Abstract
Semiconductor spin qubits have recently seen major advances in coherence time and control fidelities, leading to a singlequbit performance that is on par with other leading qubit platforms. Most of this progress is based on microwave control of single spins in devices made of isotopically purified silicon. For controlling spins, the exchange interaction is an additional key ingredient which poses new challenges for highfidelity control. Here, we demonstrate exchangebased singlequbit gates of twoelectron spin qubits in GaAs double quantum dots. Using careful pulse optimization and closedloop tuning, we achieve a randomized benchmarking fidelity of (99.50±0.04)% and a leakage rate of 0.13% out of the computational subspace. These results open new perspectives for microwavefree control of singlettriplet qubits in GaAs and other materials.
Introduction
While semiconductor spin qubits have been pioneered with GaAsbased quantum dot devices^{1,2,3,4,5,6,7}, the adoption of isotopically purified silicon to avoid decoherence from nuclear spins has led to coherence times approaching one second^{8,9} and control fidelities above 99.9%^{10,11,12}, thus meeting the requirements for scalable quantum computing regarding singlequbit performance. These results are generally achieved with resonant microwave control of individual spins via electric or magnetic fields. An additional key requirement is to controllably couple multiple qubits. A natural and widespread approach is to use the exchange interaction between tunnelcoupled electron spins, which can also be used to manipulate qubits encoded in two or more electron spins^{13,14}. Advantages of this approach include a short gate duration and the avoidance of microwaves, which is a considerable simplification regarding power dissipation, complexity of control systems and addressability in the context of scaling to large qubit numbers. Exchangebased twoqubit gates of individual spins as well as twoelectron singlequbit gates have reached fidelities up to about 98%^{15,16} before our work. However, qubit control via the exchange interaction is also associated with certain challenges like the need for strong driving well beyond the rotating wave approximation, nonlinear coupling to control fields, a susceptibility to charge noise that scales with the interaction strength^{17}, and a high sensitivity to the detailed shape of baseband control pulses.
To address these difficulties, we numerically optimized control pulses for exchangebased singlequbit gates with fidelities approaching 99.9% in previous work^{18}. Remaining inaccuracies in these optimized pulses can be removed by a closedloop gate set calibration protocol (GSC), which allows the iterative tuneup of gates using experimental feedback^{18,19}. In comparison to automated calibration of singlespin^{12} and superconducting qubits^{20,21}, GSC extracts tomographic information about all unitary degrees of freedom to improve convergence. In addition to recalibration of drifting parameters, this method is suited for tuneup of gates with initial infidelities larger than 10%^{18}. Here, it allows us to optimize roughly an order of magnitude more parameters than before^{12,20,21} to fully leverage the degrees of freedom provided by our hardware.
With this approach we achieve accurate control of GaAsbased singlettriplet qubits encoded in twoelectron spins with a fidelity of 99.50 ± 0.04% (in contrast to a preliminary preprint^{22} of the present study we used a device with a more representative charge noise level compared to other groups^{17}). For comparison, stateoftheart singlespin control in GaAs leads to a fidelity of 96%^{23}. Furthermore, our result is in the range required by certain quantum error correction schemes^{24,25} and validates corresponding simulations^{18}, which also predict a similar performance for twoqubit control^{26}. A comparable fidelity of 99.6% has also recently been demonstrated for singlettriplet singlequbit gates in Si^{27}. In addition, we demonstrate a low leakage rate of 0.13% out of the subspace of valid qubit states, an important consideration for any qubit encoded in multiple spins^{28}. A similar leakage rate of 0.17% has been observed in an isotopically purified Si device using a threespin encoding^{29}. Besides addressing the challenges of exchangebased qubit control, these results also open new perspectives for GaAsbased devices.
Results
Singlettriplet qubit
The S − T_{0} spin qubit^{14} used in this work is illustrated in the middle of the figure below and can be described by the Hamiltonian \(H=\frac{\hslash J(\epsilon )}{2}{\sigma }_{x}+\frac{\hslash \Delta {B}_{z}}{2}{\sigma }_{z}\) in the \((\left\uparrow \downarrow \right\rangle =\left0\right\rangle ,\left\downarrow \uparrow \right\rangle =\left1\right\rangle )\) basis, where arrows denote electron spin up and down states. J(ϵ) denotes the exchange splitting between the singlet \(\left{\rm{S}}\right\rangle =(\left\uparrow \downarrow \right\rangle \left\downarrow \uparrow \right\rangle )/\sqrt{2}\) and s_{z} = 0 triplet state \(\left{{\rm{T}}}_{0}\right\rangle =(\left\uparrow \downarrow \right\rangle +\left\downarrow \uparrow \right\rangle )/\sqrt{2}\), while ΔB_{z} is the magnetic field gradient across both dots from different nuclearspin polarizations^{2}. The remaining triplet states, \(\left{{\rm{T}}}_{+}\right\rangle =\left\uparrow \uparrow \right\rangle\) and \(\left{{\rm{T}}}_{}\right\rangle =\left\downarrow \downarrow \right\rangle\), represent undesirable leakage states. J(ϵ) is manipulated by the detuning ϵ, the potential difference between both dots. We use standard state initialization and readout based on electron exchange with the lead, Pauli blockade and charge sensing (see methods). For singlequbit operations, ϵ is pulsed on a nanosecond timescale using an arbitrary waveform generator (AWG) whereas ΔB_{z} is typically stabilized at 2π ⋅ (42.1 ± 2.8 MHz) by dynamic nuclear polarization (DNP)^{5}. The resulting dynamics are illustrated in Fig. 1. In our simulations we use the experimentally motivated model \(J(\epsilon )={J}_{0}\exp (\epsilon /{\epsilon }_{0})\)^{17} to capture the nonlinear relation between control voltage and exchange coupling.
Numerical pulse optimization
To experimentally implement accurate singlequbit π/2 rotations around the x and yaxis (denoted by π/2_{x} and π/2_{y}), we use a control loop adapted from ref. ^{18} (illustrated in Fig. 2a, b) in conjunction with numerically optimized control pulses. To obtain a reasonably accurate system model for the numerical optimization procedure, we measure the step response of our electrical setup, J_{0}, ϵ_{0}, and ΔB_{z}. In addition, we determine the coherence properties of the qubit to construct a noise model including quasistatic hyperfine noise, quasistatic charge noise, and white charge noise. The details of the noise and control model are discussed further below and in Supplementary Notes 1 to 6. Next, we use this model to numerically optimize pulses consisting of N_{seg} piecewise constant nominal detuning values ϵ_{j}, j = 1…N_{seg} to be programmed into the AWG with a segment duration of 1 ns. The last four to five segments are set to the same baseline level \({\epsilon }_{\min }\) for all gates to minimize errors arising from pulse transients of previous pulses. We choose \({\epsilon }_{\min }\) such that \(J({\epsilon }_{\min })\ll \Delta {B}_{z}\). Typical optimized pulse profiles \({\epsilon }_{j}^{g},j=1\ldots {N}_{{\rm{seg}}}\) for two gates g = π/2_{x} and g = π/2_{y} are shown in Fig. 2a.
Experimental gate set calibration
Since our control model does not capture all effects to sufficient accuracy to directly achieve highfidelity gates, these pulses need to be refined using experimental feedback. Hence, error information about the gate set is extracted in every iteration of our control loop. Standard quantum process tomography can not be applied to extract this information as it requires wellcalibrated gates, which are not available before completion of the control loop. We solve this bootstrap problem with a selfconsistent method that extracts eight error syndromes S_{i}, i = 1…8 in each iteration^{18}. The first six syndromes are primarily related to overrotation and offaxis errors while the remaining two syndromes are proxies for decoherence. A syndrome S_{i} is measured by preparing \(\left0\right\rangle\), applying the corresponding sequence U_{i} of gates from Table 1, and determining the probability \(p(\left0\right\rangle )\) of obtaining the state \(\left0\right\rangle\) by measuring the sequence 10^{3} to 10^{4} times. For perfect gates, the first six syndromes^{30} should yield \(p(\left0\right\rangle )=0.5\), corresponding to \({S}_{i}=<{\sigma }_{z}> =0\). The last two syndromes should yield \(p(\left0\right\rangle )=0\) (S_{i} = −1). Deviations of S_{i} from the expected values indicate decoherence and systematic errors in the gate set. To make our method less sensitive to state preparation and measurement (SPAM) errors, we also prepare and readout a completely mixed state with measurement result S_{M}, and a triplet state \(\left{{\rm{T}}}_{0}\right\rangle\), which yields the measurement result S_{T} after correcting for the approximate contrast loss of the triplet preparation (see Supplementary Note 12). GSC then minimizes the norm of the modified error syndromes \(\tilde{{S}_{i}}={S}_{i}{S}_{{\rm{M}}}\) for i = 1…6 and \(\tilde{{S}_{i}}={S}_{i}{S}_{{\rm{T}}}\) for i ∈ {7, 8}.
For swift convergence, we start the control loop with numerically optimized pulses \({\epsilon }_{j}^{g}\), which theoretically implement the desired operations without systematic errors and with minimal decoherence by partially decoupling from slow noise (similar to dynamically corrected gates^{31}). First, we scale these pulses by ±10% in 2% increments and measure, which scaling achieves the lowest \(\tilde{{S}_{i}}\). GSC then optimizes the best pulses by minimizing \(\tilde{{S}_{i}}\) with the Levenberg–Marquardt algorithm (LMA). In each LMA iteration, we use finite differences to experimentally estimate derivatives \(d\tilde{{S}_{i}}/d{\epsilon }_{j}^{g}\), which are subsequently used to calculate updated pulse amplitudes \({\epsilon }_{j}^{g}\). Note that ΔB_{z} is not calibrated since the control via DNP is more involved than adjusting ϵ.
Convergence
Pulses with N_{seg} ≥ 24 lead to reliable convergence in several calibration runs, typically within 15 iterations as shown in Fig. 2c. Thus, all experiments were performed using 36 ns long gates. An exception is the extraction of gate trajectories in Fig. 2d, e, where 24 ns long gates were used. We have chosen the calibration algorithm such that it only adjusts those segments \({\epsilon }_{j}^{g}\), which are not at the baseline, resulting in 50 free parameters for the 36 ns long π/2_{x} and π/2_{y} gates shown in Fig. 2a. Sometimes, moderate charge rearrangements in our sample lead to a deterioration of the optimized gates. As a remedy we run GSC again, resulting in slightly different gates than before. While the initial tuneup took about 2 days, recalibration from previously tunedup gates only takes on the order of minutes to hours, depending on how much the sample tuning changed in between. To visualize the experimental gates, we perform selfconsistent quantum state tomography (QST)^{32} and extract state information after each segment \({\epsilon }_{j}^{g}\). As seen in Fig. 2d, e, the qubit state trajectories for model and experiment closely resemble each other, indicating that the GSCtuned pulses remain close to the optimum found in simulations.
Fidelity and leakage benchmarking
In order to determine the gate fidelity \({\mathcal{F}}\), we apply randomized benchmarking (RB) after completion of GSC. In RB, the fidelity is obtained by applying sequences of randomly chosen Clifford gates, here composed of π/2_{x} and π/2_{y} gates, to the initial state \(\left0\right\rangle\). The last Clifford operation of each sequence is chosen such that \(\left0\right\rangle\) would be recovered if the gates were perfect^{33}. For imperfect gates, the return probability \(p(\left0\right\rangle )\) decays as a function of sequence length and the decay rate indicates the average error per gate. In addition, we also apply an extended RB protocol, which omits the last Clifford from each RB sequence^{28}. Without leakage, averaging over many randomly chosen sequences should yield \(p(\left0\right\rangle )=50 \%\). However, for nonzero leakage we expect a single exponential decay of \(p(\left0\right\rangle )\) as a function of increasing sequence length since the additional leakage states have the same readout signature as \(\left1\right\rangle\) (see methods).
We indeed find such a decay law, indicated in blue in Fig. 3. A joint fit of the standard (red) and leakage detection (blue) RB data yields \({\mathcal{F}}=99.50\pm 0.04 \%\) and an incoherent gate leakage rate \({\mathcal{L}}=0.13\pm 0.03 \%\)^{28} (in the same GSC run we observed fidelities of 99.4% and higher six times after different iterations in separate RB runs). Since our pulses operate close to the S − T_{+} transition while \(\left{{\rm{T}}}_{}\right\rangle\) is far away in energy, leakage should predominantly occur into the \(\left{{\rm{T}}}_{+}\right\rangle\) level. In another sample we observed 0.4 ± 0.1% leakage and much higher fast charge noise levels (indicated by a low echo time of \({T}_{2}^{{\rm{echo}}}=183\ {\rm{ns}}\) for exchange oscillations)^{22}. This suggests that leakage is predominantly driven by charge noise.
Agreement with theoretical results
We previously predicted fidelities approaching 99.9% for GaAsbased S − T_{0} qubits^{18} with the best reported noise levels^{5,17}. To make a samplespecific comparison, we measured \({T}_{2}^{* }\) and \({T}_{2}^{{\rm{echo}}}\) by performing free induction decay and echo experiments for both hyperfine and exchange driven oscillations at several detunings. For J(ϵ) = 2π ⋅ 119 MHz we find that 726 coherent exchange oscillations are possible within \({T}_{2}^{{\rm{echo}}}=6.1\ \mu\)s. This is larger than 585 oscillations reported in ref. ^{17} for a comparable charge noise sensitivity dJ/dϵ ≈ 2π ⋅ 160 MHz/mV. Based on these measurements, we construct a noise model including white charge noise and contributions from quasistatic hyperfine and charge noise (see Supplementary Note 3) to evaluate the fidelities of the numerically optimized gates used as a starting point for GSC (see Supplementary Note 6). Without systematic errors, their theoretical fidelities of 99.86% (π/2_{x}) and 99.38% (π/2_{y}) correspond to an average fidelity of 99.62%. The good agreement with the experimentally observed average fidelity of 99.50 ± 0.04%, which includes residual systematic errors, indicates that this noise model can be used to obtain good estimates of the achievable fidelities. Using this model we estimate that speeding up the gate pulses by a factor 6 would increase the fidelity at least to 99.8% by further reducing the effect of ΔB_{z} fluctuations. Additional improvements are possible by using barrier control^{34} or reducing charge noise and residual systematic errors.
Discussion
One implication of our results is that the unavoidable presence of nuclear spins in GaAs spin qubits, which is often thought of as prohibitive for their technological prospects, actually does not preclude the fidelities required for faulttolerant quantum computing. While the overhead associated with mitigating nuclearspininduced decoherence remains a disadvantage^{5}, GaAs quantum dot devices currently tend to be more reproducible and require less challenging lithographic feature sizes. Furthermore, they avoid the complication of several neardegenerate conduction band valleys and are better suited for the conversion between spin states and flying photonic qubits due to the direct band gap^{35,36}. Given that GaAs has certain advantages over Si (but also other disadvantages), our study contributes to a wellfounded comparison of the two material systems. While the measured infidelities are somewhat lower than achieved in some Si devices, further improvement is possible. Moreover, twoqubit gates will be a more decisive factor for scalability. Our simulations based on the same type of model validated by the present experiments predict twoqubit fidelities of 99.9%^{26}.
Independent of the host material, a strength of S − T_{0} qubits are gate durations of a few tens of nanoseconds. These gates require only subgigahertz electrical control, in contrast to the manipulation frequencies typically used for singlespin qubits, which often lead to slower gate speeds on the order of one microsecond. Furthermore, these relaxed control hardware requirements could substantially facilitate the adoption of integrated cryogenic control electronics.
Although driven by the needs of GaAsbased S − T_{0} qubits, we expect that our approach is equally viable for other qubit types—specifically if many free gate parameters need to be tuned, if it is not intuitive how these parameters affect the qubit operations or if the qubit control model is not very accurate. As such, future work will focus on implementing exchangemediated twoqubit gates^{26} in GaAs or Si, and experimentally studying the factors determining the success rate of GSC. With appropriate parallelization and simplifications (e.g., fixing the gradients used by GSC), our method could also be adapted for the calibration of larger qubit arrays.
Methods
Qubit system
This work was performed using two different samples. The first sample is identical to the one in ref. ^{22} and was used to establish the calibration routine and measure the Bloch sphere trajectories. The gate fidelities were obtained in a second sample with lower charge noise.
We work in a dilution refrigerator at an electron temperature of about 130 mK. A lateral double quantum dot is defined in the twodimensional electron gas of a doped, molecularbeam epitaxygrown GaAs/AlGaAsheterostructure by applying voltages to metallic surface gates. For both samples, we use the same gate layout as ref. ^{7} and ref. ^{37} shown in Fig. 2b with two dedicated RF gates (yellow) for controlling the detuning. As we only apply RF pulses to these gates and no DC bias, we can perform all qubit operations without the need for bias tees, which reduces pulse distortions.
Quantum gates are performed in the (1,1) charge configuration, where one electron is confined in the left and one in the right quantum dot. In this regime, the computational subspace is defined by the s_{z} = 0 triplet state \(\left{{\rm{T}}}_{0}\right\rangle\) and the spin singlet state \(\left{\rm{S}}\right\rangle\). The other s_{z} = ±1 (1,1) triplet states \(\left\uparrow \uparrow \right\rangle\) and \(\left\downarrow \downarrow \right\rangle\) are split off energetically via the Zeeman effect by applying an external magnetic field of 500 mT.
We always readout and initialize the dot in the \((\left\uparrow \downarrow \right\rangle ,\left\downarrow \uparrow \right\rangle )\) basis by pulsing slowly from (0,2) to (1,1) and thus adiabatically mapping singlet \(\left{\rm{S}}\right\rangle\) and triplet \(\left{{\rm{T}}}_{0}\right\rangle\) to \(\left\uparrow \downarrow \right\rangle\) and \(\left\downarrow \uparrow \right\rangle\) (see Supplementary Note 7).
Readout calibration
For measuring the quantum state, we discriminate between singlet and triplet states by Pauli spin blockade. Using spin to charge conversion^{2}, the resistance of an adjacent sensing dot depends on the spin state and can be determined by RFreflectometry^{38}. In this manner, we obtain different readout voltages for singlet and triplet states but can not distinguish between \(\left{{\rm{T}}}_{0}\right\rangle\) and the triplet states \(\left{{\rm{T}}}_{\pm }\right\rangle\).
The measured voltages are processed in two ways. First, binning on the order of 10^{4} consecutive single shot measurements yields bimodal histograms where the two peak voltages roughly correspond to the singlet and triplet state. Second, the measured voltages are averaged over many repetitions of a pulse to reduce noise.
For the benchmarking experiments (which were performed using the second sample), we linearly convert the averaged voltages to probabilities \(p(\left0\right\rangle )\). The parameters of the linear transformation are obtained by fitting the histograms with a model that takes the decay of \(\left{{\rm{T}}}_{0}\right\rangle\) to \(\left{\rm{S}}\right\rangle\) during the readout phase into account^{4}.
In the first sample we also observed considerable excitation of \(\left{\rm{S}}\right\rangle\) to \(\left{{\rm{T}}}_{0}\right\rangle\). Thus, we modify the histogram fit model for the selfconsistent state tomography data (which was obtained with the first sample) by introducing the excitation of \(\left{\rm{S}}\right\rangle\) to \(\left{{\rm{T}}}_{0}\right\rangle\) as an additional fit parameter.
For GSC, the averaged voltages U_{i} corresponding to the error syndromes S_{i} do not need to be explicitly converted to probabilities \(p(\left0\right\rangle )\). Since mixed and triplet state reference voltages U_{M} and U_{T} are measured alongside the error syndromes, it is attractive to directly minimize the norm of \(\tilde{{U}_{i}}={U}_{i}{U}_{{\rm{M}}}\) for i = 1…6 and \(\tilde{{U}_{i}}={U}_{i}{U}_{{\rm{T}}}\) for i ∈ {7, 8}.
Note that RB is insensitive to state preparation and measurement (SPAM) errors. In addition, we decrease the sensitivity of GSC to SPAM errors by using reference measurements for a completely mixed and a triplet state. Therefore, our readout calibration does not need to be especially accurate or precise.
Further information regarding readout can be found in Supplementary Notes 7–11.
Data availability
The benchmarking data that support the key findings of this study are available in the supplementary information files. Various other raw data are available from the corresponding author upon reasonable request.
Code availability
The code used for this study is available from the corresponding author upon reasonable request.
References
 1.
Elzerman, J. M. et al. Singleshot readout of an individual electron spin in a quantum dot. Nature 430, 431–435 (2004).
 2.
Petta, J. R. et al. Coherent manipulation of coupled electron spins in semiconductor quantum dots. Science 309, 2180–2184 (2005).
 3.
Hanson, R., Petta, J. R., Tarucha, S. & Vandersypen, L. M. K. Spins in fewelectron quantum dots. Rev. Mod. Phys. 79, 1217–1265 (2007).
 4.
Barthel, C., Reilly, D., Marcus, C., Hanson, M. & Gossard, A. Rapid singleshot measurement of a singlettriplet qubit. Phys. Rev. Lett. 103, 160503 (2009).
 5.
Bluhm, H., Foletti, S., Mahalu, D., Umansky, V. & Yacoby, A. Enhancing the coherence of a spin qubit by Operating it as a feedback loop that controls its nuclear spin bath. Phys. Rev. Lett. 105, 216803 (2010).
 6.
Nowack, K. C. et al. Singleshot correlations and twoqubit gate of solidstate spins. Science 333, 1269–1272 (2011).
 7.
Shulman, M. D. et al. Demonstration of entanglement of electrostatically coupled singlettriplet qubits. Science 336, 202–205 (2012).
 8.
Veldhorst, M. et al. An addressable quantum dot qubit with faulttolerant controlfidelity. Nat. Nanotechnol. 9, 981–985 (2014).
 9.
Muhonen, J. T. et al. Storing quantum information for 30 seconds in a nanoelectronic device. Nat. Nanotechnol. 9, 986–991 (2014).
 10.
Yoneda, J. et al. A quantumdot spin qubit with coherence limited by charge noise and fidelity higher than 99.9%. Nat. Nanotechnol. 13, 102–106 (2018).
 11.
Muhonen, J. T. et al. Quantifying the quantum gate fidelity of singleatom spin qubits in silicon by randomized benchmarking. J. Phys. Condens. Matter 27, 154205 (2015).
 12.
Yang, C. H. et al. Silicon qubit fidelities approaching incoherent noise limits via pulse engineering. Nat. Electron. 2, 151–158 (2019).
 13.
DiVincenzo, D. P., Bacon, D., Kempe, J., Burkard, G. & Whaley, K. B. Universal quantum computation with the exchange interaction. Nature 408, 339–342 (2000).
 14.
Levy, J. Universal quantum computation with spin1/2 pairs and Heisenberg exchange. Phys. Rev. Lett. 89, 147902 (2002).
 15.
Huang, W. et al. Fidelity benchmarks for twoqubit gates in silicon. Nature 569, 532–536 (2019).
 16.
Nichol, J. M. et al. Highfidelity entangling gate for doublequantumdot spin qubits. npj Quantum Inf. 3, 1–4 (2017).
 17.
Dial, O. E. et al. Charge noise spectroscopy using coherent exchange oscillations in a singlettriplet qubit. Phys. Rev. Lett. 110, 146804 (2013).
 18.
Cerfontaine, P., Botzem, T., DiVincenzo, D. P. & Bluhm, H. Highfidelity singlequbit gates for twoelectron spin qubits in GaAs. Phys. Rev. Lett. 113, 150501 (2014).
 19.
Cerfontaine, P., Otten, R. & Bluhm, H. Selfconsistent calibration of quantumgate sets. Phys. Rev. Appl. 13, 044071 (2020).
 20.
Egger, D. J. & Wilhelm, F. K. Adaptive hybrid optimal quantum control for imprecisely characterized systems. Phys. Rev. Lett. 112, 240503 (2014).
 21.
Kelly, J. et al. Optimal quantum control using randomized benchmarking. Phys. Rev. Lett. 112, 240504 (2014).
 22.
Cerfontaine, P. et al., Feedbacktuned noiseresilient gates for encoded spin qubits. Preprint at http://arxiv.org/abs/1606.01897 (2016).
 23.
Yoneda, J. et al. Fast electrical control of single electron spins in quantum dots with vanishing influence from nuclear spins. Phys. Rev. Lett. 113, 267601 (2014).
 24.
Fowler, A., Stephens, A. & Groszkowski, P. Highthreshold universal quantum computation on the surface code. Phys. Rev. A 80, 052312 (2009).
 25.
Rispler, M., Cerfontaine, P., Langrock, V. & Terhal, B. M. Towards a realistic GaAsspin qubit device for a classical errorcorrected quantum memory. Physical Review A Preprint at https://arxiv.org/abs/2002.05136 (2020).
 26.
Cerfontaine, P., Otten, R., Wolfe, M. A., Bethke, P. & Bluhm, H. Highfidelity gate set for exchangecoupled singlettriplet qubits. Phys. Rev. B 101, 155311 (2020).
 27.
Takeda, K., Noiri, A., Yoneda, J., Nakajima, T. & Tarucha, S. Resonantly driven singlettriplet spin qubit in silicon. Phys. Rev. Lett. 124, 117701 (2020).
 28.
Wallman, J. J., Barnhill, M. & Emerson, J. Robust characterization of loss rates. Phys. Rev. Lett. 115, 060501 (2015).
 29.
Andrews, R. W. et al. Quantifying error and leakage in an encoded Si/SiGe tripledot qubit. Nat. Nanotechnol. 14, 747–750 (2019).
 30.
Dobrovitski, V. V., de Lange, G., Ristè, D. & Hanson, R. Bootstrap tomography of the pulses for quantum control. Phys. Rev. Lett. 105, 077601 (2010).
 31.
Khodjasteh, K. & Viola, L. Dynamically errorcorrected gates for universal quantum computation. Phys. Rev. Lett. 102, 080501 (2009).
 32.
Takahashi, M., Bartlett, S. D. & Doherty, A. C. Tomography of a spin qubit in a double quantum dot. Phys. Rev. A 88, 022120 (2013).
 33.
Magesan, E., Gambetta, J. M. & Emerson, J. Scalable and robust randomized benchmarking of quantum processes. Phys. Rev. Lett. 106, 180504 (2011).
 34.
Martins, F. et al. Noise suppression using symmetric exchange gates in spin qubits. Phys. Rev. Lett. 116, 116801 (2016).
 35.
Kim, D. et al. Optically loaded semiconductor quantum memory register. Phys. Rev. Appl. 5, 1–18 (2016).
 36.
Joecker, B. et al. Transfer of a quantum state from a photonic qubit to a gatedefined quantum dot. Phys. Rev. B 99, 205415 (2019).
 37.
Botzem, T. et al. Quadrupolar and anisotropy effects on dephasing in twoelectron spin qubits in GaAs. Nat. Commun. 7, 11170 (2016).
 38.
Reilly, D. J., Marcus, C. M., Hanson, M. P. & Gossard, A. C. Fast singlecharge sensing with a rf quantum point contact. Appl. Phys. Lett. 91, 89–92 (2007).
Acknowledgements
This work was supported by the Alfried Krupp von Bohlen und Halbach Foundation, DFG grants BL 1197/21, BL 1197/41, BO 3140/41, and SFB 1277. P.C. acknowledges support by Deutsche Telekom Stiftung. We thank Robert P. G. McNeil for fabricating the gate structures and ohmics of the second sample. Open access funding provided by Projekt DEAL.
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The first sample’s heterostructure was grown with molecularbeamepitaxy by D.B. and D.S. The second sample’s heterostructure was grown by J.R., A.L., and A.D.W.; T.B. fabricated the first sample and setup the experiment. S.S.H. developed the driver for the digitizer hardware used for data acquisition. T.B. and P.C. developed the feedback software and conducted the experiment. H.B., T.B., and P.C. analyzed the data and cowrote the paper. H.B. and P.C. developed the theoretical models.
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Cerfontaine, P., Botzem, T., Ritzmann, J. et al. Closedloop control of a GaAsbased singlettriplet spin qubit with 99.5% gate fidelity and low leakage. Nat Commun 11, 4144 (2020). https://doi.org/10.1038/s41467020178653
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