# Fidelity benchmarks for two-qubit gates in silicon

## Abstract

Universal quantum computation will require qubit technology based on a scalable platform1, together with quantum error correction protocols that place strict limits on the maximum infidelities for one- and two-qubit gate operations2,3. Although various qubit systems have shown high fidelities at the one-qubit level4,5,6,7,8,9,10, the only solid-state qubits manufactured using standard lithographic techniques that have demonstrated two-qubit fidelities near the fault-tolerance threshold6 have been in superconductor systems. Silicon-based quantum dot qubits are also amenable to large-scale fabrication and can achieve high single-qubit gate fidelities (exceeding 99.9 per cent) using isotopically enriched silicon11,12. Two-qubit gates have now been demonstrated in a number of systems13,14,15, but as yet an accurate assessment of their fidelities using Clifford-based 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 gate-defined quantum dots, we demonstrate Bell state tomography with fidelities ranging from 80 to 89 per cent, and two-qubit randomized benchmarking with an average Clifford gate fidelity of 94.7 per cent and an average controlled-rotation 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 frequencies16,17, together with advanced pulsing techniques18, should therefore enable much higher fidelities in the near future.

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## Data availability

The data that support the findings of this study are available from the corresponding author upon reasonable request. Source Data for Figs. 1, 2, 3, 4 and Extended Data Figs. 1, 2 are available with the online version of the paper.

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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 (W911NF-13-1-0024 and W911NF-17-1-0198), 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 Grant-in-Aid for Scientific Research by MEXT, NanoQuine, FIRST, and the JSPS Core-to-Core Program.

### Reviewer information

Nature thanks Jason Petta and the other anonymous reviewer(s) for their contribution to the peer review of this work.

## Author information

Authors

### 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 28Si 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 co-authors. A.M., A.L. and A.S.D. planned and supervised the experiment.

### Corresponding authors

Correspondence to W. Huang or A. S. Dzurak.

## Ethics declarations

### Competing interests

The authors declare no competing interests.

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## Extended data figures and tables

### Extended Data Fig. 1 Single-qubit coherence properties in the (1, 1) regime.

Blue data corresponds to Q1 and red data corresponds to Q2. We have characterized the single-qubit coherence properties $${T}_{2}^{\ast }$$ and $${T}_{2}^{{\rm{Hahn}}}$$, and measured their control fidelities via single-qubit randomized benchmarking. All data are acquired with the frequency feedback protocol described in Extended Data Fig. 2. a, Spin-up 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, Spin-down 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, Single-qubit randomized benchmarking with the other qubit initialized in the $$\left|\downarrow \right\rangle$$ state. Only the frequencies f1↓ and f2↓ 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 $${P}_{\uparrow }=A(1-2{r}_{{\rm{C}}})+B$$, and the Clifford gate fidelity is given by $${F}_{{\rm{Clifford}}}=1-{r}_{{\rm{C}}}$$. The single-qubit Clifford gates are on average composed of 1.875 primitive π/2-pulses, so the π/2-pulse 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%. Source data

### 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 on-resonance by applying a low-power (26 dB lower than the typical operating power) π-rotation. If the spin-up probability is above the threshold of 50% of the readout visibility, the experiment will continue. If the spin-up 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 fidelities37,38,39. b, c, Resonance frequency fluctuations $${\rm{\Delta }}f={f}_{1\downarrow }-{f}_{{\rm{avg}}}$$ of f1↓ (b) and $${\rm{\Delta }}f={f}_{2\downarrow }-{f}_{{\rm{avg}}}$$ of f2↓ (c) during the measurement period. We subtracted the average values of the respective frequencies favg 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 B0 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 Javg 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 600-kHz 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 single-electron 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 29Si nuclei that locally couple to the quantum dots. Source data

## Supplementary information

### Supplementary Information

This file contains Supplementary Text, Supplementary Figures 1-4 and additional references.

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Huang, W., Yang, C.H., Chan, K.W. et al. Fidelity benchmarks for two-qubit gates in silicon. Nature 569, 532–536 (2019). https://doi.org/10.1038/s41586-019-1197-0

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