Abstract
Spin qubits in quantum dots define an attractive platform for quantum information because of their compatibility with semiconductor manufacturing, their long coherence times, and the ability to operate above one Kelvin. However, despite demonstrations of SWAP oscillations, the integration of this twoqubit gate together with singlequbit control to create a universal gate set as originally proposed for single spins in quantum dots has remained elusive. Here, we show that we can overcome these limitations and execute a multitude of native twoqubit gates, together with singlequbit control, in a single device, reducing the operation overhead to perform quantum algorithms. We demonstrate singlequbit rotations, together with the twoqubit gates CROT, CPHASE, and SWAP, on a silicon double quantum dot. Furthermore, we introduce adiabatic and diabatic composite sequences that allow the execution of CPHASE and SWAP gates on the same device, despite the finite Zeeman energy difference. Both twoqubit gates can be executed in less than 100 ns and, by theoretically analyzing the experimental noise sources, we predict control fidelities exceeding 99%, even for operation above one Kelvin.
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Introduction
Twoqubit gates are at the heart of quantum information science, as they may be used to create entangled states with a complexity beyond what is classically simulatable^{1}, and ultimately may enable the execution of practically relevant quantum algorithms^{2}. Optimizing twoqubit gates is therefore a central aspect across all qubit platforms^{3}. In quantum dot systems, twoqubit gates can be naturally implemented using the exchange interaction between spin qubits in neighboring quantum dots^{4}. Pulsing the interaction drives SWAP oscillations, where the spin states in the quantum dots are being exchanged, when the exchange energy is much larger than the Zeeman energy difference of the qubits^{4,5,6}, while it results in controlledphase (CPHASE) oscillations, where only the phase information is exchanged, when the Zeeman energy difference is much larger than the exchange energy^{7}. Singlequbit gates need also to be implemented to access the full twoqubit Hilbert space, and this requires distinguishability between the qubits. This is commonly obtained through the spinorbit coupling^{8,9} or by integrating nanomagnets^{10,11}, causing significant Zeeman energy differences. Realizing a highfidelity SWAPgate in this scenario would require extremely large values of exchange interaction. For this reason, the CPHASE operation has been the native gate in experimental demonstrations of twoqubit logic when the exchange interaction is pulsed^{12,13,14}. An alternative implementation of twoqubit logic can be realized by driven rotations, which become state dependent in the presence of exchange interaction and can be used to realize controlledrotation (CROT) operations^{15,16,17,18,19}. Driving rotations can also be used to realize a resonant SWAP gate^{20}, which can be used to perform state swapping. While universal quantum logic can be obtained by combinations of singlequbit rotations and an entangling twoqubit operation^{21}, the ability to directly execute a multitude of twoqubit gates would reduce the number of operations required to execute practical algorithms.
Here, we demonstrate on the same device the implementation of the CROT, SWAP, and CPHASE, which are all essential gates in quantum computing and error correction applications. SWAP operations can in particular be useful in large quantum dot arrays, providing a mean to achieve beyond nearestneighbor connectivity. We overcome the limitations imposed by the finite Zeeman energy difference between the qubits by introducing control sequences, which also allows the execution of the CPHASE and the SWAP in short time scales and a predicted highfidelity. Moreover, we demonstrate these operations at temperatures exceeding one Kelvin. The cooling power at these elevated temperatures is much larger and thereby more compatible with the operation of classical electronics, such that quantum integrated circuits based on standard semiconductor technology become feasible^{22,23,24}.
Results and discussion
Silicon quantum dot device
The experimental twoqubit system is based on electron spin states confined in a silicon double quantum dot as schematically shown in Fig. 1a. The silicon double quantum dot is fabricated using an overlapping gate architecture on a silicon wafer with an isotopically enriched ^{28}Si epilayer of 800 ppm residual concentration of ^{29}Si^{19,25}. In order to obtain an optimal exchange coupling between the electrons, qubits Q1 and Q2 are defined with N_{Q1} = 1 and N_{Q2} = 5, where N is the charge occupancy. Spin readout is performed at the (1,5)(2,4) charge anticrossing, where the \(\left\uparrow \downarrow \right\rangle\) tunnels to the singlet (2,4) charge state, while the other sp in states are blocked because of the Pauli exclusion principle. By using an adiabatic pulse from the (2,4) to the (1,5) region, we initialize the system in the \(\left\downarrow \uparrow \right\rangle\) state. Because of the limited sensitivity of the singleelectrontransistor (SET) that we use for charge readout, we average the singleshot readout traces and subtract a reference signal. We therefore obtain a current signal, proportional to the probability to have a blocked state. More details about the readout scheme can be found in ref. ^{19}. We note that the readout fidelity can be further improved, even at these higher temperatures^{26}, but here we focus on the coherent control (details on the reconstruction are in Supplementary Note 1: Reconstruction of the spin state probabilities). We perform spin manipulation via electron spin resonance (ESR) using an onchip aluminum microwave antenna. All measurements have been performed in a dilution refrigerator at a temperature of T_{fridge} = 1.05 K and with an external magnetic field of B_{ext} = 250 mT.
Readout on Pauli spin blockade is relatively insensitive to temperature, since it does not rely on any external reservoir. However, a finite temperature can still affect qubit readout in the form of an enhanced relaxation^{27}. Furthermore, the initialization fidelity can also be lowered due to a non zero population of the excited valley states in the (2,4) charge configuration. By taking into account the two singlet and the three triplet states and estimating a valley splitting of E_{vs} = 300 μeV from previous works^{19,27}, we compute a total population of the ground singlet (2,4) charge state of 87%. This initialization fidelity can be pushed beyond 99% with a valley splitting E_{vs} > 550 μeV (see Supplementary Note 2: Temperature effects on readout and initialization). Similar valley splitting values have already been measured in SiMOS samples^{9}.
Qubit characterization
We control the exchange interaction J via the detuning ϵ between the two quantum dots and we measure couplings from J = 2 MHz up to J = 45 MHz, as shown in Supplementary Fig. 1a. By fitting the exchange spectrum we extract a Zeeman energy difference between the two qubits ΔE_{z} = 11 MHz, which originates from the electron gfactor variations due to spin orbit coupling. This frequency difference is large enough to have a negligible impact on qubit control fidelities. The fitting suggests a negligible dependence of ΔE_{z} on detuning, further supported by the small magnetic field applied and the absence of external magnetic gradients. Figure 1b shows the four resonance frequencies of the twoqubit system when J = 3 MHz. At this value of exchange interaction we tune the πrotation times to be t_{CROT} = 660 ns such that we synchronize the Rabi oscillations of the target transition with the closest offresonant transition in order to suppress crosstalk^{28}. From Ramsey experiments on frequencies f_{1} and f_{4} we measure dephasing times \({T}_{2,{{{{{{{\rm{Q1}}}}}}}}}^{* }=2.3\,\upmu\)s and \({T}_{2,{{{{{{{\rm{Q2}}}}}}}}}^{* }=2.9\,\upmu\)s. The CarrPurcellMeiboomGill pulse sequence can extend the coherence times, by filtering out the low frequency noise. As shown in Fig. 1c, we measure a maximum T_{2,Q1} = 63 μs and T_{2,Q2} = 44 μs when 15 refocusing pulses are applied, setting benchmarks for the coherence time of quantum dot spin qubits at temperatures above one Kelvin.
When the exchange interaction is set to a nonzero value, it is possible to realize the CROT via driven rotations since the resonance frequency of one qubit depends on the state of the other qubit. This CROT gate is a universal twoqubit gate and equivalent to a CNOT gate up to single qubit phases^{19}. Figure 1d, e shows controlled rotations by setting both configurations of target and control qubits.
Hot, fast, and highfidelity CPHASE gates
An alternative way to achieve a universal gate set is through the implementation of the CPHASE gate. Moving in detuning energy toward the (1,5)(2,4) charge anticrossing lowers the energy of the antiparallel \(\left\downarrow \uparrow \right\rangle\) and \(\left\uparrow \downarrow \right\rangle\) states with respect to the parallel \(\left\downarrow \downarrow \right\rangle\) and \(\left\uparrow \uparrow \right\rangle\) spin states. Therefore, pulsing the detuning for a time t results in a phase gate on the target qubit conditional on the spin state of the control qubit. When the total phase ϕ = \({\phi }_{\left\downarrow \uparrow \right\rangle }\) + \({\phi }_{\left\uparrow \downarrow \right\rangle }\) = (2n + 1)π with n integer, a CPHASE gate is realized^{7}. A highfidelity implementation of such a gate requires a Zeeman energy difference between the two qubits much larger than the exchange interaction, in order to suppress the evolution of the exchange gate^{4}. This condition is conveniently met in devices with micromagnets^{13,14,15}, where the CPHASE is the most natural choice as native twoqubit gate.
In our system, ΔE_{z} is comparable in magnitude to the accessible J (see Supplementary Note 3: Exchange spectrum), due to the small B_{ext} applied. This means that a detuning pulse will also cause the \(\left\downarrow \uparrow \right\rangle\) and \(\left\uparrow \downarrow \right\rangle\) states to undergo SWAP rotations. While these rotations occur along a tilted angle due to the nonzero ΔE_{z}, they can still reduce the fidelity of the CPHASE gate. In order to avoid unwanted SWAP rotations we implement an adiabatic detuning pulse, by ramping ϵ to the desired value instead of changing it instantaneously (see schematic in Fig. 2j). In this way, a highfidelity CPHASE gate can still be realized with an arbitrarily small ΔE_{z} at the cost of a longer gate time. In Fig. 2a–f we change the duration of a detuning pulse in between a Ramseylike experiment on Q1, with and without a π pulse applied to Q2. The frequency of the oscillations of Q1 depends strongly on the spin state of Q2, thereby demonstrating a controlled phase operation. Because of the finite Zeeman energy difference, the antiparallel \(\left\downarrow \uparrow \right\rangle\) state shifts significantly more in energy than the \(\left\uparrow \downarrow \right\rangle\) state. Consequently, the oscillations in Fig. 2c are significantly faster than in Fig. 2f. Similarly, the decay time in Fig. 2e is significantly longer than in Fig. 2b because of the lower sensitivity to electrical noise. In Fig. 2g–i the pulse time is calibrated such that the total phase ϕ = 3π. We measure this in a Ramseylike experiment where we probe the phase acquired by the target qubit for different control qubit states. From Fig. 2h, i, we can observe that the resulting oscillations are nicely outofphase, which demonstrates the CPHASE gate. We achieve a gate time t_{CPHASE} = 152 ns, which is mostly limited by the adiabatic ramps which take t_{r} = 60 ns. From a comparison with simulations, we find that the contribution of both ramps to the total phase ϕ is approximately 1.7π.
This gate time can be significantly sped up with the implementation of a geometric CPHASE gate, that does not require adiabaticity^{29}. For the implementation of this gate we synchronize the unwanted exchange oscillations with the total gate duration, i.e., our gate performs a CPHASE evolution while the exchange oscillations performs a complete cycle. For a perfectly diabatic pulse the condition for the exchange interaction is:
where \({J}_{{{{{{{{\rm{res}}}}}}}}}\) is the residual exchange interaction at the point where we perform CROT gates (see Supplementary Note 4: Twoqubit gate simulations).
Figure 2k, l shows the experimental implementation of the geometric CPHASE gate. We sweep the amplitude of the detuning pulse and monitor the spin state probabilities (see Supplementary Note 1: Reconstruction of the spin state probabilities) during exchange oscillations, and the total phase acquired by the antiparallel spin states. We notice that, when ϵ ≈ 68 mV, the antiparallel spin states execute a 2π rotation, while acquiring a total phase shift of π. At this value of detuning we measure J ≈ 10 MHz (see Supplementary Note 3: Exchange spectrum) and therefore in agreement with Eq. (1). The total gate time is reduced here to t_{CPHASE} = 67 ns.
Hot, fast, and highfidelity SWAP operation
We now turn to the implementation of a SWAP gate, the originally proposed quantum gate for quantum dots^{4}. Despite the experimental demonstration of exchange oscillations^{5,6,30}, its implementation together with singlequbit gates is rather challenging because of the requirement of a negligible Zeeman difference between the qubits. In the following we will discuss a protocol that can overcome this problem and allow for a highfidelity SWAP gate, even in the presence of a finite ΔE_{z}.
In order to observe SWAP oscillations, we implement a sequence where we initialize in the \(\left\downarrow \uparrow \right\rangle\) state and pulse ϵ for a time t. Clear exchange oscillations between the \(\left\downarrow \uparrow \right\rangle\) and the \(\left\uparrow \downarrow \right\rangle\) state are visible when the detuning pulse is diabatic (see Fig. 3a, b), where the oscillation frequency is \({f}_{{{{{{{{\rm{SWAP}}}}}}}}}=\sqrt{{J}^{2}+\Delta {E}_{{{{{{{{\rm{z}}}}}}}}}^{2}}\). As we make the pulse more adiabatic by ramping ϵ, the oscillations disappear and the regime becomes suitable for a CPHASE implementation as discussed before. Even when the detuning pulse is perfectly diabatic, we do not obtain a perfect SWAP due to the finite ΔE_{z}. Instead, the spin states rotate in the Bloch sphere around the tilted axis of rotation \({{{{{{{\boldsymbol{r}}}}}}}}={(J,0,\Delta {E}_{{{{{{{{\rm{z}}}}}}}}})}^{T}\), similar to what happens for offresonant driving. Figure 3c, d show that when starting in the \(\left\downarrow \uparrow \right\rangle\) state, a maximum \(\left\uparrow \downarrow \right\rangle\) state probability of 64% is obtained in t_{SWAP} = 18 ns, which is in agreement with our simulated predictions (see Supplementary Note 4: Twoqubit gate simulations).
Composite pulse sequences^{31,32} can correct for the tilted axis of rotation. It is possible to achieve full population transfer with an exchange sequence consisting of alternating diabatic and adiabatic exchange pulses. The corresponding time evolution operators in the odd parity subspace are:
for a diabatic and an adiabatic pulse respectively (see Supplementary Note 4: twoqubit gate simulations). Here \({{{{{{{\boldsymbol{\sigma }}}}}}}}=(\hat{X},\hat{Y},\hat{Z})\) is the vector consisting of the Pauli matrices, Φ_{r,z} = Jt_{r,z}/2 the accumulated entangling phase during the pulse, and \({\theta }_{{{{{{{{\rm{r}}}}}}}},{{{{{{{\rm{z}}}}}}}}}={t}_{{{{{{{{\rm{r}}}}}}}},{{{{{{{\rm{z}}}}}}}}}\sqrt{{J}^{2}+\Delta {E}_{{{{{{{{\rm{z}}}}}}}}}^{2}}/2\) the angle of rotation. The condition for a SWAP gate is \({U}_{{{{{{{{\rm{tot}}}}}}}}}={U}_{{{{{{{{\rm{r}}}}}}}}}{U}_{{{{{{{{\rm{z}}}}}}}}}{U}_{{{{{{{{\rm{r}}}}}}}}}{U}_{{{{{{{{{\rm{z}}}}}}}}}_{2}}{U}_{{{{{{{{{\rm{r}}}}}}}}}_{2}}\cdots \equiv \hat{X}\). The number of necessary pulses depends on the angle of rotation; obviously a minimal pulse sequence requires ∣ΔE_{z}∣ ≤ J. In the typical regime of operation for devices with micromagnets, where J < ΔE_{z}, a multistep sequence is required. In the limit J ≪ ΔE_{z} many steps are necessary and the pulse sequence becomes gradually an ac signal giving rise to the acSWAP gate^{20}. Furthermore, it is essential to include the global phase which corresponds to a conditional phase evolution in the full twoqubit space and needs to vanish when implementing a SWAP gate. This protocol is highly versatile and can also produce maximally entangling gates, i.e., \(\sqrt{{{{{{{{\rm{SWAP}}}}}}}}}\) if \({U}_{{{{{{{{\rm{tot}}}}}}}}}\equiv {{{{{{{\rm{i}}}}}}}}\hat{X}/2\) and iSWAP for \({U}_{{{{{{{{\rm{tot}}}}}}}}}\equiv {{{{{{{\rm{i}}}}}}}}\hat{X}\). While finding an optimal sequence for such a composition can be done in general following the procedure of ref. ^{32}, here we extend these considerations into a multiqubit space, which gives rise to additional constraints.
A possible minimal length solution for a SWAP gate is sketched in Fig. 3e and the trajectory of the qubit state is seen in the inset. In the experiment, we calibrate the exchange interaction at all stages of the pulse, fix the time of the diabatic pulses to 12 ns, and sweep the length of the adiabatic pulse t_{corr} in order to find the best point. Figure 3f shows how the four spin probabilities change when sweeping t_{corr}. We find an optimal t_{corr} = 62 ns and the four spin state probabilities for a total pulse duration t_{SWAP} = 88 ns are plotted in Fig. 3g. The SWAP probability exceeds 90%, where the remaining error is dominated by miscalibrations, inaccuracies in the gates needed to reconstruct the spin state probabilities, and statepreparationandmeasurement (SPAM) errors. We note here that constructing a SWAP gate out of CPHASEs and CROTs would result in a gate time significantly slower than the sequence discussed here. A SWAP gate can be compiled using 3 consecutive CROT gates, which would give a total SWAP time of ≈2 μs. A SWAP gate compiled from the much faster CPHASE gate requires 11 primitive operations^{33}, which include 8 singlequbit gates and would therefore give an even larger overhead. Therefore, the composite exchange sequence can improve the gate time by more than one order of magnitude.
Hot twoqubit gate performance
In order to assess the performance of our twoqubit gates, we perform time dependent simulations of the Heisenberg hamiltonian, based on the exchangedetuning curve that me measure experimentally (see Supplementary Note 3: Exchange spectrum). Additionally, we also take into account the effects of the finite bandwidth of the setup (300 MHz) on the pulse generation. Table 1 shows the fidelities associated with the twoqubit gates CROT, CPHASE, and SWAP. Here, F_{ideal} represents the simulated fidelities taking into account the relevant parameters, but neglecting any decoherence. We find F_{ideal} > 99% for all gates except the SWAP, which is limited in fidelity by the finite ΔE_{z}. At the elevated temperatures discussed in this work, thermal noise can impact the gate performances in the form of a larger charge noise amplitude^{27}, which couple to the spin as an effective magnetic noise via the exchange interaction. Therefore, we have also modeled the decoherence by adding stochastic fluctuations of the detuning, sampled from a 1/f noise spectrum (see Supplementary Note 4: Twoqubit gate simulations). By fitting the experimental data in Fig. 2b, e, we conclude that our model is able to reproduce the decoherence with good agreement. Based on these simulations we determine F_{noise}. The fidelity of the CROT and the CPHASE gate are significantly affected by the noise, due to the relatively long gate times, and we find that the predicted CROT fidelity F_{noise} = 89% is close to the experimentally measured fidelity F = 86%^{19}. The SWAP, diabatic CPHASE, and composite SWAP are less affected by the noise, and in particular, we predict that both the diabatic CPHASE and composite SWAP can be executed with fidelities above 99%. While experimental data will be needed to validate these predictions, these results showcases how a multitude of native twoqubit gates can be executed with highfidelities and remarkable gate speeds. The limiting factor to the fidelities is the charge noise, as we have to significantly pulse the detuning to control the exchange interaction. Significant improvements can be expected by keeping the detuning at zero and instead pulsing the tunnel coupling, as this scheme is to first order insensitive to charge noise.
The ability to execute a diverse set of highfidelity twoqubit gates define silicon quantum dots as a versatile platform for quantum information. The low magnetic field operation and the small Zeeman energy difference between qubits is furthermore beneficial for the realization of scalable qubit tiles, as it supports highfidelity shuttlers and onchip resonators for longdistance qubit links. Moreover, the ability to execute quantum logic at temperatures exceeding one Kelvin provides a pathway to quantum integrated circuits that host both the qubits and their control circuitry for scalable quantum hardware.
Methods
The experiments have been performed in a Bluefors refrigerator with a base temperature T_{base} ≈ 0.45 K with a 3 Tesla magnet. In the experiments we make use of d.c. voltages and a.c. voltages. The d.c. voltages are supplied via batterypowered voltage sources and filtered through Cupowder filters and 30 Hz and 150 kHz filters. The a.c. voltages are supplied through a biastee that is on the sample printed circuit board with a cutoff frequency of 3 Hz. Pulses are generated by an arbitrary wave form generator (Keysigh M3202A) with 14 bit resolution and 1 GS/s. Microwave signals are applied via a Keysight E8267D.
Data availability
The measurement and analysis code and the underlying data is available in the Zenodo repositories (10.5281/zenodo.7179960).
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Acknowledgements
We thank S. de Snoo and S.G.J. Philips for software developments and the Veldhorst group for useful discussions. M.V. acknowledges support through a Vidi and an NWO grant, both associated with the Netherlands Organization of Scientific Research (NWO). Research was sponsored by the Army Research Office (ARO) and was accomplished under Grant No. W911NF 1710274. The views and conclusions contained in this document are those of the authors and should not be interpreted as representing the official policies, either expressed or implied, of the Army Research Office (ARO), or the U.S. Government. The U.S. Government is authorized to reproduce and distribute reprints for Government purposes notwithstanding any copyright notation herein.
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L.P. performed the experiments, M.R. developed the theoretical models. H.G.J.E. and W.I.L.L. were responsible for the device fabrication. J.S.C. supervised the wafer growth. L.P. and M.R. analyzed the data with input from L.M.K.V. and M.V. L.P., M.R., and M.V. wrote the manuscript with input from all authors. M.V. conceived and supervised the project.
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Petit, L., Russ, M., Eenink, G.H.G.J. et al. Design and integration of singlequbit rotations and twoqubit gates in silicon above one Kelvin. Commun Mater 3, 82 (2022). https://doi.org/10.1038/s43246022003049
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DOI: https://doi.org/10.1038/s43246022003049
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