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Quantum tomography of an entangled three-qubit state in silicon

Nature Nanotechnology volume 16, pages 965–969 (2021)Cite this article

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Abstract

Quantum entanglement is a fundamental property of coherent quantum states and an essential resource for quantum computing1. In large-scale 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 circuits2,3, trapped ions4 and nitrogen-vacancy centres in diamond5. Among the candidates for large-scale quantum computing devices, silicon-based spin qubits offer an outstanding nanofabrication capability for scaling-up. Recent studies demonstrated improved coherence times6,7,8, high-fidelity all-electrical control9,10,11,12,13, high-temperature operation14,15 and quantum entanglement of two spin qubits9,11,12. Here we generated a three-qubit Greenberger–Horne–Zeilinger state using a low-disorder, fully controllable array of three spin qubits in silicon. We performed quantum state tomography16 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 silicon-based spin qubit platforms for multiqubit quantum algorithms.

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Fig. 1: Device and experimental setup.
Fig. 2: Single-qubit and controlled phase operations.
Fig. 3: Three-qubit entanglement generation and measurement.

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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 Q-LEAP) 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.

Author information

Author notes

  1. Jun Yoneda

    Present address: Tokyo Tech Academy for Super Smart Society, Tokyo Institute of Technology, Tokyo, Japan

Authors and Affiliations

  1. Center for Emergent Matter Science (CEMS), RIKEN, Wako-shi, Saitama, Japan

    Kenta Takeda, Akito Noiri, Takashi Nakajima, Jun Yoneda, Takashi Kobayashi & Seigo Tarucha

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  1. Kenta Takeda
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  3. Takashi Nakajima
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Contributions

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 co-authors. S.T. supervised the project.

Corresponding authors

Correspondence to Kenta Takeda or Seigo Tarucha.

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The authors declare no competing interests.

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Peer review information Nature Nanotechnology thanks Joseph Kerckhoff and the other, anonymous, reviewer(s) for their contribution to the peer review of this work.

Publisher’s note Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Extended data

Extended Data Fig. 1 Initialization and measurement protocol.

Initialization and readout procedure. The spin readout and initialization for Q1 is performed near the (111)–(011) boundary, whereas for Q2 and Q3 it is performed near the (111)–(110) boundary. Note that Q2 cannot be directly read out through the reservoir as the co-tunnelling 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 Q3 initialization, 750 μs for Q1 initialization, 300 μs for Q3 readout and 750 μs for Q1 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 Q3 is compromised for the finite T1 relaxation times of Q1 and Q2 and it results in imperfect spin-down 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 spin-down after the readout stage.

Extended Data Fig. 2 Single-qubit characterization.

All qubits are initialized to spin-down 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 best-fit values. a–d, T1 measurements. First, a spin-up state is prepared using an X pulse. Then we vary the waiting time of tw at the single-qubit manipulation point before performing single-shot measurement. In this T1 measurement, all three spins are sequentially read out and therefore the visibilities of Q1 and Q2 are decreased by T1 relaxation during the readout stage. Note that the visibility of Q3 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 xy-plane in the Bloch sphere. After an evolution time of tevol, another π/2 pulse is applied to project the spin state in the z-axis 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↑(tevol) = Aexp(−(tevol/T2echo)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 (Q1), 2.75 ± 0.10 (Q2) and 2.61 ± 0.09 (Q3).

Source data

Extended Data Fig. 3 Additional randomized benchmarking measurements.

a, Measurement result with an asymmetric readout condition. F1 (F0) is the spin-up probability when the recovery Clifford gate is chosen so that the ideal final spin state is spin-up (-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 F1 and no pulse is applied in the case of F0. 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.

Source data

Extended Data Fig. 4 Randomized benchmarking with detuned microwave frequency.

a, Randomized benchmarking sequence for Q2 fidelity measurement. b, Randomized benchmarking measurement result of Q2 with a frequency detuning of 0.6 MHz. For each of the control bit (Q3) 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 best-fit values. c, d, Similar randomized benchmarking measurement performed for Q3. The errors represent the estimated standard errors for the best-fit values.

Source data

Extended Data Fig. 5 Measurements of exchange interactions.

All errors represent the estimated standard errors for the best-fit values. a, b, Controlled-rotation for Q1 and Q2. The measurement is performed to probe \(J_{12}^\prime\). First, Q1 and Q2 are initialized to spin-down. To prepare a spin-up control qubit (Q2) state, an X pulse is applied. After tuning on \(J_{12}^\prime\) by a gate voltage pulse, a low-power Gaussian microwave pulse (truncated at ±2σ) is applied to induce a controlled-rotation. The filled (open) circles show the measured spin-up probabilities with the control qubit spin-down (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 Q2 and Q3. 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 Q1 with different control qubit (Q2) states. The difference of qubit frequency detuning is equivalent to \(J_{12}^{{\mathrm{off}}}\). f, Ramsey measurement result when Q2 is spin-down. The red circles are the measured Q1 spin-up 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 Q2 is spin-up. 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.

Source data

Extended Data Fig. 6 Bell state tomography using Q2 and Q3.

As a benchmark of our two-qubit CZ gate, we perform Bell state tomography on Q2 and Q3. The experiment is a reduced version of the three-qubit 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−).

Source data

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

Source Data Fig. 1

Numerical data used to generate Fig. 1.

Source Data Fig. 2

Numerical data used to generate Fig. 2.

Source Data Fig. 3

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.

Source Data Extended Data Fig. 4

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 three-qubit state in silicon. Nat. Nanotechnol. 16, 965–969 (2021). https://doi.org/10.1038/s41565-021-00925-0

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