Integrated $^{9}$Be$^{+}$ multi-qubit gate device for the ion-trap quantum computer

TL;DR

This paper reports the successful experimental implementation of a high-fidelity two-qubit entangling gate in a $^9$Be$^+$ ion trap using microwave near-fields, advancing scalable quantum computing architectures.

Contribution

It introduces a novel microwave conductor design embedded in a surface-electrode trap that enables high-fidelity multi-qubit gates in ion-trap quantum computers.

Findings

Achieved 98.2% gate fidelity limited by technical imperfections.

Designed a microwave conductor that produces a high magnetic field gradient.

Demonstrated a scalable, self-contained module for entangling gates.

Abstract

We demonstrate the experimental realization of a two-qubit M{\o}lmer-S{\o}rensen gate on a magnetic field-insensitive hyperfine transition in $^9$Be$^+$ ions using microwave-near fields emitted by a single microwave conductor embedded in a surface-electrode 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 $98.2\pm1.2\,\%$ and is limited by technical imperfections, as is confirmed by a comprehensive numerical error analysis. The conductor design can potentially simplify the implementation of multi-qubit gates and represents a self-contained, scalable module for entangling gates within the quantum CCD architecture for an ion-trap quantum computer.

Tables

Table 1. Table 1: Infidelity contributions from different sources of imperfections. The infidelity values result from numerical simulations of the quantum dynamics according to Eq.( 2 ) including the respective noise effect with a strength given by the measured parameter specified in the second column.

Effect	Parameter	Infidelity
Mode instability	$\sqrt{⟨{\left({\delta }_{ϵ}/\delta \right)}^2⟩}=1.1\times {10}^{-2}$	$1.3\times {10}^{-2}$
	$0.3\mathrm{Hz}\mu s^{-1}$ ‘chirp’ for $600\mu s$
[2.5pt] Spectator mode	$\Delta \nu =2\pi \times 42.5\mathrm{kHz}$	$5.2\times {10}^{-3}$
[2.5pt]	with ${\overline{n}}_{\mathrm{r1}}=0.27$
Motional heating	${\gamma }_h={\dot{\overline{n}}}_{\mathrm{r2}}=28s^{-1}$	$3.8\times {10}^{-3}$
[2.5pt] Off-resonant	- (measured infidelity
[2.5pt] scattering loss	following ballance_high-fidelity_2017 ; sepiol_high-fidelity_2016 )
Qubit decoherence	${\tau }_d=1/{\gamma }_d>0.5s$
[2.5pt] Pulse shape	see main text	$6.3\times {10}^{-4}$
ACZS fluctuations	$\sqrt{⟨{\left({\Delta }_{ϵ}/\Delta \right)}^2⟩}=8\times {10}^{-4}$	$1.1\times {10}^{-4}$
[2.5pt] Rabi frequency imbalance	$\frac{{\Omega }_R-{\Omega }_B}{{\Omega }_B}=2.33\times {10}^{-2}$	$4.1\times {10}^{-6}$

Equations

$$H_{\mathrm{MS}}=\frac{\Omega }{2}\sum _{j=1}^N({\sigma }_j^{+}+{\sigma }_j^{-})(a_{r2}{e}^{\mathrm{i}\delta t}+a_{r2}^{†}{e}^{-\mathrm{i}\delta t}),$$

$$\dot{\rho }=-\mathrm{i}[H,\rho ]+{\mathcal{L}}_h\rho +{\mathcal{L}}_d\rho$$

$${\tilde{H}}_{\mathrm{MS}}=\frac{1}{2}\sum _{j=1}^N({\Omega }^{\mathrm{B}}(t){\sigma }_j^{+}{a}_{r2}^{†}{e}^{-\mathrm{i}\delta t}+{\Omega }^{\mathrm{R}}(t){\sigma }_j^{+}{a}_{r2}{e}^{\mathrm{i}\delta t})+H.c.$$

Full text

Following the proposal by Cirac et al. cirac_quantum_1995 , trapped atomic ions have shown to be a promising and pioneering platform for implementing elements of quantum information processing (QIP) blatt_entangled_2008 ; monroe_scaling_2013 . Qubits are encoded in the internal states of individual ions and shared motional modes are used as a ‘quantum bus’ for multi-qubit operations. Towards a large-scale universal quantum processor based on trapped-ion qubits, the ‘Quantum Charge-Coupled Device’ (QCCD) wineland_experimental_1998 ; kielpinski_architecture_2002 is considered a possible scalable hardware implementation. It relies on microfabricated multi-zone ion trap arrays in which quantum information is processed in dedicated zones interconnected via ion transport. While some key requirements such as high-fidelity ion transport blakestad_high-fidelity_2009 and fault-tolerant single-qubit gates brown_single-qubit-gate_2011 ; harty_high-fidelity_2014 have already been demonstrated in multiple setups, high-fidelity multi-qubit gates gaebler_high-fidelity_2016 ; ballance_high-fidelity_2016 below the fault-tolerant threshold still remain challenging. In this context, entangling gates driven by microwave fields mintert_ion-trap_2001 ; ospelkaus_trapped-ion_2008 represent a technically less demanding alternative to laser-induced 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 ozeri_errors_2007 and experimental fidelities ospelkaus_microwave_2011 ; khromova_designer_2012 ; weidt_trapped-ion_2016 ; harty_high-fidelity_2016 are approaching the fidelities of the best laser-driven gates gaebler_high-fidelity_2016 ; ballance_high-fidelity_2016 . Here we focus on the near-field microwave ospelkaus_trapped-ion_2008 gate approach, where the leading sources of infidelity in implementations so far comprise the spatio-temporal stability of the microwave near-field pattern ospelkaus_microwave_2011 ; warring_techniques_2013 or fluctuating AC Zeeman shifts harty_high-fidelity_2016 ; sepiol_high-fidelity_2016 . We note that in the latter work the error contribution arising from fluctuating AC Zeeman shifts has been reduced to $\lesssim 0.1\,$% through the use of a dynamical decoupling scheme harty_high-fidelity_2016 .

In this letter, we realize a two-qubit gate using a tailored microwave conductor embedded in a surface-electrode trap optimized to produce high oscillating magnetic near-field gradients and low residual fields at the ion position, thus directly addressing the main sources of error in previous near-field gates. The gate is realized on a field-independent hyperfine qubit in 9Be+ ions, a promising ion species for scalable QIP home_complete_2009 ; tan_demonstration_2013 ; negnevitsky_repeated_2018-1 , following the Mølmer-Sørensen (MS) sorensen_quantum_1999 ; molmer_multiparticle_1999 ; milburn_ion_2000 ; solano_deterministic_1999 protocol. The implementation is based on an optimized single-conductor design which can be thought of as the prototype of a scalable multi-qubit gate module for an ion-trap quantum computer based on surface-electrode trap arrays. The measured gate fidelity of $98.2\pm 1.2\,\%$ is purely limited by technical imperfections, in agreement with a numerical analysis.

The surface-electrode trap was fabricated at the PTB cleanroom facility employing the single-layer method as detailed in bautista-salvador_multilayer_2019 on an AlN substrate NoteX . Gold electrodes are about $11\,\mathrm{\mu m}$ thick and separated by $5\,\mathrm{\mu m}$ gaps. Aiming to remove potential organic residuals on top of the electrode surfaces, the trap was cleaned in an ex-situ dry-etching process before being installed in a UHV vacuum chamber at room-temperature. Electrical connectivity is provided by wire bonding to a printed-circuit board for DC signal filtering and signal routing.

Fig. 1 shows a top view of the trap center. Besides the microwave conductor MWM, which produces a magnetic field quadrupole wahnschaffe_single-ion_2017 suitable for driving multi-qubit gates, the electrode layout includes two microwave conductors labeled MWC to generate an oscillating magnetic field for global spin state manipulation. The ions are radially and axially confined at an ion-to-electrode distance of about $70\,\mathrm{\mu m}$ using one radio frequency (RF) and 10 DC control electrodes, respectively. With an applied RF voltage of $150\,\mathrm{V_{pp}}$ at $\Omega_{\mathrm{RF}}\simeq 2\pi\times 88.2\,\mathrm{MHz}$, the motional mode frequencies of the radial rocking modes of a two-ion 9Be*+* crystal were measured to be $(\omega_{\mathrm{r1}},\omega_{\mathrm{r2}})\simeq 2\pi\times(6.275,6.318)\,\mathrm{MHz}$.

The static magnetic field $\bm{\mathrm{B_{0}}}$ defining the quantization axis at an angle of $45\,^{\circ}$ with respect to the trap axis is produced by a hybrid setup consisting of two permanent magnet assemblies and a pair of compensation coils hakelberg_hybrid_2018 . At the ion position, this setup generates a magnetic field of $\left|\bm{\mathrm{B_{0}}}\right|=22.3\,\mathrm{mT}$ forming a first-order magnetic field insensitive qubit langer_long-lived_2005 on the hyperfine levels $\mathrm{{}^{2}S_{1/2}}\left|F=1,\,m_{\mathrm{F}}=1\right>\equiv\left|\uparrow\right>$ and $\mathrm{{}^{2}S_{1/2}}\left|F=2,\,m_{\mathrm{F}}=1\right>\equiv\left|\downarrow\right>$ with an unperturbed transition frequency of $\omega_{\mathrm{0}}\simeq 2\pi\times 1082.55\,\mathrm{MHz}$, cf. Fig. 2. Here, $F$ and $m_{\mathrm{F}}$ 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 wahnschaffe_single-ion_2017 . State initialization is done via optical pumping to the $\left|2,\,2\right>$ state (also referred to as the bright state) and subsequent Doppler cooling on the closed-cycle transition $\mathrm{{}^{2}S_{1/2}}\left|2,2\right>$ $\leftrightarrow$ $\mathrm{{}^{2}P_{3/2}}\left|m_{\mathrm{J}}=\frac{3}{2},m_{\mathrm{I}}=\frac{3}{2}\right>$ (where $m_{\mathrm{J}}$ and $m_{\mathrm{I}}$ are the projections of the total electronic and nuclear angular momenta onto the quantization axis). Resolved sideband cooling is performed by a pair of counter-propagating Raman beams aligned along the $\hat{z}$ direction. Each sideband cooling cycle consists of a global $\pi$ 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 $\pi$ rotations on the qubit transition and laser pulses on the $\mathrm{{}^{2}S_{1/2}}\left|1,1\right>$ $\leftrightarrow$ $\mathrm{{}^{2}P_{3/2}}\left|\frac{1}{2},\frac{3}{2}\right>$ transition similar to warring_techniques_2013 . Employing interleaved sideband cooling sequences on both radial rocking modes of a two-ion crystal, we measure a mean phonon number $\bar{n}$ of the modes to be $\bar{n}_{\mathrm{r1}}\simeq 0.27$ and $\bar{n}_{\mathrm{r2}}\simeq 0.11$ assuming a thermal distribution. The heating rate was determined to be $\dot{\bar{n}}_{\mathrm{r2}}\simeq 28\,\mathrm{s^{-1}}$. In an earlier measurement with similar radial mode frequencies, the heating rates of a single-ion’s low-frequency (LF) and high-frequency (HF) radial mode were found to be about $\dot{\bar{n}}_{\mathrm{LF}}\simeq 116\,\mathrm{s^{-1}}$ and $\dot{\bar{n}}_{\mathrm{HF}}\simeq 122\,\mathrm{s^{-1}}$, respectively.

To perform a MS two-qubit entangling gate on the qubit transition using near-field microwaves ospelkaus_trapped-ion_2008 , we subsequently initialize the qubits in $\left|\uparrow\uparrow\right>$ and apply a bichromatic microwave current to MWM at the frequencies $\omega_{\mathrm{RSB}}=\omega_{\mathrm{0}}+\Delta-(\omega_{\mathrm{r2}}+\delta)$ and $\omega_{\mathrm{BSB}}=\omega_{\mathrm{0}}+\Delta+(\omega_{\mathrm{r2}}+\delta)$. Here $\delta$ is the gate detuning from the high frequency rocking mode of $N=2$ ions at $\omega_{\mathrm{r2}}$ and $\Delta$ 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

[TABLE]

where $\Omega$ is the gate Rabi frequency, $a_{r2}$ ($a_{r2}^{\dagger}$) represents the mode’s annihilation (creation) operator and we define $\sigma^{\pm}=1/2(\sigma^{\mathrm{x}}\pm\mathrm{i}\sigma^{\mathrm{y}})$ with $\sigma^{\mathrm{x}}$ and $\sigma^{\mathrm{y}}$ being the Pauli matrices. Following sorensen_entanglement_2000 ; sackett_experimental_2000 , we apply the interaction on $\left|\uparrow\uparrow\right>$ in order to produce the maximally entangled state $\left|\Psi\right>=1/\sqrt{2}(\left|\uparrow\uparrow\right>+\mathrm{i}\left|\downarrow\downarrow\right>)$ at time $\tau=\pi\sqrt{K}/\Omega$ (where $K$ is an integer number) and calculate the resulting state preparation fidelity $\mathcal{F}\equiv\left<\Psi\right|\rho\left|\Psi\right>=1/2(P_{\uparrow\uparrow}+P_{\downarrow\downarrow})+\left|\rho_{\uparrow\uparrow,\downarrow\downarrow}\right|$ by determining the far off-diagonal element $\rho_{\uparrow\uparrow,\downarrow\downarrow}$ of the system’s density matrix $\rho$ as well as the population probabilities in $\left|\downarrow\downarrow\right>$, $\left|\uparrow\downarrow\right>$ and $\left|\downarrow\uparrow\right>$, and $\left|\uparrow\uparrow\right>$ given by $P_{\downarrow\downarrow}$, $P_{\uparrow\downarrow,\downarrow\uparrow}$ and $P_{\uparrow\uparrow}$, respectively.

Experimentally, we infer these quantities by adding a $\pi/2$ analysis pulse after the gate operation and observe the global ion fluorescence as a function of the analysis pulse’s phase $\phi_{\mathrm{a}}$ employing fluorescence detection on the closed-cycle transition for $400\,\mathrm{\mu s}$. Prior to detection, we transfer the population in $\left|\uparrow\right>$ back to the bright state and shelve the population in $\left|\downarrow\right>$ to $\left|1,\,-1\right>$ (also referred to as the dark state) using a sequence of $\pi$ rotations on the transitions ‘I’ and ‘II’-‘IV’, respectively. The populations are determined by repeating the experiment 200 times for each value of $\phi_{\mathrm{a}}$ 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 two-ion Ramsey experiment which we assume to produce an ideal outcome following the same procedure of ospelkaus_microwave_2011 (see online methods). To account for off-resonant 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 langer_high_2006 . By consequence, we expect the resulting populations to compensate imperfect state preparation and detection within the present level of accuracy. Finally, $\rho_{\uparrow\uparrow,\downarrow\downarrow}$ can be deduced by calculating the parity $\Pi(\phi_{\mathrm{a}})=P_{\downarrow\downarrow}(\phi_{\mathrm{a}})+P_{\uparrow\uparrow}(\phi_{\mathrm{a}})-P_{\uparrow\downarrow+\downarrow\uparrow}(\phi_{\mathrm{a}})$ while $\phi_{\mathrm{a}}$ is varied and extracting the magnitude $\left|A_{\Pi}\right|$ of the parity oscillation equal to $\left|2\,\rho_{\uparrow\uparrow,\downarrow\downarrow}\right|$ sackett_experimental_2000 . Fig. 3 shows the population and parity signal after performing the two-qubit gate operation using the MWM conductor. From sinusoidal fits to the extracted populations (solid lines), we calculate a corresponding gate fidelity of $98.2\pm 1.2\%$. The error on the fidelity is derived from the uncertainties in the population fits. In more detail, we apply a power of $\sim 5.5\,\mathrm{W}$ to each sideband tone, resulting in gate Rabi frequency of $\Omega/2\pi=1.071\,\mathrm{kHz}$ and an inferred near-field gradient of around $19\,\mathrm{T\,m^{-1}}$. For a single sideband the corresponding residual magnetic field at the ion position is highly suppressed by the optimized conductor geometry, resulting in an on-resonance Rabi frequency of about $\Omega_{\mathrm{c}}/2\pi\sim 450\,\mathrm{kHz}$. In case the bichromatic gate drive is applied, the accompanied differential AC Zeeman shift is measured to be $\Delta/2\pi=4.37\,\mathrm{kHz}$ and is predominantly caused by $\sigma$-components of the residual magnetic fields as the induced shifts of the $\pi$-components mainly cancel each other assuming an equal power in both sideband tones. In order to suppress off-resonant spin excitations, we adiabatically shape the envelope of the gate pulse at its beginning and end with a $2\,\mathrm{\mu s}$ long cumulative error function using a microwave envelope stabilization circuit based on a fast analog multiplier and a digital PI controller hannig_highly_2018 with the regulator setpoint generated by an arbitrary waveform generator bowler_arbitrary_2013 . 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 $\tau=808\,\mathrm{\mu s}$ and $\delta/2\pi=3.4\,\mathrm{kHz}$, respectively, corresponding to $K=3$ loops in motional phase space. Here the mismatch to the theoretically predicted detuning at $\delta_{\mathrm{theory}}/2\pi=3.71\,\mathrm{kHz}$ was tracked back to a systematic frequency offset from an independent radial mode frequency measurement as well as a radial mode frequency ‘chirp’ of $0.3\,\mathrm{Hz\,\mu s^{-1}}$ during the gate pulse. Qualitatively, a similar effect is also observed in other experiments using near-field gradients and appears to be inherent to warm-up processes in the microwave-generating structures sepiol_high-fidelity_2016 . This hypothesis is supported by our observation of a saturation behavior of the ‘chirp’ at around $1\,\mathrm{ms}$. Consequently, we reduce the impact of the ‘chirp’ by preceding the gate with a $400\,\mathrm{\mu s}$ long warm-up 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

[TABLE]

where the Hamiltonian is $H=\tilde{H}_{\mathrm{MS}}+H_{\mathrm{m}}+H_{\mathrm{z}}+H_{\mathrm{spec}}$, and

[TABLE]

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, $H_{\mathrm{z}}=\Delta_{\epsilon}(t)/2\sum_{j}\sigma^{\mathrm{z}}_{j}$ gives the uncompensated AC Zeeman shift resulting from shot-to-shot microwave power fluctuations, general $\Omega^{\mathrm{B}}$ and $\Omega^{\mathrm{R}}$ allow an imbalance in the two sideband Rabi frequencies, and time dependencies of the pulse shape are taken into account by the time dependent parameters. Couplings via additional, off-resonant motional modes are included by the term $H_{\mathrm{spec}}=\Omega_{r1}/2\sum_{j=1}^{N}(\sigma^{+}_{j}+\sigma^{-}_{j})(a_{r1}e^{\mathrm{i}(\Delta\nu+\delta)t}+a_{r1}^{\dagger}e^{-\mathrm{i}(\Delta\nu+\delta)t})$ whereby we limit ourselves to the nearest mode only (with $\Omega_{r1}\simeq\Omega$), which contributes the largest error of this kind. In addition to the unitary dynamics, motional heating to a thermal state with $\overline{n}_{\mathrm{th}}\gg 1$ and qubit decoherence are considered by the Lindblad terms sorensen_entanglement_2000 $\mathcal{L}_{\mathrm{h}}\rho=\gamma_{\mathrm{h}}(\mathcal{D}[a_{r2}]\rho+\mathcal{D}[a_{r2}^{\dagger}]\rho)$ with the heating rate $\gamma_{\mathrm{h}}$ in phonons per second and $\mathcal{L}_{\mathrm{d}}\rho=\gamma_{\mathrm{d}}/2\sum_{j}\mathcal{D}[\sigma^{\mathrm{z}}_{j}]\rho$ with the decoherence rate $\gamma_{\mathrm{d}}$ respectively, where $\mathcal{D}[\hat{O}]\rho=\hat{O}\rho\hat{O}^{\dagger}-\hat{O}^{\dagger}\hat{O}\rho/2-\rho\hat{O}^{\dagger}\hat{O}/2$.

Table 1 lists contributions of the different error sources to the infidelity $1-\mathcal{F}$. 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 noauthor_qutip:_2016 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 $2\pi\times 101\,\mathrm{Hz}$ 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 $0.3\,\mathrm{Hz}\,\mu\mathrm{s}^{-1}$ within the first $600\,\mu\mathrm{s}$ 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 johnson_active_2016 ; harty_high-fidelity_2013 , the ‘chirp’ can be reduced by e.g. longer warm-up 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 low frequency rocking mode. The measured frequency spacing was $\Delta\nu=\omega_{\mathrm{r2}}-\omega_{\mathrm{r1}}=2\pi\times 42.5\,\mathrm{kHz}$ and ground state cooling to $\bar{n}_{\mathrm{r1}}=0.27$ was applied, resulting in an infidelity of $5.2\times 10^{-3}$ from the competing gate dynamics. As this effect scales inversely with the radial mode splitting, it can be suppressed by engineering the trap potentials or suitable pulse sequences choi_optimal_2014 ; milne_phase-modulated_2018 . Heating of the motion and decoherence of the qubits contribute $\sim 0.1\%$ and $<0.1\%$ infidelity, respectively. Again, both effects do not represent a fundamental limit to the gate performance and can be improved experimentally hite_100-fold_2012 ; langer_long-lived_2005 . Off-resonant scattering on carrier transitions can lead to undesired excitations inside and outside the qubit-manifold and thus contribute a gate error. Here, an excitation other than on the qubit transition is much less probable due to the higher frequency difference of the driving field which is $>150\,\mathrm{MHz}$ detuned from the next spectator transition. Direct simulation of this effect was not performed due to the vastly different time-scales of the gate dynamics ($\sim\mathrm{kHz}$) and the carrier processes ($\sim\mathrm{GHz}$) which would have considerably increased the runtime of the numerical simulations. We instead performed direct measurements on a single qubit ballance_high-fidelity_2017 ; sepiol_high-fidelity_2016 to evaluate the extent of this error which is then quantified to be $<2.3\times 10^{-3}$. Infidelities below $6.3\times 10^{-4}$ 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 shot-to-shot fluctuations of the power, and accordingly of the AC Zeeman shift, to an extent that the simulated infidelity of $1.1\times 10^{-4}$ contributes only insignificantly. The same applies to the imbalance of Rabi frequencies, cf. Table 1.

In conclusion, we have demonstrated a microwave-driven two-qubit gate with 9Be*+* ions using a single microwave conductor with an optimized design embedded in a surface-electrode ion trap. The design of the MWM conductor has been developed to generate a high magnetic near-field gradient with low residual field at the ion position, thus suppressing AC Zeeman shift fluctuations, an inherent error source of the near-field approach, to a simulated infidelity contribution of $\sim 10^{-4}$. 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 two-qubit gate infidelity of $1.8\pm 1.2\,\%$. In addition to technical modifications to the apparatus, more elaborate gate schemes employing Walsh modulation hayes_coherent_2012 or continuous dynamic decoupling harty_high-fidelity_2016 can be applied in order to increase the gate fidelity as required for fault-tolerant quantum computation. In the future, the MWM conductor design can be used as an entangling gate unit of a ‘QCCD’ architecture purely employing microwave-driven quantum gates. Moreover, the conductor design can also be integrated into a scalable multilayer trap hahn_multilayer_2018 ; bautista-salvador_multilayer_2019 .

Data availability

The datasets generated during and/or analysed during the current study are available from C. Ospelkaus (christian.ospelkaus@iqo.uni-hannover.de) on reasonable request.

Acknowledgements

We acknowledge funding from DFG through CRC 1227 DQ-mat, 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.

Competing interests

The authors declare that there are no competing interests.

Author contributions

HH and GZ performed the measurements and analyzed the data. ABS produced the ion trap. MS and KH contributed the numerical simulations. CO devised the experiment plan. All authors participated in the error analysis and the realization of the manuscript.