Introduction

Quantum computers have the potential to transform our ability to solve currently intractable problems, with implications for critical sectors such as finance, defence and pharmaceuticals. The benefits of quantum computation are already expected to be observed with "noisy intermediate scale quantum” (NISQ) devices1, which possess tens to hundreds of qubits. However, it is widely acknowledged that the algorithms2 expected to provide the most significant advantage over their classical counterparts will require error correction, where information is encoded in logical qubits and processors need millions of physical qubits to operate3,4. Scalability is thus an unavoidable objective for any quantum computation technology platform.

Electron spin qubits in gate-defined silicon quantum dot (QD) devices are a leading platform for realising large-scale quantum computers. Silicon QDs exhibit relatively long coherence times5, are able to operate at temperatures above 1 K6,7, and can utilise traditional very large-scale integration (VLSI) fabrication processes8,9. The feasibility of universal quantum computing in silicon has been established through the demonstration of high-fidelity single5,10,11 and two12,13,14 qubit gates, with the focus now on scaling up to NISQ devices and beyond to fault-tolerant systems8.

A critical requirement along this path is the ability to deliver microwave signals, which are needed to control spin qubits, across the entire quantum chip. Current methods for controlling electron spin qubits in devices include direct magnetic driving using on-chip transmission lines (TL)15,16 and electrically driven spin resonance (EDSR)6,17,18,19,20,21, both of which have proven useful in small scale (1–10 qubit) device demonstrations. However, as the number of qubits scales up, heating and design complexity issues will need to be resolved22,23 for these approaches to be practical.

Global control24 was an early technique proposed for spin qubits, offering a scalable solution to the problem of delivering microwave signals. Here, a global and uniform magnetic field is applied across the entire quantum chip in order to drive each qubit8,25,26. Qubit manipulation is activated by locally applied electric fields to shift their individual resonance frequencies into and out of resonance with the global field27. This technique differs the from the well-established TL15 and EDSR17-based approaches by applying a single microwave field to all qubits (potentially millions26) in the quantum processor, instead of just a few. It does not suffer the same scalability concerns as those approaches, since no high frequency lines or microwave currents directly pass through the chip.

Despite the appeal of global control, owing to its simplicity, single spin resonance using a global field was only achieved recently. A 3D microwave dielectric resonator (DR) made from potassium tantalate (KTaO3 or KTO)26,28,29 was used to create a global, off-chip magnetic field and perform electron spin resonance (ESR) of single spins in a planar SiMOS device26. The microwave magnetic field generated by the DR produced incoherent mixing of spin states in a double quantum dot (DQD), which was detected with a single electron transistor through a process known as spin to charge conversion30. However, the coherent control of spin qubits using a global field has remained an outstanding challenge that must to be addressed to fully establish the feasibility of applying this technique in large-scale quantum computers.

Here we present the coherent control of single electron spin qubits in a planar SiMOS DQD device using a global magnetic microwave field generated off-chip by a KTO dielectric resonator. We measure Rabi frequencies and report on the coherence properties of both spin qubits. We compare the noise spectrum seen by the qubits in this DR-driven device to that observed by traditional TL-driven qubits and conclude that this new scalable approach does not diminish the performance of the qubits. This work demonstrates the potential of global control using dielectric resonators as a scalable qubit control technique.

Results

Single spin resonance

The device studied here employs a nominally identical stack as reported previously26 (see Fig. 1a), which consists of a 0.7 × 0.55 × 0.3 mm rectangular KTO prism positioned above a silicon quantum nanoelectronic device. A 0.2 mm thick low-loss dielectric sapphire spacer is positioned between the DR and device to isolate them from one another. The fundamental mode of the DR produces an alternating magnetic field (B1) out of the qubit plane and in a direction perpendicular to the DC magnetic field (B0), as shown in Fig. 1a. We use the B1 field to control the spin state of the qubits via magnetic resonance. A coaxial loop coupler, through which the microwave power is inductively coupled to the DR, is placed above the stack26 (not shown).

Fig. 1: Device stack and electron spin resonance (ESR).
figure 1

a A 3D render of the global control device stack used in our experiments, including the silicon quantum nanoelectronic device (bottom, black), sapphire dielectric spacer (middle, blue) and potassium tantalate (KTO) dielectric microwave resonator (top, pink). b A schematic cross-section through the measured silicon QD device showing the 3D structure of the gates, enriched silicon-28 substrate (merged with panel c) and insulating oxide layers. c Steps for preparing the isolated mode measurements, depicted with conduction band profiles at the interface of the 28Si substrate. The preparation consists of three phases: loading (i), isolation (ii) and initialisation (iii) (see text for more details). d Charge stability diagram measured in isolated mode. Four charge transitions occur while the detuning voltage ϵ is swept from −0.6 to 0.4 V, indicating that there are 4 electrons in the double dot system. Black geometric symbols (explained in panel e) show the voltage configurations used in the ESR experiments. e Pulsing scheme for the ESR measurements. For a detailed explanation, please see Methods. f Even state probability as a function of fMW and B0, revealing two ESR peaks that shift with magnetic field, consistent with two spin qubits occupying the double dot system. The peaks are labelled Qubit 1 and Qubit 2. An S11 reflection measurement (pink circles) probed via the coaxial loop coupler is superimposed over the 2D map. The ESR pulse duration is fixed at 1.5 μs. g Even state probability as a function of B0 when fMW corresponds to the centre frequency of the DR resonance (dotted lines with arrows in panel f). The pulse duration is sufficiently long to make the ESR drive incoherent (25μs), causing the spin states to become completely mixed and resulting in a peak amplitude of Peven = 0.5 for both resonances.

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The qubit device is a metal-oxide-semiconductor (MOS) DQD formed in an isotopically enriched silicon-28 substrate (50 ppm residual 29Si), whose cross-sectional view is depicted in Fig. 1b. This is in contrast to previous work26 which employed a natural silicon substrate with a ~4.7% abundance of 29Si nuclei, which produced strong dephasing of the electron spins. Isotopic enrichment is a well-established technique for enhancing the coherence times of spin qubits in silicon devices31,32. A scanning electron microscope (SEM) image of an identical device to the one measured is also provided in Supplementary Note 1. The QDs are electrostatically defined by a palladium (Pd) multi-layer gate stack architecture in which different layers are electrically isolated by atomic-layer-deposited (ALD) aluminium oxide (AlOx), since Pd does not form its own native oxide33. A thermally grown SiO2 layer above the silicon substrate prevents any current leakage between the gate electrodes and the substrate. The device consists of a single electron transistor (SET), which is used as a charge sensor, with a top gate (ST) for tuning its charge accumulation and sensitivity, two plunger gates (D1–D2) for forming the quantum dots and setting their charge occupations, two barrier gates that control the coupling between the dots (J) or between dot 2 and the SET island (SETB) and confinement gates (CB1-CB2) to laterally confine the dots (not shown in Fig. 1b, see the SEM image in the Supplementary Information).

Measurements in this work have been carried out in a configuration referred to here as isolated mode, where the electrons inside the double dot system are electrically isolated from the nearby electron reservoirs, as employed elsewhere6. The steps required to prepare this configuration are depicted in Fig. 1c and entail three main stages. In the loading phase (i), electrons are introduced to the double dot system via the SET, which is also coupled to an electron reservoir (not shown). The number of electrons loaded can be tuned using the D1, D2 and J gates. The system is then isolated (ii) by raising the potential underneath the gate SETB. Finally, the desired charge occupation is initialised (iii) by setting the plunger gate (D1 and D2) voltages appropriately. Figure 1d shows a charge stability diagram measured in isolated mode with a double lock-in technique34. The four vertical blue lines indicate that 4 electrons are trapped inside the DQD system, with tunnelling between the dots depending on the value of the voltage detuning (ϵ = VD1 − VD2). A more positive ϵ favours electron occupation under gate D1, whilst a more negative ϵ favours occupation under D2. The absence of additional lines on both sides of the diagram (also not observed on a measurement over a larger VJ range, see Supplementary Fig. 3) confirm that only 4 electrons have been loaded in the DQD system. The charge configurations are labelled as (N2, N1) where N1(2) refers to the number of electrons under D1(2).

In the following spin measurements we focus on the (3,1) charge configuration (emphasised in Fig. 1d), which provides an equivalent spin state to (1,1) since the first two electrons under D2 form a spin-zero closed shell and do not interact with the remaining electrons in the system (see panel iii in Fig. 1c). This is preferred rather than having only 2 electrons in total, since the orbital splitting is generally an order of magnitude larger than valley splitting in SiMOS devices35. Figure 1e depicts the pulse scheme applied during the measurements. The system is first initialised in a spin singlet state \((\left|\downarrow \uparrow \right\rangle +\left|\uparrow \downarrow \right\rangle )/\sqrt{2}\) (see Supplementary Note 4). Following this, the system is plunged into the middle of the (3,1) region – and converted to a \(\left|\downarrow \uparrow \right\rangle\) state—while a microwave signal is applied to the loop coupler for a period of time in order to flip the spins in the DQD with the global B1 field. If the frequency of the B1 field matches one of the qubit frequencies, i.e. fMW = gμBB0/h (where g is the electron g-factor, μB the Bohr magneton and h Planck’s constant), the resonant spin qubit will flip between the \(\left|\uparrow \right\rangle\) and \(\left|\downarrow \right\rangle\) states. The resulting DQD spin state is then translated into a charge state in the Pauli Spin Blockade (PSB) region, measured using the SET, as either a singlet or triplet state30. However, we have confirmed that the blockade is restricted to even parity spin states instead of all triplet states (in what is known as parity readout36). Therefore, the readout effectively unveils an even or odd parity spin state.

In Fig. 1f we plot the measured DQD even state probability (Peven) after applying the measurement sequence (Fig. 1e), as we scan both the B0 field and the microwave frequency fMW. Electron spin resonance signatures are detected for the spins in both dots (diagonal features in Fig. 1f), where it is observed that the qubit frequencies shift linearly as a function of B0, confirming the features are indeed electron spin resonance (ESR) peaks. Note that for SiMOS spin qubit devices, the qubit frequencies are different due to the variability in spin-orbit effects caused by the imperfections (e.g. surface roughness) in the Si-SiO2 interface37. By monitoring how the qubit frequencies change due to Stark shifts from voltages applied to the different gates (see analysis in the Supplementary Note 5), we deduce that the spin resonance signal at the higher frequency (for a fixed B0) corresponds to the spin under gate D1, whilst the other resonance belongs to the spin under D2. We thus label the resonances as ‘Qubit 1’ and ‘Qubit 2’, respectively. We perform an S11 reflection measurement on the dielectric resonator via the loop coupler and superimpose it on Fig. 1f (pink circles). The amplitude of the ESR peaks are correlated with the S11 DR resonance, indicating that the B1 field generated by the KTO resonator indeed drives the qubits26. In Fig. 1g we show a one-dimensional slice of the qubit resonances taken at the DR centre frequency, measured along the black dotted arrows marked in Fig. 1f.

These results are consistent with recent observations in a natural silicon device26, confirming that this new spin control technology is device independent. Next, we exploit the long spin coherence times available in enriched 28Si to enable coherent qubit control using the off-chip B1 field.

Coherent qubit control

We measure the even state probability Peven as a function of the applied ESR pulse length (tp), with fMW and B0 chosen to satisfy the resonance condition fMW = gμBB0/h for each qubit. The result is plotted in Fig. 2a and clearly demonstrates coherent Rabi oscillations. Repeating this measurement as a function of fMW for a fixed B0 (see Fig. 2b, c) we observe Rabi chevron patterns for both qubits, where faster driving occurs as fMW becomes detuned from resonance with the qubits and is accompanied by a reduction in the oscillation visibility. The qubit frequency drifts seen in these Chevron patterns is a phenomenon commonly observed in spin qubit systems38 and not specifically related to the physics of the dielectric cavity.

Fig. 2: Coherent control.
figure 2

a Coherent Rabi oscillations for both qubits. The Qubit 1 measurement is offset by 1.0 for clarity. b, c Rabi chevrons for Qubit 1 (b) and Qubit 2 (c). The DC magnetic field is tuned to shift the qubits close to the DR centre frequency. d Rabi frequency vs. MW power applied to the coaxial loop coupler. For low powers, the Rabi frequencies of both qubits are linearly proportional to the square root of the power, as expected (see text for more details). The Rabi frequencies begin to saturate at an input power of ~20 μW, as discussed in the text. We plot the shift in the DR frequency from its low-power value (green triangles) as a function of the applied microwave power, measured in a continuous wave experiment. e, f Rabi frequencies vs. qubit frequencies for Qubit 1 (e) and Qubit 2 (f). For each data point (solid circles), the DC magnetic field is tuned appropriately in order to shift the qubits to the desired frequency, then fRabi at that qubit frequency is measured. The error bars are not shown since they all lie within the extent of the data point markers. The DR S11 measurement from Fig. 1f is superimposed (grey line) on these panels for ease of comparison. The region of enhanced Rabi frequencies overlaps with the DR response, confirming that magnetic resonance via the KTO DR is the primary mechanism for driving spin rotations in these regions. Black solid lines are Lorentzian fits to the fRabi distributions.

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The relation between the B1 field generated by the DR and the applied microwave power is given as \({B}_{1}=C\sqrt{{P}_{{{{\rm{MW}}}}}}\), where C is the conversion factor and PMW is the power26,29. Therefore, we expect fRabi = gμBB1/h to have a linear dependence on \(\sqrt{{P}_{{{{\rm{MW}}}}}}\). This is investigated in Fig. 2d, where for low powers fRabi indeed increases linearly with respect to \(\sqrt{{P}_{{{{\rm{MW}}}}}}\) with an average conversion factor extracted from the slopes of this plot of \(C\approx 11\,\mu {{{\rm{T}}}}/\sqrt{\mu {{{\rm{W}}}}}\).

For powers exceeding PMW ≈ 20μW ( ≈ − 17 dBm) the Rabi frequency begins to saturate for both qubits. We probe the shift in the resonance frequency of the KTO resonator (relative to its low-power value) as a function of PMW and find that the change is minimal (<1 MHz) over the range of powers measured, which would not suffice to explain the saturation as discussed below. We therefore conclude that the observed saturation in fRabi is not caused by power-induced shifts of the DR frequency, for example due to microwave heating of the KTO dielectric material29. Qualitatively similar behaviour has been observed before in devices very different from the one investigated here, including devices with microwave drive applied by broadband on-chip antennae and made with different materials stacks (Aluminium gates with thermally oxidised insulation between the gates). While in some cases a small stray electric-field drive might be present (which is known to cause a saturation at high amplitudes39), the underlying cause for the saturation of the magnetic drive remains unexplained. Further work is warranted to understand the mechanism behind the Rabi frequency saturation.

Next we explore how fRabi varies with respect to the qubit resonance frequencies (fqubit) in order to quantify the effect of the resonator more accurately. We have measured Rabi oscillations for different fqubit values (setting B0 = hfqubit/gμB) around the fundamental mode of the DR, and plot fRabi against fqubit for both qubits in Fig. 2e, f. It is clear from these measurements that fRabi is enhanced for qubit frequencies corresponding with the DR resonance (as can be seen from the superimposed S11 measurement), which implies that the qubits are primarily driven by the KTO resonator in these regions. Compared to the off-resonant drive, fRabi is enhanced at the centre of the DR resonance by a factor of 3.3 for Qubit 1 and 2 for Qubit 2. We note that a factor 3.3 enhancement in B1 corresponds to a >10 times lower power requirement for a given field strength. There is also a residual drive that is present across all qubit frequencies, which we believe could originate from several sources. Microwaves may couple into a broadband transmission line (unused in this experiment, see Supplementary Fig. 2) that is terminated 200 nm from the DQD, producing a weak B1 field across a large bandwidth. In addition, microwave currents that are induced in the metal gate electrodes may produce magnetic or electric fields that result in weak ESR or EDSR40. Especially electric drive—combined with a location-dependent spin-orbit interaction strength41—could explain the frequency domain response differences between the qubits. See Supplementary Note 7 for additional data supporting this hypothesis. Finally, we note that the data in Fig. 2e, f also indicate that a <1 MHz shift from the DR resonance should result in a less than 5% drop in the Rabi frequency, which implies that the Rabi frequency saturation observed in Fig. 2d cannot be explained solely by the shift in resonator frequency.

Coherence time measurements

We investigate the coherence times of the qubits by performing Ramsey free induction decay and Hahn echo experiments. The data measured in these experiments are shown in Fig. 3a, b for Qubits 1 and 2, respectively. To extract the \({T}_{2}^{{{{\rm{* }}}}}\) times, we fit the Ramsey data to exponential decay functions of the form \({P}_{{{{\rm{even}}}}}=A{e}^{-{(t/{T}_{2}^{{{{\rm{* }}}}})}^{n}}+B\), where the parameters A and B are related to the measurement visibility and n is the decay exponent, typically ranging between 1 and 2 (ref. 42). The fits reveal \({T}_{2}^{{{{\rm{* }}}}}=2.33\pm 0.35\,\mu\)s (n = 1.13) for Qubit 1 and \({T}_{2}^{{{{\rm{* }}}}}=1.79\pm 0.20\,\mu\)s (n = 1.27) for Qubit 2. Similarly, the Hahn echo measurement results are fit with the function \({P}_{{{{\rm{even}}}}}=A(1-{e}^{-{(t/{T}_{2}^{{{{\rm{Hahn}}}}})}^{n}})+B\), yielding \({T}_{2}^{{{{\rm{Hahn}}}}}=9.5\pm 1.0\,\mu\)s (n = 1.86) for Qubit 1 and and \({T}_{2}^{{{{\rm{Hahn}}}}}=12.1\pm 1.4\,\mu\)s (n = 1.67) for Qubit 2.

Fig. 3: Coherence time measurements.
figure 3

a, b Spin coherence times measured via Ramsey free induction decay (\({T}_{2}^{* }\)) and Hahn echo (\({T}_{2}^{Hahn}\)) experiments for Qubit 1 (a) and Qubit 2 (b). The Hahn echo data are offset by −0.1 for clarity. c CPMG noise spectroscopy measurements taken from three devices having a similar gate layout (Orange: Our device—Qubit 1, Blue: Our device, Qubit 2, Purple: Device A—Pd gate electrodes with an ALD-AlOx insulator, Green: Device B—Al gate electrodes with a thermal-AlOx insulator). S(ω) is the power spectral density of the qubit frequency noise. The error bars indicate the 95% confidence interval bounds for the S(ω) fits.

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The reported spin coherence times are 1-2 orders of magnitude smaller than the best measured values in silicon MOS devices5. In order to determine if this is related to the DR, we make a comparison of the coherence and noise properties for typical silicon qubit devices constructed with palladium gates and aluminium gates. We first compare Carr-Purcell-Meiboom-Gill (CPMG) noise spectroscopy measurements43 from our device and two other devices (one containing palladium gates and the other aluminium gates) having nearly identical gate layouts, but where the qubit control signals are delivered by conventional on-chip TLs (Fig. 3c). The TL device with Al gate electrodes (green circles) has a thermally grown AlOx gate insulator, while the other TL device (purple circles) is made from exactly the same materials as the current device (Pd gates with an ALD-AlOx insulator), whose noise spectroscopy data is also shown. The results demonstrate that the devices with Pd gate electrodes and ALD insulators have a similar noise floor, which is an order of magnitude higher than that of the Al gate device, despite the residual 29Si concentration (50 ppm) being considerably lower in the Pd devices than the one containing Al gates (800 ppm). We note that the spectra obtained have a frequency dependency weaker than the charge-noise-induced 1/f trend. The frequency range reported here typically has a stronger white noise component due to instrumental thermal noise. The lower frequency range, however, would be expected to be dominated by 1/f noise created by charges in the oxide44. We could not observe this effect due to the limited coherence time in our device, which sets the minimum frequency accessible by this technique. Comparing the \({T}_{2}^{\mathrm{CPMG}}\) times across all devices (see Supplementary Note 6) shows that the coherence of the Pd gate based qubits are all comparable and considerably reduced relative to the Al gate based qubits.

We believe that the higher decoherence of the qubits in the Pd devices is most likely related to the materials used, where charge noise arising from the ALD-grown AlOx gate oxide layers45 is a potential source. The slight difference in the noise power spectral densities of the Pd devices (with and without DR, see Fig. 3c) could be due to device variability, or perhaps because the oxide charges are disturbed in the DR experiment over a large area as compared to a small area in a conventional TL-based one. The variability in the coherence properties will be investigated in the future by measuring additional devices. Ultimately, we believe that moving to an Al device with thermally grown AlOx should substantially improve the qubit coherence times.

Discussion

We have demonstrated, for the first time, the coherent control of spin qubits in a nanoelectronic device using a globally applied magnetic field, achieving Rabi frequencies greater than 1 MHz for an input power of approximately 20 μW. We also report a comparative analysis of the coherence and noise properties of devices where ESR is driven via on-chip transmission lines (local control) to those employing an off-chip dielectric resonator (global control). The coherence times of the local and global control devices made from identical materials (Pd gates with an ALD oxide) are within a factor of 3, but differ by an order of magnitude from that measured in a local control device made with an Al gate stack, leading us to suspect the device materials as the origin of noise. In future work, we plan to apply this off-chip DR control technique to a device made with Al gate electrodes and thermally grown AlOx gate insulators, which we expect will result in longer spin coherence times.

Another avenue for improvement is the DR quality factor (Qi = 780), which is currently limited by losses in the device. The material-limited quality factor for KTO DRs is approximately two orders of magnitude larger (Qi > 60, 000)26—reaching this limit would mean the power could be reduced a hundredfold for a given B1 amplitude, with corresponding lower levels of undesired disturbance, for example due to unintended microwave loops or resonances in the device.

The improvements in global control hardware developed here should be accompanied by efforts to design and optimise pulse protocols for implementing high-fidelity single qubit gate operations46. Such a protocol has already been experimentally demonstrated in a SiMOS spin qubit device equipped with on-chip TL control47. In addition, qubit operations such as initialisation, readout and two-qubit entangling gates must be harmonised with the presence of the continuously driven microwave field generated by a high-Q resonator. Recent work in this direction48 shows that high-fidelity gates should be possible with realistic experimental parameters.

Our work shows that delivering microwave signals to spin qubits in a quantum processor—something that has so far been seen as a major challenge and drawback of the platform—can be elegantly resolved by means of global control using a KTO dielectric resonator. Demonstrating off-chip coherent control of spin qubits brings the prospect of large-scale spin-based quantum computers one important step closer.

Methods

Pulsing scheme for the ESR measurements

A diagram for the scheme is depicted in Fig. 1e, please refer to this figure for the following discussion. The DQD is initialised as a \(\left|\downarrow \uparrow \right\rangle\) state in the (3,1) charge configuration point near the (3,1)–(4,0) transition. It is then pulsed deeper into the (3,1) region (square) and a square microwave pulse is applied to the dielectric resonator, generating an alternating magnetic field, B1, which can rotate the spins if they are in resonance with the field. Readout is performed in the Pauli Spin Blockade region (triangle) and reveals if the system is in an even or odd state (see the text for more detail). Finally, a diabatic ramp from (4,0) (circle) to (3,1) (star) is applied in order to both re-initialise the DQD in a \(\left|\downarrow \uparrow \right\rangle\) state (see Supplementary Note 4) and implement gate-level feedback. The gate-level feedback protects the measurements from charge-noise-induced charge transition shifts by monitoring the transition during the ramp and applying a correction to the detuning (ϵ)6.