Abstract
Fault-tolerant quantum computers that can solve hard problems rely on quantum error correction1. One of the most promising error correction codes is the surface code2, which requires universal gate fidelities exceeding an error correction threshold of 99 per cent3. Among the many qubit platforms, only superconducting circuits4, trapped ions5 and nitrogen-vacancy centres in diamond6 have delivered this requirement. Electron spin qubits in silicon7,8,9,10,11,12,13,14,15 are particularly promising for a large-scale quantum computer owing to their nanofabrication capability, but the two-qubit gate fidelity has been limited to 98 per cent owing to the slow operation16. Here we demonstrate a two-qubit gate fidelity of 99.5 per cent, along with single-qubit gate fidelities of 99.8 per cent, in silicon spin qubits by fast electrical control using a micromagnet-induced gradient field and a tunable two-qubit coupling. We identify the qubit rotation speed and coupling strength where we robustly achieve high-fidelity gates. We realize Deutsch–Jozsa and Grover search algorithms with high success rates using our universal gate set. Our results demonstrate universal gate fidelity beyond the fault-tolerance threshold and may enable scalable silicon quantum computers.
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Data availability
All data in this study are available from the Zenodo repository at https://doi.org/10.5281/zenodo.5508362.
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Acknowledgements
We thank the Microwave Research Group in Caltech for technical support. This work was supported financially by Core Research for Evolutional Science and Technology (CREST), Japan Science and Technology Agency (JST) (JPMJCR15N2 and JPMJCR1675), MEXT Quantum Leap Flagship Program (MEXT Q-LEAP) grant numbers JPMXS0118069228, JST Moonshot R&D grant number JPMJMS2065, and JSPS KAKENHI grant numbers 16H02204, 17K14078, 18H01819, 19K14640 and 20H00237. T.N. acknowledges support from JST PRESTO grant number JPMJPR2017.
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A.N. and K.T. fabricated the device and performed the measurements. T.N. and T.K. contributed the data acquisition and discussed the results. A.S. and G.S. developed and supplied the 28silicon/silicon-germanium heterostructure. A.N. wrote the manuscript with inputs from all co-authors. S.T. supervised the project.
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Extended data figures and tables
Extended Data Fig. 1 Detuning dependence of EDSR spectra.
a, Stability diagram around the (1,1) charge state. b, Quantum circuit for producing c. The microwave frequency of the \({\rm{\pi }}\) CROT on Q1 is varied to measure EDSR spectra. c, Detuning dependence of EDSR spectra of Q1. The detuning axis and its origin are shown as the white arrow and square in a. Three black symbols show the conditions where the dephasing times \({T}_{2,1,\downarrow }^{\ast }\) shown in d–f are measured. d–f, Ramsey fringes of Q1 when Q2 is spin-down measured at the detuning\(\,=-0.009\) V (d), \(0\) V (e), and \(0.009\) V (f). The integration time is 87 s for all of the traces. The errors in \({T}_{2,1,\downarrow }^{\ast }\) represent the estimated standard errors for the best-fit values. We observe longer (shorter) \({T}_{2,1,\downarrow }^{\ast }\) when the slope of the EDSR frequency against the detuning is smaller (larger), indicating the detuning charge noise limits \({T}_{2,1,\downarrow }^{\ast }\) at the charge-symmetry point where a finite slope exists due to the micromagnet-induced gradient field. A similar tendency is also observed in all the \({T}_{2,{\rm{m}},\sigma }^{\ast }\).
Extended Data Fig. 2 Qubits characterizations.
a, b, Sequences to measure spin relaxation times for Q1 when Q2 is spin-down, \({T}_{1,1,\downarrow }\) (a) and -up, \({T}_{1,1,\uparrow }\) (b). c, Spin-up probability as a function of the wait time. All of the traces do not show a decaying property indicating that spin relaxation is negligible for both qubits. The purple (magenta) curve is obtained using the sequence shown in a (b). The roles of Q1 and Q2 are swapped to measure the data for Q2. Each trace is offset by \(0.45\) for clarity. All of the measurements are performed with \(J=18.85\) MHz and \({f}_{{\rm{R}}}=4.867\) MHz. d, e, Ramsey sequences to measure dephasing times for Q1, \({T}_{2,1,\downarrow }^{\ast }\) and \({T}_{2,1,\uparrow }^{\ast }\). f, Ramsey fringes of Q1 and Q2 fitted with Gaussian decaying oscillation functions. The integration time is 87 s for all of the traces. The errors represent the estimated standard errors for the best-fit values. Each trace is offset by \(0.6\) for clarity. g, h, Echo sequences to measure echo times for Q1, \({T}_{2,1,\downarrow }^{{\rm{echo}}}\) and \({T}_{2,1,\uparrow }^{{\rm{echo}}}\). The phase of the final \({\rm{\pi }}/2\) rotation is varied and the amplitude of the measured oscillation as a function of the phase is plotted in i. i, Echo amplitudes as a function of the evolution time. The exponent of the decay is \(1.5,\,1.2,\,1.8\), and \(1.6\) for \({T}_{2,1,\downarrow }^{{\rm{echo}}}\), \({T}_{2,1,\uparrow }^{{\rm{echo}}}\), \({T}_{2,2,\downarrow }^{{\rm{echo}}}\), and \({T}_{2,2,\uparrow }^{{\rm{echo}}}\). The errors represent the estimated standard errors for the best-fit values. Each trace is offset by \(0.2\) for clarity. j, k, Measurement of Rabi decay time for Q1, \({T}_{2,1,\downarrow }^{{\rm{Rabi}}}\), and \({T}_{2,1,\uparrow }^{{\rm{Rabi}}}\). We measure Rabi oscillations by varying microwave burst time \({t}_{{\rm{burst}}}\) from \(0.01\) μs to \(0.41\) μs with a separation of \(0.01\) μs. Rabi oscillations for longer \({t}_{{\rm{burst}}}\) (offset by \(20\), \(40\), and \(80\) μs) are also measured and the amplitudes of the oscillations are plotted in l. l, Rabi oscillation amplitude as a function of the microwave burst time with decaying fits. The decay follows \({R}_{m,\sigma }(t)=\exp \,(-t/{T}_{2,m,\sigma }^{{\rm{Rabi}}})W(t)\) where \(W(t)={(1+{t}^{2}/{({f}_{{\rm{R}}}{({T}_{2,m,\sigma }^{\ast })}^{2})}^{2})}^{-1/4}\) represents the effect of dephasing32. From the fit, we extract the Rabi decay during a \({\rm{\pi }}/2\) CROT as \({D}_{m,\sigma }={R}_{m,\sigma }(t=1/(4{f}_{{\rm{R}}}))\). The errors represent the estimated standard errors for the best-fit values. Each trace is offset by \(0.5\) for clarity.
Extended Data Fig. 3 Single-tone single-qubit gate performance.
a, b, Quantum circuits of single-tone single-qubit Clifford-based randomized benchmarking for Q1 when Q2 is spin-down (a) and -up (b). c, Single-tone single-qubit primitive gate fidelities \({F}_{{\rm{p}},m,\sigma }\) assessed by the Clifford-based randomized benchmarking. The purple (magenta) curve is obtained using the sequence shown in a (b). The roles of Q1 and Q2 are swapped to measure the data for Q2. \({f}_{{\rm{R}}}=4.867\) MHz and \(J=18.85\) MHz \(=\sqrt{15}{f}_{{\rm{R}}}\) are used. Each trace is offset by \(0.15\) for clarity. The uncertainty in the gate fidelities are obtained by a Monte Carlo method4. The obtained fidelities are consistent with those obtained in Fig. 2c as \({F}_{{\rm{p}},m}\approx {F}_{{\rm{p}},m,\downarrow }{F}_{{\rm{p}},m,\uparrow }\). d, Rabi frequency dependence of single-tone single-qubit primitive gate infidelities. Since the control qubit state is fixed in this measurement, the off-resonant rotation does not matter so that \({f}_{{\rm{R}}}\) can be varied under a fixed \(J\) of \(32.0\) MHz. Therefore, the impact of \({f}_{{\rm{R}}}\) on the single-qubit gate performance is assessed without involving the effect of \(J\). We find that the fidelities depend on \({f}_{{\rm{R}}}\) and the best values are obtained at \({f}_{{\rm{R}}}=2\)–\(5\) MHz. Around the best condition, the fidelities are uniformly high suggesting that the fidelity is mostly limited by pulse imperfections and calibration errors rather than dephasing and Rabi decay effects. The uncertainty in the gate fidelities are obtained by a Monte Carlo method4.
Extended Data Fig. 4 Two-qubit gate fidelity extraction.
a, Number of Clifford gates \(n\) dependence of the projection state probability \({{\rm{P}}}_{\uparrow \uparrow }\)4,16. The ideal final state is spin-up for both qubits. To extract gate fidelity, we need to measure the saturation value of \({{\rm{P}}}_{\uparrow \uparrow }\) with a large \(n\) (Methods). The uncertainty in the gate fidelity is obtained by a Monte Carlo method4. b, Gate fidelity extraction from the sequence fidelity \({F}_{{\rm{t}}}\). In addition to the data in a, we measure another data set where the final ideal state is spin-down for both qubits and then obtain \({F}_{{\rm{t}}}\) as shown in blue (Methods). The saturation value of \({F}_{{\rm{t}}}\) is almost zero (\({F}_{{\rm{t}}}(271)=-\,0.007\)) as expected. Gate fidelity extraction using only the data up to \(n=62\) is shown in red. The uncertainty in the gate fidelities are obtained by a Monte Carlo method4. The trace is offset by \(0.1\) for clarity. The obtained gate fidelities agree well with that obtained in the standard protocol in a. The uncertainty in the fidelity is larger in a due to the uncertainty of the saturation value of \({{\rm{P}}}_{\uparrow \uparrow }\). \({f}_{{\rm{R}}}=5.732\) MHz and \(J=22.2\) MHz are used.
Extended Data Fig. 5 Estimation of two-qubit primitive gate infidelity by resonance frequency noise.
a, Time dependence of \(\Delta J/2=(\Delta {f}_{1,\uparrow }-\Delta {f}_{1,\downarrow })/2\) (blue), \(\Delta {f}_{1}=(\Delta {f}_{1,\uparrow }+\Delta {f}_{1,\downarrow })/2\) (purple), and \(\Delta {f}_{2}=(\Delta {f}_{2,\uparrow }+\Delta {f}_{2,\downarrow })/2\) (orange) extracted from repeated Ramsey fringe measurements (Methods). \(J\) is fixed at \(18.85\) MHz. Each trace is offset by \(0.25\) MHz for clarity. Single-qubit frequency noises (\(\Delta {f}_{1}\) and \(\Delta {f}_{2}\)) are larger than that of the exchange noise \(\Delta J/2\). b, Simulation of a two-qubit primitive gate infidelity by the frequency noises obtained in a (Methods). c, Similar to b but the case with inserting an idle time for both qubits to remove the controlled-phase accumulation during the CROT when switching \(J\) on and off18,31.
Extended Data Fig. 6 Detuning dependence of the two-qubit gate performance.
a, Detuning dependence of \(J\). \(J\) at the charge-symmetry point (detuning \(=\,0\) mV) is \(18.85\) MHz. b, Detuning dependence of the two-qubit primitive gate fidelity \({F}_{{\rm{p}}}\) (indigo circles) and the Rabi decay during the \({\rm{\pi }}/2\) CROT (colored squares) obtained similarly to Fig. 1f. Around the charge-symmetry point, we reproducibly obtain \({F}_{{\rm{p}}}\) higher than \(99\)%. In large positive and negative detuning, \({F}_{{\rm{p}}}\) sharply drops mainly due to the fast Rabi decay. The uncertainty in the gate fidelity is obtained by a Monte Carlo method4. The errors in the Rabi decay represent the estimated standard errors for the best-fit values.
Extended Data Fig. 7 Measurement error calibration in state tomography.
Typical joint probabilities measured with preparing \(|\uparrow \uparrow \rangle \), \(\tilde{|\uparrow \downarrow }\rangle \), \(\tilde{|\downarrow \uparrow }\rangle \), and \(|\downarrow \downarrow \rangle \). At \(J\,=18.85\) MHz, \(\tilde{|\downarrow \uparrow }\rangle =0.9995|\downarrow \uparrow \rangle +0.0310|\uparrow \downarrow \rangle \).
Extended Data Fig. 8 Output state of Deutsch–Jozsa algorithm and Grover search algorithm.
a–c, Real part of the measured density matrix for the final output states for \({f}_{0}\) (a), \({f}_{1}\) (b), and \({f}_{3}\) (c) in the Deutsch–Jozsa algorithm (Fig. 4a). d–f, Real part of the measured density matrix for the final output states for \({f}_{10}\) (d), \({f}_{01}\) (e), and \({f}_{00}\) (f) in the Grover search algorithm (Fig. 4b). The absolute values of the matrix elements for the imaginary parts are less than \(0.055\) (a), \(0.056\) (b), \(0.040\) (c), \(0.111\) (d), \(0.072\) (e), and \(0.081\) (f). The uncertainty in the state fidelities \(F\) are obtained by a Monte Carlo method16,19,41.
Extended Data Fig. 9 Bell state tomography.
a, Quantum circuit for the Bell state tomography. After the first \({\rm{\pi }}/2\) rotation, \({\rm{Z}}\text{-}{{\rm{CNOT}}}_{2}\) and \(-{{\rm{Z}}}_{2}/2\) \(({{\rm{Z}}}_{2}/2)\) are applied for b (c), \({{\rm{CNOT}}}_{2}\) and \({{\rm{Z}}}_{2}/2\) \((-{{\rm{Z}}}_{2}/2)\) are applied for d (e). \({I},\,{X}/2,\,{Y}/2,\,\)and \({X}\) acting on both qubits at the end change the measurement axis to implement the state tomography (Methods). b–e, Real part of the measured density matrix for the prepared Bell states for \({\Phi }^{-}\) (b), \({\Phi }^{+}\) (c), \({\Psi }^{-}\) (d), and \({\Psi }^{+}\) (e), respectively. The absolute values of the matrix elements for the imaginary parts are less than \(0.038\) (b), \(0.093\) (c), \(0.100\) (d), \(0.113\) (e). The uncertainty in the state fidelities \(F\) are obtained by a Monte Carlo method16,19,41.
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Noiri, A., Takeda, K., Nakajima, T. et al. Fast universal quantum gate above the fault-tolerance threshold in silicon. Nature 601, 338–342 (2022). https://doi.org/10.1038/s41586-021-04182-y
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DOI: https://doi.org/10.1038/s41586-021-04182-y
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