooling is performed by a pair of counterpropagating Raman beams aligned along the \(\hat z\) direction. Each sideband cooling cycle consists of a global π rotation on the hyperfine transition labeled “I” in Fig. 2, followed by an optical red sideband pulse on the qubit transition and a repumping sequence to transfer all population back to the initial bright state. The repumping sequence comprises multiple microwave induced π rotations on the qubit transition and laser pulses on the ^{2}S_{1/2}1, 1〉 ↔ ^{2}P_{3/2}\(\left {\frac{1}{2},\frac{3}{2}} \right\rangle\) transition similar to ref. ^{18} Employing interleaved sideband cooling sequences on both radial rocking modes of a twoion crystal, we measure a mean phonon number \(\bar n\) of the modes \(\bar n_{{\mathrm{r}}1} \simeq 0.27\) and \(\bar n_{{\mathrm{r}}2} \simeq 0.11\) assuming a thermal distribution. The heating rate was determined to be \(\dot{\bar{n}}_{{\mathrm{r2}}} \simeq 28{\mkern 1mu} {\mathrm{s}}^{  {\mathrm{1}}}\). In an earlier measurement with similar radialmode frequencies, the heating rates of a singleion’s lowfrequency (LF) and highfrequency (HF) radial mode were found to be about \(\dot{\bar{n}}_{{\mathrm{LF}}} \simeq 116{\mkern 1mu} {\mathrm{s}}^{  {\mathrm{1}}}\) and \(\dot{\bar{n}}_{{\mathrm{HF}}} \simeq 122{\mkern 1mu} {\mathrm{s}}^{  {\mathrm{1}}}\), respectively.
To perform a MS twoqubit entangling gate on the qubit transition using nearfield microwaves,^{12} we subsequently initialize the qubits in ↑↑〉 and apply a bichromatic microwave current to MWM at the frequencies ω_{RSB} = ω_{0} + Δ − (ω_{r2} + δ) and ω_{BSB} = ω_{0} + Δ + (ω_{r2} + δ). Here, δ is the gate detuning from the highfrequency rocking mode of N = 2 ions at ω_{r2}, and Δ is the differential AC Zeeman shift of the unperturbed qubit transition induced by the bichromatic field. In the ideal case, the implemented dynamics can be described by the Hamiltonian
where Ω is the gate Rabi frequency, a_{r2} (\(a_{r2}^\dagger\)) represents the mode’s annihilation (creation) operator and we define σ^{±} = 1/2(σ^{x} ± iσ^{y}) with σ^{x} and σ^{y} being the Pauli matrices. Following,^{32,33} we apply the interaction on ↑↑〉 in order to produce the maximally entangled state \(\left {\mathrm{\Psi }} \right\rangle = 1/\sqrt 2 \left( {\left { \uparrow \uparrow } \right\rangle + {\mathrm{i}}\left { \downarrow \downarrow } \right\rangle } \right)\) at time \(\tau = \pi \sqrt K /{\mathrm{\Omega }}\) (where K is an integer number) and calculate the resulting state preparation fidelity \({\cal{F}} \equiv \left\langle {\mathrm{\Psi }} \right\rho \left {\mathrm{\Psi }} \right\rangle = 1/2(P_{ \uparrow \uparrow } + P_{ \downarrow \downarrow }) + \left {\rho _{ \uparrow \uparrow , \downarrow \downarrow }} \right\) by determining the far offdiagonal element ρ_{↑↑,↓↓} of the system’s density matrix ρ as well as the population probabilities in ↓↓〉, ↑↓〉, and ↓↑〉, and ↑↑〉 given by P_{↓↓}, P_{↑↓,↓↑}, and P_{↑↑}, respectively.
Experimentally, we infer these quantities by adding a π/2 analysis pulse after the gate operation and observe the global ion fluorescence as a function of the analysis pulse’s phase ϕ_{a} employing fluorescence detection on the closedcycle transition for 400 μs. Prior to detection, we transfer the population in ↑〉 back to the bright state and shelve the population in ↓〉 to 1, −1〉 (also referred to as the dark state) using a sequence of π rotations on the transitions “I” and “II”–“IV”, respectively. The populations are determined by repeating the experiment 200 times for each value of ϕ_{a}, and fitting the resulting histograms to a sum of weighted Poisson distributions representing 0, 1, and 2 ions bright. We calibrate the mean of the distributions to a reference twoion Ramsey experiment which we assume to produce an ideal outcome following the same procedure of^{14} (see Supplementary Methods section). To account for offresonant optical pumping effects, we modify the three Poissonians to include all depumping processes from the shelved dark state to the bright state during the detection interval.^{34} By consequence, we expect the resulting populations to compensate imperfect state preparation and detection within the present level of accuracy. Finally, ρ_{↑↑,↓↓} can be deduced by calculating the parity Π(ϕ_{a}) = P_{↓↓}(ϕ_{a}) + P_{↑↑}(ϕ_{a}) − P_{↑↓+↓↑}(ϕ_{a}), while ϕ_{a} is varied and extracting the magnitude A_{Π} of the parity oscillation equal to 2ρ_{↑↑,↓↓}.^{33} Figure 3 shows the population and parity signal after performing the twoqubit gate operation using the MWM conductor. From sinusoidal fits to the extracted populations (solid lines), we calculate a corresponding gate fidelity of 98.2 ± 1.2%. The error on the fidelity is derived from the uncertainties in the population fits. In more detail, we apply a power of ~5.5 W to each sideband tone, resulting in gate Rabi frequency of Ω/2π = 1.071 kHz and an inferred nearfield gradient of ~19 Tm^{−1}. For a single sideband, the corresponding residual magnetic field at the ion position is highly suppressed by the optimized conductor geometry, resulting in an onresonance Rabi frequency of about Ω_{c}/2π~450 kHz. In case the bichromatic gate drive is applied, the accompanied differential AC Zeeman shift is measured to be Δ/2π = 4.37 kHz and is predominantly caused by σcomponents of the residual magnetic fields as the induced shifts of the πcomponents mainly cancel each other assuming an equal power in both sideband tones. In order to suppress offresonant spin excitations, we adiabatically shape the envelope of the gate pulse at its beginning and end with a 2 μs long cumulative error function using a microwave envelope stabilization circuit based on a fast analog multiplier and a digital PI controller^{35} with the regulator setpoint generated by an arbitrary waveform generator.^{36} We optimize the gate by fixing the pulse duration to the expected value based on the gate Rabi frequency and subsequently scanning the gate detuning resulting in the highest fidelity. Following this procedure, we find an optimal gate time and detuning of τ = 808 μs and δ/2π = 3.4 kHz, respectively, corresponding to K = 3 loops in motionalphase space. Here, the mismatch to the theoretically predicted detuning at δ_{theory}/2π = 3.71 kHz was tracked back to a systematic frequency offset from an independent radialmode frequency measurement as well as a radial mode frequency “chirp” of 0.3 Hz μs^{−1} during the gate pulse. Qualitatively, a similar effect is also observed in other experiments using nearfield gradients and appears to be inherent to warmup processes in the microwavegenerating structures.^{19} This hypothesis is supported by our observation of a saturation behavior of the “chirp” at ~1 ms. Consequently, we reduce the impact of the “chirp” by preceding the gate with a 400 μs long warmup pulse with the duration chosen conservatively to avoid excessive heating of trap structures.
In order to identify current infidelity contributions in producing the maximally entangled state, we simulate the dynamics of the system using a master equation considering experimentally determined input parameters. This becomes necessary, as the exact propagator may no longer be obtained analytically in the presence of additional error sources. The master equation is given by
where the Hamiltonian is \(H = \tilde H_{{\mathrm{MS}}} + H_{\mathrm{m}} + H_{\mathrm{z}} + H_{{\mathrm{spec}}}\), and
is an extension to the ideal case presented in Eq. (1). Here, we have assumed equal Rabi frequencies and phases for both ions, which is true in the experiment to the best of our knowledge. Further, \(H_{\mathrm{m}} = \delta _\epsilon (t)a_{r2}^\dagger a_{r2}\) describes the instability of the rocking mode frequency, \(H_{\mathrm{z}} = \Delta _\epsilon (t)/2\mathop {\sum}\nolimits_j {\sigma _j^{\mathrm{z}}}\) gives the uncompensated AC Zeeman shift resulting from shottoshot microwave power fluctuations, general Ω^{B} and Ω^{R} allow an imbalance in the two sideband Rabi frequencies, and time dependencies of the pulse shape are taken into account by the timedependent parameters. Couplings via additional, offresonant motional modes are included by the term \(H_{{\mathrm{spec}}} = {\mathrm{\Omega }}_{r1}/2\mathop {\sum}\nolimits_{j = 1}^N {(\sigma _j^ + + \sigma _j^  )} (a_{r1}e^{{\mathrm{i}}({\mathrm{\Delta }}\nu + \delta )t} + a_{r1}^\dagger e^{  {\mathrm{i}}({\mathrm{\Delta }}\nu + \delta )t})\) whereby we limit ourselves to the nearest mode only (with Ω_{r1} ≃ Ω), which contributes the largest error of this kind. In addition to the unitary dynamics, motional heating to a thermal state with \(\bar n_{{\mathrm{th}}} \gg 1\) and qubit decoherence are considered by the Lindblad terms^{32} \({\cal{L}}_{\mathrm{h}}\rho = \gamma _{\mathrm{h}}({\cal{D}}[a_{r2}]\rho + {\cal{D}}[a_{r2}^\dagger ]\rho )\) with the heating rate γ_{h} in phonons per second and \({\cal{L}}_{\mathrm{d}}\rho = \gamma _{\mathrm{d}}/2\mathop {\sum}\nolimits_j {\cal{D}} [\sigma _j^{\mathrm{z}}]\rho\) with the decoherence rate γ_{d}, respectively, where \({\cal{D}}[\hat O]\rho = \hat O\rho \hat O^\dagger  \hat O^\dagger \hat O\rho /2  \rho \hat O^\dagger \hat O/2\).
Table 1 lists contributions of the different error sources to the infidelity \(1  {\cal{F}}\). These values result from numerical simulations of the quantum dynamics according to Eq. (2) considering the ideal gate dynamics with addition of the corresponding noise in the form we described above. All simulations were done with QuTiP^{37} and used a truncated Hilbert space for the motional mode. For our analysis, including the first 25 Fock states was sufficient to reach convergence given the low initial thermal distribution and the small motional displacements during the gate.
We examined the following effects, which we considered to be the most relevant, in more detail: the largest error according to our investigation results from the frequency instability of the rocking mode, which establishes the gate dynamics. This effect consists of two parts. On the one hand, normally distributed variations of the frequency with a standard deviation of \(\sqrt {\langle (\delta _\epsilon /\delta )^2\rangle } = 1.1 \times 10^{  2}\), inferred from a measured instantaneous linewidth of 2π × 101 Hz in a calibration scan directly before the gate measurement. On the other hand, a frequency “chirp” within each gate that we model by a linear increase of 0.3 Hz μs^{−1} within the first 600 μs, and subsequent constant frequency leading to in total 1.3% infidelity. While the mode fluctuations can be reduced by actively stabilizing the amplitude and frequency of the trap RF signal,^{38,39} the “chirp” can be reduced by e.g., longer warmup pulses. Simulations (see Fig. 4) also us to identify the individual contribution of each effect in view of further improvements.
The second largest contribution is caused by driving the identical spin–spin interaction via the additional lowfrequency rocking mode. The measured frequency spacing was Δν = ω_{r2}−ω_{r1} = 2π × 42.5 kHz and groundstate cooling to \(\bar n_{{\mathrm{r1}}} = 0.27\) was applied, resulting in an infidelity of 5.2 × 10^{−3} from the competing gate dynamics. As this effect scales inversely with the radialmode splitting, it can be suppressed by engineering the trap potentials or suitable pulse sequences.^{40,41} Heating of the motion and decoherence of the qubits contribute ~0.1% and <0.1% infidelity, respectively. Again, both effects do not represent a fundamental limit to the gate performance and can be improved experimentally.^{31,42} Offresonant scattering on carrier transitions can lead to undesired excitations inside and outside the qubitmanifold and thus contribute a gate error. Here, an excitation other than on the qubit transition is much less probable due to the higherfrequency difference of the driving field, which is >150 MHz detuned from the next spectator transition. Direct simulation of this effect was not performed due to the vastly different timescales of the gate dynamics (~kHz) and the carrier processes (~GHz) which would have considerably increased the runtime of the numerical simulations. We instead performed direct measurements on a single qubit^{19,43} to evaluate the extent of this error, which is then quantified to be <2.3 × 10^{−3}. Infidelities below 6.3 × 10^{−4} resulted from distortion of the pulse shape, whereby we combine here the influence of adiabatic switching on and off as well as small changes of the Rabi frequency and AC Zeeman shift during the pulses which result from power transients on the ideally rectangular signal. Stabilization of the microwave power allowed to reduce the shottoshot fluctuations of the power, and accordingly of the AC Zeeman shift, to an extent that the simulated infidelity of 1.1 × 10^{−4} contributes only insignificantly. The same applies to the imbalance of Rabi frequencies, cf. Table 1.
In conclusion, we have demonstrated a microwavedriven twoqubit gate with ^{9}Be^{+} ions using a single microwave conductor with an optimized design embedded in a surfaceelectrode ion trap. The design of the MWM conductor has been developed to generate a high magnetic nearfield gradient with low residual field at the ion position, thus suppressing AC Zeeman shift fluctuations, an inherent error source of the nearfield approach, to a simulated infidelity contribution of ~10^{−4}. In contrast, according to the presented error budget, the main contributions can all be decreased upon technical improvements; the by far biggest of these (1.3%) is consistent with the measured twoqubit gate infidelity of 1.8 ± 1.2%. In addition to technical modifications to the apparatus, more elaborate gate schemes employing Walsh modulation^{44} or continuous dynamic decoupling^{17} can be applied in order to increase the gate fidelity as required for faulttolerant quantum computation. In the future, the MWM conductor design can be used as an entangling gate unit of a “QCCD” architecture purely employing microwavedriven quantum gates. Moreover, the conductor design can also be integrated into a scalable multilayer trap.^{27,28}
Data availability
The datasets generated during and/or analyzed during the current study are available from C. Ospelkaus (christian.ospelkaus@iqo.unihannover.de) on reasonable request.
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Acknowledgements
We acknowledge funding from DFG through CRC 1227 DQmat, projects A01 and A06, and the clusters of excellence “QUEST” and “Quantum Frontiers”, from the EU QT flagship project “MicroQC” and from PTB and LUH.
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H.H. and G.Z. performed the measurements and analyzed the data. A.B.S. produced the ion trap. M.S. and K.H. contributed the numerical simulations. C.O. devised the experiment plan. All authors participated in the error analysis and the realization of the paper.
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Hahn, H., Zarantonello, G., Schulte, M. et al. Integrated ^{9}Be^{+} multiqubit gate device for the iontrap quantum computer. npj Quantum Inf 5, 70 (2019). https://doi.org/10.1038/s4153401901845
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DOI: https://doi.org/10.1038/s4153401901845
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