Abstract
Quantum entanglement is a fundamental property of coherent quantum states and an essential resource for quantum computing^{1}. In largescale quantum systems, the error accumulation requires concepts for quantum error correction. A first step toward error correction is the creation of genuinely multipartite entanglement, which has served as a performance benchmark for quantum computing platforms such as superconducting circuits^{2,3}, trapped ions^{4} and nitrogenvacancy centres in diamond^{5}. Among the candidates for largescale quantum computing devices, siliconbased spin qubits offer an outstanding nanofabrication capability for scalingup. Recent studies demonstrated improved coherence times^{6,7,8}, highfidelity allelectrical control^{9,10,11,12,13}, hightemperature operation^{14,15} and quantum entanglement of two spin qubits^{9,11,12}. Here we generated a threequbit Greenberger–Horne–Zeilinger state using a lowdisorder, fully controllable array of three spin qubits in silicon. We performed quantum state tomography^{16} and obtained a state fidelity of 88.0%. The measurements witness a genuine Greenberger–Horne–Zeilinger class quantum entanglement that cannot be separated into any biseparable state. Our results showcase the potential of siliconbased spin qubit platforms for multiqubit quantum algorithms.
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Data availability
All data in this study are available from the Zenodo repository at https://doi.org/10.5281/zenodo.4722605. Source data are provided with this paper.
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Acknowledgements
We thank the Microwave Research Group at 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 no. JPMXS0118069228, JST Moonshot R&D grant no. JPMJMS2065 and JSPS KAKENHI grant nos. 16H02204, 17K14078, 18H01819, 19K14640 and 20H00237. T.N. acknowledges support from the Murata Science Foundation Research Grant and JST, PRESTO grant no. JPMJPR2017.
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K.T. and A.N. fabricated the device and performed the measurements. T.N., J.Y. and T.K. contributed the data acquisition and discussed the results. K.T. wrote the paper with inputs from all the coauthors. S.T. supervised the project.
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Extended data
Extended Data Fig. 1 Initialization and measurement protocol.
Initialization and readout procedure. The spin readout and initialization for Q_{1} is performed near the (111)–(011) boundary, whereas for Q_{2} and Q_{3} it is performed near the (111)–(110) boundary. Note that Q_{2} cannot be directly read out through the reservoir as the cotunnelling rate between the (111) and (101) states is too small compared to the spin relaxation rate \(T_1^{  1}\). The dwell times are 450 μs for Q_{3} initialization, 750 μs for Q_{1} initialization, 300 μs for Q_{3} readout and 750 μs for Q_{1} readout. The resonant SWAP pulses are 0.25 μs long and it corresponds to an exchange Rabi frequency of 2 MHz. The resonance frequency is typically around 410 MHz. The readout dwell time of Q_{3} is compromised for the finite T_{1} relaxation times of Q_{1} and Q_{2} and it results in imperfect spindown initialization after the readout stage. Therefore, in order to increase the initialization fidelities, we use an explicit initialization stage which is redundant in the ideal case where all the three spins are initialized to spindown after the readout stage.
Extended Data Fig. 2 Singlequbit characterization.
All qubits are initialized to spindown before the manipulation stage and only one of the qubits is read out right after the manipulation stage unless noted. The exchange interactions are turned off. All errors represent the estimated standard errors for the bestfit values. a–d, T_{1} measurements. First, a spinup state is prepared using an X pulse. Then we vary the waiting time of t_{w} at the singlequbit manipulation point before performing singleshot measurement. In this T_{1} measurement, all three spins are sequentially read out and therefore the visibilities of Q_{1} and Q_{2} are decreased by T_{1} relaxation during the readout stage. Note that the visibility of Q_{3} is unaffected by the sequential readout. e–h, Ramsey interferometry measurements. First, a π/2 pulse (+2 MHz detuned from the resonance frequency) is applied to rotate the spin state to the xyplane in the Bloch sphere. After an evolution time of t_{evol}, another π/2 pulse is applied to project the spin state in the zaxis for measurement. The black solid curves are the fit with Gaussian decay. The integration time is 75.8 sec for all qubits. i–l, Hahn echo measurements. Each fitting curve is given by P_{↑}(t_{evol}) = Aexp(−(t_{evol}/T_{2}^{echo})^{γ}) + B, where A and B are the constants to account for the readout fidelities and γ is an exponent. The exponents are found to be γ = 1.79 ± 0.12 (Q_{1}), 2.75 ± 0.10 (Q_{2}) and 2.61 ± 0.09 (Q_{3}).
Extended Data Fig. 3 Additional randomized benchmarking measurements.
a, Measurement result with an asymmetric readout condition. F_{1} (F_{0}) is the spinup probability when the recovery Clifford gate is chosen so that the ideal final spin state is spinup (down). The data points at m = 0 are measured without any random Clifford gates applied. Only an X pulse is applied in the case of F_{1} and no pulse is applied in the case of F_{0}. The dashed line shows a constant 0.462, the expected saturation value derived from the readout asymmetry. b, Measurement result with a more symmetric readout condition.
Extended Data Fig. 4 Randomized benchmarking with detuned microwave frequency.
a, Randomized benchmarking sequence for Q_{2} fidelity measurement. b, Randomized benchmarking measurement result of Q_{2} with a frequency detuning of 0.6 MHz. For each of the control bit (Q_{3}) states, the measurement is performed for 16 sets of random Clifford gate sequences. The sequence fidelity shows an average of the results for the two control bit configurations. The errors represent the estimated standard errors for the bestfit values. c, d, Similar randomized benchmarking measurement performed for Q_{3}. The errors represent the estimated standard errors for the bestfit values.
Extended Data Fig. 5 Measurements of exchange interactions.
All errors represent the estimated standard errors for the bestfit values. a, b, Controlledrotation for Q_{1} and Q_{2}. The measurement is performed to probe \(J_{12}^\prime\). First, Q_{1} and Q_{2} are initialized to spindown. To prepare a spinup control qubit (Q_{2}) state, an X pulse is applied. After tuning on \(J_{12}^\prime\) by a gate voltage pulse, a lowpower Gaussian microwave pulse (truncated at ±2σ) is applied to induce a controlledrotation. The filled (open) circles show the measured spinup probabilities with the control qubit spindown (up). The solid lines are Gaussian fitting curves. From the separation of the two peaks, we obtain \(J_{12}^\prime\) = 2.75 ± 0.02 MHz. c, d, Similar controlled rotation measurement for Q_{2} and Q_{3}. We obtain \(J_{23}^\prime\) = 12.50 ± 0.02 MHz from this measurement. e, Ramsey experiment to extract \(J_{12}^{{\mathrm{off}}}\). We perform two Ramsey measurements of Q_{1} with different control qubit (Q_{2}) states. The difference of qubit frequency detuning is equivalent to \(J_{12}^{{\mathrm{off}}}\). f, Ramsey measurement result when Q_{2} is spindown. The red circles are the measured Q_{1} spinup probabilities and the black solid curve shows a fit with Gaussian decay. From the oscillation frequency of the decay curve, we extract δf_{↓} = 2.28 ± 0.01MHz. g, Measurement similar to the one in f when Q_{2} is spinup. We extract δf_{↑} = 2.21 ± 0.01MHz. Since the difference between δf_{↓} and δf_{↑} is below the stochastic fluctuation of the frequency detuning, we conclude that \(J_{12}^{{\mathrm{off}}}\) is below our detection limit. Note that each frequency error shows one standard deviation of the fitting parameter. h–j, Ramsey experiments to extract \(J_{23}^{{\mathrm{off}}}\). We obtain \(J_{23}^{{\mathrm{off}}}\) = δf_{↑}−δf_{↓} = 1.17 ± 0.01 MHz from these measurements.
Extended Data Fig. 6 Bell state tomography using Q_{2} and Q_{3}.
As a benchmark of our twoqubit CZ gate, we perform Bell state tomography on Q_{2} and Q_{3}. The experiment is a reduced version of the threequbit GHZ state tomography. The readout errors are removed using the measured readout fidelities and maximum likelihood estimation is used to reconstruct the density matrices. a, Quantum gate sequence for Bell state creation and state tomography. By modifying the phase gates after the second CZ/2 pulse, we can create all four Bell states. b–e, Real parts of the measured density matrices for four Bell states, \(\Phi ^ + = ({\uparrow \uparrow}\rangle + {\downarrow \downarrow}\rangle )/\sqrt 2\) (b), \(\Phi ^  = ({\uparrow \uparrow}\rangle  {\downarrow \downarrow} \rangle )/\sqrt 2\) (c), \(\Psi ^ + = ({\downarrow \uparrow}\rangle + {\downarrow \uparrow}\rangle)/\sqrt 2\) (d), and \(\Psi ^  = ({\downarrow \uparrow}\rangle  {\uparrow \downarrow}\rangle)/\sqrt 2\) (e). We obtain the state fidelities relative to the target states of 0.942 (Φ^{+}), 0.933 (Φ^{−}), 0.950 (Ψ^{+}) and 0.940 (Ψ^{−}), and the concurrences of 0.950 (Φ^{+}), 0.906 (Φ^{−}), 0.923 (Ψ^{+}) and 0.935 (Ψ^{−}).
Extended Data Fig. 7 Imaginary part of the experimental GHZ state.
The imaginary part is all zero for an ideal GHZ state. Here, the maximum absolute value of the matrix elements is 0.09.
Source data
Source Data Fig. 1
Numerical data used to generate Fig. 1.
Source Data Fig. 2
Numerical data used to generate Fig. 2.
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Numerical data used to generate Fig. 3.
Source Data Extended Data Fig. 2
Numerical data used to generate Extended Data Fig. 2.
Source Data Extended Data Fig. 3
Numerical data used to generate Extended Data Fig. 3.
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Numerical data used to generate Extended Data Fig. 4.
Source Data Extended Data Fig. 5
Numerical data used to generate Extended Data Fig. 5.
Source Data Extended Data Fig. 6
Numerical data used to generate Extended Data Fig. 6.
Source Data Extended Data Fig. 7
Numerical data used to generate Extended Data Fig. 7.
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Takeda, K., Noiri, A., Nakajima, T. et al. Quantum tomography of an entangled threequbit state in silicon. Nat. Nanotechnol. 16, 965–969 (2021). https://doi.org/10.1038/s41565021009250
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DOI: https://doi.org/10.1038/s41565021009250
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