Abstract
Universal quantum computation will require qubit technology based on a scalable platform^{1}, together with quantum error correction protocols that place strict limits on the maximum infidelities for one and twoqubit gate operations^{2,3}. Although various qubit systems have shown high fidelities at the onequbit level^{4,5,6,7,8,9,10}, the only solidstate qubits manufactured using standard lithographic techniques that have demonstrated twoqubit fidelities near the faulttolerance threshold^{6} have been in superconductor systems. Siliconbased quantum dot qubits are also amenable to largescale fabrication and can achieve high singlequbit gate fidelities (exceeding 99.9 per cent) using isotopically enriched silicon^{11,12}. Twoqubit gates have now been demonstrated in a number of systems^{13,14,15}, but as yet an accurate assessment of their fidelities using Cliffordbased randomized benchmarking, which uses sequences of randomly chosen gates to measure the error, has not been achieved. Here, for qubits encoded on the electron spin states of gatedefined quantum dots, we demonstrate Bell state tomography with fidelities ranging from 80 to 89 per cent, and twoqubit randomized benchmarking with an average Clifford gate fidelity of 94.7 per cent and an average controlledrotation fidelity of 98 per cent. These fidelities are found to be limited by the relatively long gate times used here compared with the decoherence times of the qubits. Silicon qubit designs employing fast gate operations with high Rabi frequencies^{16,17}, together with advanced pulsing techniques^{18}, should therefore enable much higher fidelities in the near future.
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Acknowledgements
We thank S. Bartlett, R. Harper, L. M. K. Vandersypen, T. D. Ladd and N. C. Jones for discussions. We acknowledge support from the US Army Research Office (W911NF1310024 and W911NF1710198), the Australian Research Council (CE170100012), and the NSW Node of the Australian National Fabrication Facility. The views and conclusions contained in this document are those of the authors and should not be interpreted as representing the official policies, either expressed or implied, of the Army Research Office or the US Government. The US Government is authorized to reproduce and distribute reprints for Government purposes notwithstanding any copyright notation herein. B.H. acknowledges support from the Netherlands Organization for Scientific Research (NWO) through a Rubicon Grant. K.M.I. acknowledges support from a GrantinAid for Scientific Research by MEXT, NanoQuine, FIRST, and the JSPS CoretoCore Program.
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Nature thanks Jason Petta and the other anonymous reviewer(s) for their contribution to the peer review of this work.
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Contributions
W.H. and C.H.Y. performed the experiments. K.W.C. and F.E.H. fabricated the devices. K.M.I. prepared and supplied the ^{28}Si wafer. T.T. and J.C.C.H. contributed to the preparation of the experiments. W.H., C.H.Y., M.A.F., A.L. and B.H. designed the experiment and discussed the results. R.C.C.L. assisted with the data analysis. W.H., A.L. and A.S.D. wrote the manuscript with input from all coauthors. A.M., A.L. and A.S.D. planned and supervised the experiment.
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Extended data figures and tables
Extended Data Fig. 1 Singlequbit coherence properties in the (1, 1) regime.
Blue data corresponds to Q1 and red data corresponds to Q2. We have characterized the singlequbit coherence properties \({T}_{2}^{\ast }\) and \({T}_{2}^{{\rm{Hahn}}}\), and measured their control fidelities via singlequbit randomized benchmarking. All data are acquired with the frequency feedback protocol described in Extended Data Fig. 2. a, Spinup probability as a function of wait time in the Ramsey sequence. \({T}_{2,{\rm{Q}}2}^{\ast }=10.5\pm 1\,{\rm{\mu }}{\rm{s}}\) is much shorter than \({T}_{2,{\rm{Q}}1}^{\ast }=24.3\pm 2\,{\rm{\mu }}{\rm{s}}\). b, Spindown probability as a function of wait time in the Hahn echo sequence. \({T}_{2,{\rm{Q}}2}^{{\rm{H}}{\rm{a}}{\rm{h}}{\rm{n}}}=33\pm 5\,{\rm{\mu }}{\rm{s}}\) is much shorter than \({T}_{2,{\rm{Q}}1}^{{\rm{H}}{\rm{a}}{\rm{h}}{\rm{n}}}=290\pm 40\,{\rm{\mu }}{\rm{s}}\). c, Singlequbit randomized benchmarking with the other qubit initialized in the \(\left\downarrow \right\rangle \) state. Only the frequencies f_{1↓} and f_{2↓} are used for gate operations on Q1 and Q2 (single tone randomized benchmarking), respectively. The plot shows the projected state probability with increasing number of Clifford gates. The curve is fitted with \({P}_{\uparrow }=A(12{r}_{{\rm{C}}})+B\), and the Clifford gate fidelity is given by \({F}_{{\rm{Clifford}}}=1{r}_{{\rm{C}}}\). The singlequbit Clifford gates are on average composed of 1.875 primitive π/2pulses, so the π/2pulse fidelity is extracted as \({F}_{{\rm{\pi }}/2}=1{r}_{{\rm{C}}}/1.875\). The fidelity for all ESR pulses is in excess of 99%.
Extended Data Fig. 2 Frequency tracking protocol.
a, Frequency calibration of the ESR frequencies is implemented by interleaving calibration sequences with the randomized benchmarking experiment. After acquisition of three random sequences (one sequence is repeated 125 times), we check if the ESR frequency is still onresonance by applying a lowpower (26 dB lower than the typical operating power) πrotation. If the spinup probability is above the threshold of 50% of the readout visibility, the experiment will continue. If the spinup probability is below the threshold, the resonance frequency will be recalibrated until all ESR frequencies pass the check, and the measurement will continue. More sophisticated frequency tracking schemes could also contribute to higher gate fidelities^{37,38,39}. b, c, Resonance frequency fluctuations \({\rm{\Delta }}f={f}_{1\downarrow }{f}_{{\rm{avg}}}\) of f_{1↓} (b) and \({\rm{\Delta }}f={f}_{2\downarrow }{f}_{{\rm{avg}}}\) of f_{2↓} (c) during the measurement period. We subtracted the average values of the respective frequencies f_{avg} for better visibility. Over 13 h of data acquisition, Q1 experiences multiple jumps of about 600 kHz, while the fluctuations of Q2 remain within about 300 kHz. Since the resonance frequency fluctuations of Q1 and Q2 are uncorrelated, we exclude fluctuations of B_{0} or the microwave reference clock as the cause of the frequency changes. d, Variation of exchange coupling \({\rm{\Delta }}J=J{J}_{{\rm{avg}}}\) during the measurement period. We subtracted the average value J_{avg} for better visibility. The exchange coupling is relatively stable during the experiment. If the frequency fluctuations in b and c were to originate from charge noise, it is unlikely that J would remain unaffected. Furthermore, since the Stark shift of Q1 and Q2 is approximately 30 MHz V^{−1}, a 600kHz jump would require a change of the bias voltage applied to the D1 and D2 gates of about 20 mV. Such a change in the electrostatic environment would deteriorate qubit readout via the singleelectron transistor charge sensor, but we noticed no substantial change of the readout level during the experiment. On this basis, we further exclude charge noise from being the cause of the frequency changes. We conclude that the frequency jumps are most probably caused by spin flips of residual ^{29}Si nuclei that locally couple to the quantum dots.
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Huang, W., Yang, C.H., Chan, K.W. et al. Fidelity benchmarks for twoqubit gates in silicon. Nature 569, 532–536 (2019). https://doi.org/10.1038/s4158601911970
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