Abstract
Faulttolerant quantum computers that can solve hard problems rely on quantum error correction^{1}. One of the most promising error correction codes is the surface code^{2}, which requires universal gate fidelities exceeding an error correction threshold of 99 per cent^{3}. Among the many qubit platforms, only superconducting circuits^{4}, trapped ions^{5} and nitrogenvacancy centres in diamond^{6} have delivered this requirement. Electron spin qubits in silicon^{7,8,9,10,11,12,13,14,15} are particularly promising for a largescale quantum computer owing to their nanofabrication capability, but the twoqubit gate fidelity has been limited to 98 per cent owing to the slow operation^{16}. Here we demonstrate a twoqubit gate fidelity of 99.5 per cent, along with singlequbit gate fidelities of 99.8 per cent, in silicon spin qubits by fast electrical control using a micromagnetinduced gradient field and a tunable twoqubit coupling. We identify the qubit rotation speed and coupling strength where we robustly achieve highfidelity 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 faulttolerance threshold and may enable scalable silicon quantum computers.
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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 QLEAP) 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 ^{28}silicon/silicongermanium heterostructure. A.N. wrote the manuscript with inputs from all coauthors. 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 Q_{1} is varied to measure EDSR spectra. c, Detuning dependence of EDSR spectra of Q_{1}. 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 Q_{1} when Q_{2} is spindown 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 bestfit 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 chargesymmetry point where a finite slope exists due to the micromagnetinduced 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 Q_{1} when Q_{2} is spindown, \({T}_{1,1,\downarrow }\) (a) and up, \({T}_{1,1,\uparrow }\) (b). c, Spinup 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 Q_{1} and Q_{2} are swapped to measure the data for Q_{2}. 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 Q_{1}, \({T}_{2,1,\downarrow }^{\ast }\) and \({T}_{2,1,\uparrow }^{\ast }\). f, Ramsey fringes of Q_{1} and Q_{2} 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 bestfit values. Each trace is offset by \(0.6\) for clarity. g, h, Echo sequences to measure echo times for Q_{1}, \({T}_{2,1,\downarrow }^{{\rm{echo}}}\) 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 bestfit values. Each trace is offset by \(0.2\) for clarity. j, k, Measurement of Rabi decay time for Q_{1}, \({T}_{2,1,\downarrow }^{{\rm{Rabi}}}\), 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 dephasing^{32}. 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 bestfit values. Each trace is offset by \(0.5\) for clarity.
Extended Data Fig. 3 Singletone singlequbit gate performance.
a, b, Quantum circuits of singletone singlequbit Cliffordbased randomized benchmarking for Q_{1} when Q_{2} is spindown (a) and up (b). c, Singletone singlequbit primitive gate fidelities \({F}_{{\rm{p}},m,\sigma }\) assessed by the Cliffordbased randomized benchmarking. The purple (magenta) curve is obtained using the sequence shown in a (b). The roles of Q_{1} and Q_{2} are swapped to measure the data for Q_{2}. \({f}_{{\rm{R}}}=4.867\) 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 method^{4}. 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 singletone singlequbit primitive gate infidelities. Since the control qubit state is fixed in this measurement, the offresonant 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 singlequbit 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 method^{4}.
Extended Data Fig. 4 Twoqubit 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 spinup 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 method^{4}. 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 spindown 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 method^{4}. 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 twoqubit 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. Singlequbit frequency noises (\(\Delta {f}_{1}\) and \(\Delta {f}_{2}\)) are larger than that of the exchange noise \(\Delta J/2\). b, Simulation of a twoqubit 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 controlledphase accumulation during the CROT when switching \(J\) on and off^{18,31}.
Extended Data Fig. 6 Detuning dependence of the twoqubit gate performance.
a, Detuning dependence of \(J\). \(J\) at the chargesymmetry point (detuning \(=\,0\) mV) is \(18.85\) MHz. b, Detuning dependence of the twoqubit 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 chargesymmetry 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 method^{4}. The errors in the Rabi decay represent the estimated standard errors for the bestfit 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 method^{16,19,41}.
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 method^{16,19,41}.
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Noiri, A., Takeda, K., Nakajima, T. et al. Fast universal quantum gate above the faulttolerance threshold in silicon. Nature 601, 338–342 (2022). https://doi.org/10.1038/s4158602104182y
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