Abstract
Control of entanglement between qubits at distant quantum processors using a twoqubit gate is an essential function of a scalable, modular implementation of quantum computation. Among the many qubit platforms, spin qubits in silicon quantum dots are promising for largescale integration along with their nanofabrication capability. However, linking distant silicon quantum processors is challenging as twoqubit gates in spin qubits typically utilize shortrange exchange coupling, which is only effective between nearestneighbor quantum dots. Here we demonstrate a twoqubit gate between spin qubits via coherent spin shuttling, a key technology for linking distant silicon quantum processors. Coherent shuttling of a spin qubit enables efficient switching of the exchange coupling with an on/off ratio exceeding 1000, while preserving the spin coherence by 99.6% for the single shuttling between neighboring dots. With this shuttlingmode exchange control, we demonstrate a twoqubit controlledphase gate with a fidelity of 93%, assessed via randomized benchmarking. Combination of our technique and a phase coherent shuttling of a qubit across a large quantum dot array will provide feasible path toward a quantum link between distant silicon quantum processors, a key requirement for largescale quantum computation.
Introduction
Electron spins in silicon quantum dots attract a lot of interest as a platform of quantum computation with highfidelity universal quantum control^{1,2,3}, long coherence time^{4,5,6}, capability of hightemperature operation^{7,8}, and potential scalability^{9,10,11,12}. With recent technical advances, a denselypacked array of singleelectron quantum dots works as a smallscale programmable quantum processor^{1,2,13,14}. To scale up quantum computation by wiring to such dense qubit arrays and alleviating signal crosstalk, a quantum link is highly demanded that allows to manipulate entanglement between distant quantum processors in a sparse configuration^{11,15}. A sizable exchange coupling required for twoqubit gates is, however, only achieved in qubits between nearestneighbor quantum dots^{1,2,3,6,12,13,16,17} as the coupling falls off exponentially with distance. Therefore twoqubit gates between distant quantum processors require coherent mediators such as microwave photons^{18,19,20,21}, empty and multielectron quantum dots^{22,23}, and spin chains^{24}. Another approach uses electron shuttling^{25,26,27,28,29,30} to physically move a qubit between quantum processors, bringing it wherever a twoqubit gate needs to be performed. However, a highfidelity twoqubit gate in either approach is still challenging.
Here we propose and demonstrate a shuttlingbased twoqubit gate which plays a key role in a quantum link between distant silicon quantum processors by electron shuttling. Figure 1a illustrates how this technique along with a coherent shuttling across a quantum dot array^{26,27} can be used to interconnect two distant quantum processors via an empty quantum dot array, making a quantum link between them. More specifically, a qubit in one end of a quantum processor, which we call the moving qubit, is coherently moved to near the end of the other processor, where a sizable exchange coupling with a local qubit sitting there exists (Fig. 1a). Then the moving qubit is coherently shuttled back to the original quantum dot. In contrast to previous demonstrations of a controlledphase (CZ) gate^{2,3,6,13}, our twoqubit gate between the local and moving qubits relies on dynamical switching of the exchange coupling by the shuttling processes. This technique will enable to implement the twoqubit gate between qubits at distant quantum processors when combined with shuttling across a long channel.
The experiment is performed in a tunnel coupled triple quantum dot hosting two qubits, a minimum setup to demonstrate the shuttlingbased twoqubit gate. Initially, the local and moving qubits Q_{L} and Q_{M} are in the left and right dots, respectively, where parallel quantum processing with simultaneous singlequbit gates is performed. We refer to this configuration as the sparse state (Fig. 1b). The negligible coupling between the qubits enables us to maintain the high fidelity of singlequbit gates while driving both qubits simultaneously. To perform a twoqubit gate, Q_{M} in the right dot is shuttled to the center dot, and at the same time, the exchange coupling is turned on. We refer to this state as the coupled state (Fig. 1b). The shuttlingmode exchange switching allows us to efficiently control exchange coupling with an on/off ratio above 1000. By tuning an evolution time in the coupled state, we realize a CZ gate with a fidelity of 93%. Practically, a quantum link that can couple qubits separated by ∼10 µm distance is useful for scaling up^{15}. Along with the shuttlingbased CZ gate, this requires highfidelity coherent shuttling across a large quantum dot array. With a sufficiently large interdot tunnel coupling, we demonstrate that 99.6% of the spin phase coherence is preserved in a single shuttling cycle. Then, challenges to be overcome include precise control of a large quantum dot array. A virtual gate technique is useful for tuning up such a quantum dot array in a scalable manner^{25,31}. Furthermore, a recent demonstration of conveyermode shuttling^{32} can decrease the number of control signals in a longdistance shuttling. In this approach, a qubit is shuttled by an electrostatically defined travelling potential created by an array of gate electrodes which are connected to one of the four control signal sources. Then, the number of control signals is independent of the length of shuttling channel, potentially reducing the complexity of controlling a long shuttling channel. With such technical advances, our technique can implement a quantum link between spin qubits at distant quantum processors that is useful for scaling up.
Results
The device is fabricated on an isotopically enriched silicon/silicongermanium heterostructure. Three layers of aluminum gates create confinement potentials to define the quantum dots^{9} (Fig. 1c). We operate this device in two charge configurations with two qubits: the coupled state (1,1,0) and the sparse state (1,0,1), where (l, m, n) denotes the number of electrons in the left (l), center (m) and right (n) dot. On top of the quantum dots, a cobalt micromagnet is fabricated to induce a magnetic field gradient required for electricdipole spin resonance (EDSR) control of both qubits^{33}. In addition, the field gradient makes a Zeeman energy difference of 403 MHz between the left and the center dot (Supplementary Fig. 1a, b). Compared to the Zeeman energy difference, an exchange coupling J in a range of 1–10 MHz is small, and it shifts the energy levels defined by the Zeeman energy when the two spins are antiparallel. This enables us to implement a CZ gate by simply turning on and off J^{2,12,13}.
We first demonstrate initialization, measurement, and singlequbit control of the spin qubits in the sparse state. White symbols in Fig. 1d show the gate voltage conditions used for the respective stages. Initialization and measurement are performed by energyselective tunneling between quantum dots and their adjacent reservoirs^{34,35}. Supplementary Fig. 1a, c demonstrates EDSR control of Q_{L} and Q_{M}. The resonance frequencies differ by 733.4 MHz due to the micromagnet and this is large enough to control both qubits individually. The dephasing times \({T}_{2}^{*}\) are 3 and 4 µs for Q_{L} and Q_{M}, which are enhanced by the echo sequence to 18 and 28 µs, respectively (Supplementary Fig. 2a–d). We also obtain the Rabi decay times long enough (>30 µs) for highfidelity singlequbit gates (Supplementary Fig. 2e, f). We characterize the singlequbit gate fidelities by the simultaneous Cliffordbased randomized benchmarking (Fig. 1e). We obtain highfidelity singlequbit gates (singlequbit primitive gate fidelities of F_{p,s} = 99.906 ± 0.002% for Q_{L} and 99.751 ± 0.003% for Q_{M} in Fig. 1f, g) even when the same gate sequence is applied to both qubits simultaneously, which shows that these qubits work as two independent singlequbit quantum processors.
Next, we demonstrate coherent shuttling of Q_{M} using the right and center dots (Fig. 2a). The white symbols in Fig. 2b show the two gate voltage conditions for the coupled and the sparse states. The estimated interdot tunnel coupling t_{R} between the dots is 20.2 GHz (Supplementary Fig. 4). After preparing Q_{M} in the state of either spindown or spinup, we shuttle Q_{M} back and forth by applying the pulse sequence shown in Fig. 2c and measure the final spinup probability. Figure 2d shows that the initial spin polarization decays with the number of the shuttling cycles n. We extract the spin preservation fidelity per a shuttling cycle to be F_{d} = 99.975 ± 0.012% for the spindown state and F_{u} = 99.971 ± 0.007% for the spinup state, respectively. The preservation fidelity of the phase coherence is similarly evaluated by preparing Q_{M} in the spindown and up superposition state and measuring the coherence decay (Fig. 2e). We obtain a coherence preservation fidelity per a shuttling cycle of F_{p} = 99.62 ± 0.05% (Fig. 2f). These fidelities are comparable to those reported in a silicon MOS quantum dot device^{26}. This suggests that Q_{M} can be shuttled over ~500 dots (distance of ~45 µm assuming a dot pitch of 0.09 µm) before the phase coherence decays by a factor of 1/e. We note that the phase of Q_{M} shifts when it is shuttled across dots with different Zeeman energies that originate from the micromagnetinduced gradient field and a change in the interface roughness of the heterostructure across dots^{36}. Since t_{R} is sufficiently large for adiabatic shuttling of Q_{M}, this phase shift is a deterministic coherent phase shift which can be removed by a phase gate implemented by shifting phases of subsequent control microwave pulses in zero gate time^{1,12,37}.
Then, we demonstrate switching of the exchange coupling J between Q_{L} and Q_{M} by shuttling Q_{M}. This allows us to implement a twoqubit gate between the Q_{L} and Q_{M} just by switching the operation states via coherent shuttling. To tune up J in the coupled state, we tilt the energy levels of the left and center dots by the tilt voltage V_{tilt} along the black axis in Fig. 3a. J at the coupled state is evaluated by applying the quantum circuit shown in Fig. 3c with which Q_{M} accumulates the controlled phase depending on the evolution time t_{evol} at the coupled state (Fig. 3b). The π rotations for both qubits in the middle of the phase evolution decouple quasistatic noise^{13}. Figure 3d shows J and the decoupled dephasing times for both qubits as a function of V_{tilt}. While J monotonically increases with increasing V_{tilt}, the decoupled dephasing times are barely affected between V_{tilt} = 0 V and V_{tilt} = 0.012 V and they start decreasing with increasing V_{tilt} above 0.012 V. Therefore, we use V_{tilt} = 0.012 V with J = 1.25 MHz to implement the CZ gate at the maximum performance. On the other hand, we obtain a negligibly small J of 0.9 kHz in the sparse state (Supplementary Fig. 6b). These results demonstrate that a more than one thousand switching ratio of J is obtained by coherent electron shuttling.
We now use this shuttlingmode switching of J to implement a CZ gate^{13} between Q_{L} and Q_{M}. The CZ gate is operated by tuning the evolution time in the coupled state to t_{evol} = 1/2J = 0.4 µs with singlequbit phase corrections made by shifting the phase of subsequent control pulses^{12,13,37}. We use a decoupled CZ (DCZ) gate^{12,13} to suppress dephasing during the controlledphase evolution (Fig. 4a). To verify the construction of the DCZ gate, we measure the phase of Q_{M} after the DCZ gate (Fig. 4b) using the quantum circuit shown in Fig. 4a. The obtained controlled phase is 1.00 ± 0.01 π from which we demonstrate an execution of the DCZ gate. We note that the CZ gate can be implemented by the DCZ gate followed by singlequbit gates acting on both qubits (Fig. 4c). From the results, we demonstrate that the CZ gate is appropriately operated between Q_{L} and Q_{M}.
Finally, we execute twoqubit randomized benchmarking to characterize the CZ gate^{38,39}. The blue circles in Fig. 4f show the averaged sequence fidelity F_{t} (Methods) measured by the Clifford sequence shown in Fig. 4d. From the decay of the sequence fidelity, we extract a twoqubit Clifford gate fidelity F_{C} = 88.02 ± 0.06% (Methods). The CZ gate fidelity is characterized with an additional measurement (Fig. 4e) where the CZ gate is interleaved between each randomly chosen Clifford gate. By comparing the decay of sequence fidelities between with (red circles in Fig. 4f) and without the interleaved CZ gates, we extract the CZ gate fidelity of F_{CZ} = 92.72 ± 0.18% (Methods). The obtained fidelity is mostly limited by dephasing due to the slow controlledphase accumulation of 0.4 µs compared to the decoupled dephasing times of ~7 µs (Supplementary Fig. 8). Application of a barrier gate pulse in addition to the shuttling pulse would further improve the CZ gate fidelity by increasing J around the chargesymmetry point (Supplementary Note 1). In addition, the fluctuations of EDSR resonance frequencies during the data acquisition contribute to the obtained infidelity of the CZ gate. We calibrate these parameters in every ~2 h and the total data acquisition takes ~10 h. More frequent autocalibration during the measurement^{37} would further improve the gate fidelity.
Discussion
We also emphasize that the shuttlingmode exchange switching is beneficial for local qubit operations. A highfidelity twoqubit gate requires large (~10 MHz) exchange coupling for a short gate time^{1,2,3}. Except when operating the twoqubit gate, on the other hand, the coupling must be strictly turned off to below ~10 kHz to maintain the demonstrated high fidelity^{5,40} of singlequbit gates (Supplementary Fig. 9). This is because the residual coupling induces a qubit energy shift conditional on neighboring qubit states, which decreases the singlequbit gate fidelity^{2,12}. Typical residual coupling is a few tens of kHz^{2,3} in the conventional scheme where the exchange coupling is switched by tilting the energy levels of quantum dots^{6,41,42} and/or by modifying the potential barrier between quantum dots^{16,17,43,44,45}. The shuttlingmode operation naturally enables to switch the exchange coupling with a high enough on/off ratio of above 1000. We note that controllability of the coupling by the conventional schemes has been improved recently to the on/off ratio of 1000 in an advanced device structure^{46} but an even larger on/off ratio may be required for further enhancing the gate fidelity. The shuttlingmode exchange switching can be used together with the conventional technique to improve the exchange controllability and thus favorable not only for linking distant quantum processors but also for implementing highfidelity local qubit operations.
In summary, we demonstrate a CZ gate between silicon spin qubits by coherent shuttling of one of the qubits for linking distant quantum processors. The coherent shuttling allows us to shuttle a qubit while preserving its spin phase by 99.6% and simultaneously switch on and off the exchange coupling. The shuttlingmode exchange switching allows us to implement the CZ gate with a fidelity of 93% accompanied with a high on/off ratio of more than one thousand. Even higher gate fidelity will be achieved by an additional barrier gate pulse. These results demonstrate key technologies for a shuttlingbased quantum link between distant quantum processors and thereby open a path to realization of largescale spinbased quantum computation.
Methods
Measurement setup
The sample is cooled down in a dry dilution refrigerator (Oxford Instruments Triton) to the electron temperature of ∼60 mK. The dc gate voltages are supplied by a 24channel digitaltoanalog converter (QDevil ApS QDAC), which is lowpass filtered at a cutoff frequency of 800 Hz. The voltage pulses applied to the P1, P2, and P3 gate electrodes are generated by an arbitrary waveform generator (Tektronix AWG5014C). The output of the arbitrary waveform generator is lowpass filtered at a cutoff frequency of 100 MHz, which limits the time required for the electron shuttling to ∼3 ns. By inserting a ramp time for the shuttling pulse, we find that the preservation fidelity of spin phase coherence monotonically decreases with increasing the ramp time. Therefore, we omit the ramp time all through the experiments. The EDSR microwave pulses are generated using an I/Q modulated signal generator (Anritsu MG3692C with a Marki microwave MLIQ0218 I/Q mixer) and applied to the bottom screening gate. The I/Q modulation signals are generated by another arbitrary waveform generator (Tektronix AWG70002A) triggered by the arbitrary waveform generator used for generating the gate voltage pulses.
Sequence fidelity and gate fidelity extraction in randomized benchmarking
The sequence fidelity of singlequbit randomized benchmarking is obtained by the following procedure^{1,4,12,13}. We measure two data sets of spinup probability \({{{{{{\rm{P}}}}}}}_{\uparrow }\left(L\right)\) and \({{{{{{\rm{P}}}}}}}_{\uparrow }^{{\prime} }\left(L\right)\) as a function of the number of Clifford gates L. Here, the recovery Clifford gate is chosen so that the final ideal state is spinup for \({{{{{{\rm{P}}}}}}}_{\uparrow }\left(L\right)\) and spindown for \({{{{{{\rm{P}}}}}}}_{\uparrow }^{{\prime} }\left(L\right)\). Then the sequence fidelity F_{s} (L) is obtained from \({F}_{{{{{{\rm{s}}}}}}}\left(L\right)={{{{{{\rm{P}}}}}}}_{\uparrow }\left(L\right){{{{{{\rm{P}}}}}}}_{\uparrow }^{{\prime} }\left(L\right)={{A}_{{{{{{\rm{s}}}}}}}p}_{{{{{{\rm{s}}}}}}}^{L}\), where p_{s} is the depolarizing parameter and A_{s} is the constant which absorbs the state preparation and measurement errors. We average 24 random sequences, each of which are repeated 1000 times to measure F_{s} (L). The Clifford gate fidelity F_{C,s} is obtained by F_{C,s} = (1 + p_{s}) / 2. Since a Clifford gate contains 1.875 primitive gates on average, we extract the primitive gate fidelity F_{p,s} as \({F}_{{{{{{\rm{p}}}}}},{{{{{\rm{s}}}}}}}=1(1{F}_{{{{{{\rm{C}}}}}},{{{{{\rm{s}}}}}}})/1.875\).
Similarly, the sequence fidelity of twoqubit randomized benchmarking is extracted by the following procedure^{1}. We measure \({{{{{{\rm{P}}}}}}}_{\uparrow \uparrow }(L)\) (\({{{{{{\rm{P}}}}}}}_{\uparrow \uparrow }^{{\prime} }(L)\)) as a function of L with the recovery Clifford gate chosen so that the final ideal state is spinup (spindown) for both qubits. Here \({{{{{{\rm{P}}}}}}}_{\uparrow \uparrow }\) and \({{{{{{\rm{P}}}}}}}_{\uparrow \uparrow }^{{\prime} }\) is the joint probability of spinup in both qubits. Then the sequence fidelity F_{t} (L) is extracted from \({F}_{{{{{{\rm{t}}}}}}}\left(L\right)={{{{{{\rm{P}}}}}}}_{\uparrow \uparrow }\left(L\right){{{{{{\rm{P}}}}}}}_{\uparrow \uparrow }^{{\prime} }\left(L\right)={A}_{{{{{{\rm{t}}}}}}}{p}_{{{{{{\rm{t}}}}}}}^{L}\), where p_{t} is the depolarizing parameter and A_{t} is the constant which absorbs the state preparation and measurement errors. We average 50 random sequences each of which are repeated 2000 times to measure F_{t} (L). The twoqubit Clifford gate fidelity is obtained by \({F}_{{{{{{\rm{C}}}}}}}=(1+3{p}_{{{{{{\rm{t}}}}}}})/4\).
The CZ gate fidelity is obtained as follows^{1,38}. We first measure F_{t} (L) by applying random Clifford gates (Fig. 4d) and obtain the depolarizing parameter p_{ref} as a reference. We also measure F_{t} (L) by applying the CZ gate between each random Clifford gates (Fig. 4e) and obtain the depolarizing parameter p_{CZ}. Then we extract the CZ gate fidelity as \({F}_{{{{{{\rm{CZ}}}}}}}=(1+3{p}_{{{{{{\rm{CZ}}}}}}}/{p}_{{{{{{\rm{ref}}}}}}})/4\).
The errors of the gate fidelities are obtained by a Monte Carlo method^{1,38}. We fit the resulting fidelity distribution by the Gaussian distribution and extract its standard deviation.
Data availability
The data that support findings in this study are available from the Zenodo repository at https://doi.org/10.5281/zenodo.7033594.
Change history
01 November 2022
A Correction to this paper has been published: https://doi.org/10.1038/s41467022342362
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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 Nos. JPMXS0118069228, JST Moonshot R&D Grant Number JPMJMS226B, and JSPS KAKENHI grant Nos. 16H02204, 17K14078, 18H01819, 19K14640, and 20H00237. T.N. acknowledges support from JST PRESTO Grant Number JPMJPR2017.
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A.N. and T.N. conceived the project. 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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Noiri, A., Takeda, K., Nakajima, T. et al. A shuttlingbased twoqubit logic gate for linking distant silicon quantum processors. Nat Commun 13, 5740 (2022). https://doi.org/10.1038/s4146702233453z
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DOI: https://doi.org/10.1038/s4146702233453z
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