Implementation of an advanced dressing protocol for global qubit control in silicon

The successful implementation of full scale quantum computing is expected to drive technological progress in a multitude of areas. However, the realization of a fault-tolerant quantum computer sets strict requirements on qubit fidelities.^1,2 At most, an error rate of 1% is tolerable for a two-dimensional qubit array using the surface code.^3,4 Depending on the error rate, the number of physical qubits required for error correction is expected to exceed millions, underpinning the need for the qubits to be not only highly robust against errors but also readily scalable. Silicon quantum dots, in particular those based on silicon metal-oxide-semiconductor (SiMOS) devices, are promising candidates for qubits due to their long coherence times and their compatibility with current semiconductor manufacturing capabilities.^5–10 However, even with isotopically enriched silicon substrates, residual nuclear spins^11,12 and spin–orbit coupling due to the oxide interface^13,14 reduce both the coherence time and the homogeneity of the spin qubit properties. It is, therefore, important that the qubit control scheme is resilient to qubit variability, and that careful considerations have been made about the engineering constraints for a scaled-up architecture.

Global control strategies, amenable to large-scale operation and manufacturing, have been discussed in recent works.^15–18 The control strategy proposed by Kane in his original 1998 paper¹⁹ involves a global microwave field that is, by default, off resonance with the qubits, and local controls are applied to tune the individual qubits into resonance to perform gate operations.²⁰ Another option is to collectively drive all qubits on-resonance by default, to operate them as dressed qubits.^16,21–23 The global dressing field provides continuous decoupling of the two-level systems from environmental noise, enabling longer coherence times.^24,25 The control resources in a global control architecture are shared among several qubits, relaxing the engineering constraints significantly. Recently, coherent control was demonstrated with an off-chip dielectric resonator,¹⁸ which significantly simplifies the chip layout. The key challenge to the scalability of this scheme is the spread in resonance frequencies of the qubits, making them susceptible to crosstalk.²⁶ To make dressed qubits more robust to Larmor frequency deviations caused by local charge and nuclear spin fluctuation (typically around 200 kHz for quantum dot spin qubits¹¹) and amplitude fluctuations of the driving field, the SMART protocol has been proposed,²⁷ which employs a custom-designed modulation of the global microwave field.

Here, we use a SiMOS quantum dot device for a proof-of-principle demonstration of the SMART protocol. In Fig. 1(a), an illustration of a large-scale architecture is shown, with shared control resources among a two-dimensional array of qubits. This shows the scale-up potential of the SMART protocol. We confirm the increased robustness of qubits operated in this regime against time-dependent fluctuations in qubit frequency and microwave amplitude, as predicted in Ref. 27. The experiments are performed on a device identical to the one shown in Fig. 1(b). A single quantum dot is formed under gate G1, while gate G2 is pulsed to control the qubit through Stark shift of the gyromagnetic ratio.⁶ We confirm the optimal modulation conditions from theoretical predictions and demonstrate coherent control with fidelity estimated by randomized benchmarking. Throughout the paper, we compare the dressed and SMART qubit protocol performance exclusively because of the shared compatibility with global control and the scalability potential.

In this work, we investigate a qubit continuously driven by a resonant a.c. magnetic field that is amplitude-modulated by a sinusoid. The SMART protocol is defined by the Hamiltonian in the dressed basis ${|zρ⟩,|z¯ρ⟩}$ (see the supplementary material for basis transformation),

where h is the Planck constant, $ΩR$ is the root mean square (RMS) Rabi frequency, $fmod$ is the modulation frequency, and $Δν(t)$ is the frequency detuning (see the supplementary material for derivation of $Hρ cos$ from the laboratory frame). One period of the sinusoidal modulation will from now on be denoted $Tmod$⁠. The $2$ factor accounts for the RMS power difference between dressed (continuous drive) and SMART protocol (sinusoidal modulation). For a fair comparison between the two, the average power is an important factor, setting the heating in a full-scale architecture.

Similar to Eq. (1), we can modulate the global field with a sine instead of a cosine. We will refer to these two variants as SMART(cos) and SMART(sin), the difference between the two being the relative phase between the global field modulation and the local modulation $Δν(t)$⁠.

For a qubit driven by a single sinusoid, there exist optimal modulation conditions that increase the robustness to noise. These conditions are given by the product of the amplitude and the period of the sinusoid and are found using the Magnus expansion and a geometric formalism^27,29 discussed in recent papers^30,31 (see the supplementary material for more details). The conditions are as follows:

where $ropt$ is the optimal rotation power resulting from certain combinations of modulation amplitude and period and j_i is the i-th root of the zeroth order Bessel function.

For spin qubits in SiMOS, shift in qubit frequency caused by the electric field controlled by the top gates can be used for local control (namely, Stark shift control), as shown in Fig. 1(a). The same top gate is used to form the quantum dot and tune it for initialization and readout. Tuning and Stark shift control of an undressed SiMOS spin qubit is shown in Fig. 2(a). After loading a single electron, the electron transition frequency shifts with applied d.c. offsets in top gate voltages, $VG1$ and $VG2$⁠, inside the dot [Figs. 2(b)–2(d)]. This corresponds to the $Δν(t)$ term in Eq. (1) and is exploited for local addressing in the SMART protocol. The effect from $VG1$ and $VG2$ is different in magnitude, and we use $VG2$ throughout this paper due to higher stability/linearity. We construct our SMART qubit gates by taking advantage of filter functions, and single-qubit gates are implemented by modulating $Δν(t)$ according to these filter function results (see the supplementary material). We find that two-axes control for the dressed and SMART protocol requires sinusoidal modulation of $Δν(t)$ at the Rabi frequency and at multiples of $fmod$⁠, respectively.

Measurement of qubits in the SMART qubit protocol is based on initializing a spin $|↓⟩$ state and reading out the spin $|↑⟩$ probability by energy-dependent tunneling to a reservoir, detected using a single electron transistor (SET). Spin-to-charge conversion is used for spin readout³² at the first electron transition [Fig. 2(a)]. This means that in this experiment, the global microwave field needs to be turned off for the duration of the initialization and readout steps. This could be circumvented using a readout scheme based on the Pauli spin blockade.^17,27 We use the frequency feedback protocol described in the supplementary material for the majority of the experiments.

In terms of scalability, the SMART protocol requires a single local oscillator (LO) at the order of GHz and two intermediate frequency channels (IF) at the order or MHz, shared among all qubits in the array [see Fig. 1(a)]. The LO is required for the high frequency microwave signal and the two IFs for amplitude modulation of both the microwave signal and the voltage level of the top gates. Two switches per top gate, corresponding to the modulation of the top gate voltage using the first and second harmonic (x- and y-gate), are ideally enough for universal control. Compared to conventional frequency addressing, where frequency crowding can cause unwanted rotations, we expect a significant reduction in crosstalk. Since all controls applied to the device for the SMART qubit protocol are a.c., one can use high/bandpass filters to reduce d.c. and high frequency noise.

The coherence time of a driven spin qubit cannot be directly compared to that of a bare spin qubit as they are not sensitive to the same type of noise. Driven spin qubits are specifically susceptible to noise at the frequency of the driving field.^27,33 This can be seen from the peaks at different frequencies in the calculated filter function (see the supplementary material). The quantization axis for a dressed qubit is along the driving field axis $zρ$⁠. For the SMART(cos) protocol, on the other hand, the qubit quantization axis is along $xρ$ and along the w-axis for the SMART(sin) protocol. This can be explained from the direction of the first non-zero term in the Magnus expansion series (see Table I in Sec. IV and supplementary material). We refer to the noise-induced decay time of information along the quantization axis as T₁ time and along the transverse direction as T₂ time. We stress that T₁ and T₂ here are not equivalent to the relaxation and decoherence axes for bare qubits. Due to this difference in relaxation and coherence axes, we define a new parameter $Tdecay$ as the measured coherence time of the spin rotation resulting from the driving field, when initialized to $|y¯ρ⟩$ (driving field along $zρ$⁠). This choice of initialization is made because $|y¯ρ⟩$ is simultaneously orthogonal to the T₁ axis of the bare, dressed, and SMART qubit protocol. Note that for a driven qubit, we need a tensor description of the decay time, relating different initializations and different measurement bases. By using $Tdecay$⁠, we compare the most limiting coherence metric (see data in the supplementary material). We note that more extensive noise spectroscopy/tomography should be completed to get a better picture of the noise mechanics in driven qubit systems. This is left for future work.

In Figs. 3(a)–3(c), $Tdecay$ is measured for the SMART protocol. After initializing the qubit to $|y¯ρ⟩$ via a $π/2$ rotation about $zρ$ [denoted $Xrot$ in Figs. 3(d)–3(f), which is a rotation about x in the rotating basis, or equivalently $zρ$ in the dressed basis], the modulated driving about the same axis is turned on for a certain wait time, $twait$⁠, followed by a final $π/2$ rotation about $zρ$ [$Xrot−twait−Xrot$⁠, see Fig. 3(f) right panel]. The spin is then measured at increasing multiples of $Tmod$⁠, where the modulated driving itself ideally equates to an identity operation after every period. By keeping $ΩR$ fixed and varying the period $Tmod$⁠, the 2D map in Fig. 3(a) is acquired. The data are fitted according to an exponential decay $∝e(−t/Tdecay)$⁠, resulting in the decay rate plotted in Fig. 3(b). As expected, the decay rate as a function of $Tmod$ closely resembles the absolute value of the zeroth order Bessel function [according to Eq. (2)] plotted for comparison. The maximum coherence time measured for the SMART protocol is $TdecaySMART=2(1)$ ms, which is obtained at $fmod=41.19$ kHz [$Tmod=24.28$ μs in Fig. 3(a)]. For comparison, the bare spin qubit (⁠$T2*$⁠) and the dressed qubit coherence times are measured to be $16(3)$ μs and $235(38)$ μs in the same device [Figs. 3(d) and 3(e)], respectively. The dressed data are very sensitive to changes in precession frequency, causing fluctuation in the driving frequency; therefore, a fit of the envelope is shown in addition to the fit of the oscillations, which shows a coherence time of $820(150) μ$s. Note that we are using mw amplitudes differing by a factor of $2$ when comparing the dressed and SMART protocol to ensure equal RMS power.

The improvement in coherence time with distinct periods of the microwave field modulation, as apparent by the four peaks in Fig. 3(a) and corresponding dips in Bessel function in Fig. 3(b), agrees with the theoretical prediction originating from the geometric formalism.^27,29 For a fixed RMS modulation amplitude $ΩR$⁠, we rewrite Eq. (2), such that the optimal modulation periods for a given $ΩR$ are given by

This expression agrees with the decay rates fitted to the Bessel function in Fig. 3(b). For the remaining part of this paper, we will focus on the first peak from the left in Fig. 3(a) where $Tmod=6.76$μs and $Tdecay=1.6(6)$ ms. This corresponds to the $j1=2.404 826$ solution of Eq. (3). The first peak is chosen because it requires shorter gate times than the rest of the peaks. The space curves for all four peaks in Fig. 3(a) [corresponding to the roots of the Bessel function in Fig. 3(b)] are shown in Fig. 3(c). We see that all solutions give the shape of a figure eight, with additional windings for higher roots. These curves correspond to first and second order noise cancelation according to the geometric formalism.

The width of the four peaks in Fig. 3(a) demonstrates the robustness to microwave strength variation (offsets in $ΩR$⁠). From Eq. (2), it can be seen why this is relevant because the product of $ΩR$ and $Tmod$ must equal certain values in order to achieve high robustness. Microwave sources generally produce stable output amplitude levels; hence, temporal microwave power fluctuation is not a prevalent source of noise. However, in a large-scale chip, the power density may vary at different positions. Moreover, the electric field of the microwaves may cause a dot-dependent resonant driving contribution due to the interface-induced spin–orbit coupling. Therefore, it is important that the protocol is robust against microwave amplitude variations.

If a microwave resonator is used to provide the global field, another limitation is set by the global field modulation frequency. A 6 GHz resonator with a Q-factor of 10⁴ only has a 600 kHz bandwidth. The resonator bandwidth must be in the range of the modulation frequency (MHz) in order to successfully create qubits with the SMART protocol. This can be relaxed by choosing a larger $Tmod$ [peaks to the right in Fig. 3(a)] at the expense of longer gate times.

In Fig. 3(a), $Tmod$ is swept on the horizontal axis. However, we see from Eq. (2) that we can similarly keep $Tmod$ fixed and sweep $ΩR$ to get similar results, as it is the product $TmodΩR$ that must equal $ji/2$ to achieve optimal modulation conditions (see the supplementary material).

We employ a simple single-tone sinusoidal driving field for most measurements in this work. However, more complex modulation shapes can potentially contribute to higher robustness. In Fig. 4, we investigate, experimentally and by simulations, a multi-tone driving field using the first and third harmonic of a cosine modulated microwave field according to

where θ determines the ratio of the two harmonics, while keeping the total power fixed. We name this variant SMART(3rd), and examples of different modulation shapes are shown in Fig. 4(a).

The experimental data in Fig. 4(b) are acquired in a similar fashion to the Ramsey experiment in Fig. 3, but at a fixed wait time of 400 μs and $Tmod=40$ μs, for a range of driving field amplitudes $ΩR2$ and ratios of the two tones as defined by θ in Eq. (4). The four peaks observed in Fig. 3(a) correspond to the case of θ = 0, for which the relative amplitude of the third harmonic is zero. The same color coding with triangular markers has been used in Fig. 3. The high spin-up probability regions (⁠$Tdecay$ $> 400$ μs) in Fig. 4(b) indicate the more ideal modulation parameters for the multi-tone drive. These regions closely resemble the simulated data in Fig. 4(c).

One of these regions of high robustness is at $θ=−0.675 45$ rad, as indicated by the red lines and yellow star marker in Figs. 4(b) and 4(c). We show the corresponding modulation shape in Fig. 4(d). We further investigate this special case by recording Ramsey data for a range of amplitudes. We plot the experimental data in Fig. 4(e) and the corresponding simulations in Fig. 4(f). The maximum $Tdecay$ in Fig. 4(e) is $4(1)$ ms. The width of the peak represents high resilience to amplitude fluctuations at the order of 10% of $ΩR$⁠. This is an improvement compared to the peaks in Fig. 3(a) (see the supplementary material for side-by-side comparison). From theory, we find that the noise cancelation order provided by this multi-tone driving field corresponds to the third order (see Sec. IV).

The use of even more complex or optimally shaped, arbitrary driving fields might result in even better performance. See the supplementary material for how the filter function formalism can be used to determine the required control frequency and phase for arbitrary driving fields. From a scalability perspective, however, more complex driving fields would require more complex control modulations and electronics.

In order to perform rotations using the SMART qubit protocol, we employ modulated Stark shift control of the spin via the top gate, according to the theory developed in Refs. 24 and 27 and earlier experimental demonstration on dressed qubits.²⁴ The Stark shift modulation is in line with the filter function results (see the supplementary material), which determine the effective rotation axis when $Δν(t)$ is modulated at different frequencies. A typical Stark shift region for the device was shown in Fig. 2, where we measure a Stark shift magnitude of −55 MHz/V for gate G2.

To confirm the rotation axes predicted in Ref. 27, we perform process tomography.³⁴ In order to completely reconstruct the 2 × 2 density matrix, six spin projections are acquired³⁵ (see the supplementary material for more details). We demonstrate the two variants of the SMART qubit protocol—SMART(cos) and SMART(sin), with a cosine and a sine modulated global field, respectively—and compare them to the dressed qubit. In Fig. 5, we show the results for the gates $X,Y,V$⁠, and $W$⁠. The $V$ and $W$ gates constitute rotations about an alternative, diagonal set of rotation axes (see top right insert) that can be used for the SMART qubit protocol.²⁷ These axes were also predicted from the filter function formalism (see the supplementary material), where the first harmonic results in rotation about an axis pointing between $+X$ and −Y (V) and the second harmonic about an axis between $+X$ and $+Y$ (W). The individual panels contain the measured superoperator matrices as well as details on the pulse sequences and modulation shapes. For comparison, we plot the ideal superoperators to the far right. All measured superoperator matrices are in agreement with the ideal matrices with good fidelity. The space curve geometry of the gates is also shown, where dressed and SMART gates correspond to rotations of the circular and figure-eight space curves about the axis of rotation.

In order for the comparison between the dressed and the SMART qubit protocols to be fair, the same global field RMS power is used. For rotations, Stark shift amplitudes are chosen such that the gate duration of the two qubits is approximately the same. Here, we use gate times for the SMART qubit protocol of $7×Tmod$ and for the dressed qubit $10/ΩR$⁠, as shown in Fig. 5. For larger available Stark shift amplitudes, shorter gate duration can be used. The rotating wave approximation (RWA) must be taken into account here, as discussed in Refs. 27 and 36.

In this work, we focus on two-axis control using gates in the Clifford group. However, we note that we can perform arbitrary rotation angles about arbitrary axes by tuning the amplitude, number of periods, and frequency of the Stark shift modulation. We expect these types of gates to show similar improvement in noise robustness.

In order to assess the performance of the SMART qubit protocol, we carry out randomized benchmarking. Here, we determine the average Clifford gate fidelity $FC$ and the noise coherence $ξC$³⁷ according to the state purity. Coherent noise can generally be reduced by improved pulse calibration and incoherent noise by pulse shaping.³⁵ The 24 Clifford gates are generated using the dressed basis gate set ${X,Y,±X,±Y}$⁠. The results for the dressed and SMART qubit protocol are presented in Fig. 6. The datasets are acquired in an interleaved fashion, with and without artificial detuning noise added to the G2 gate as illustrated in Fig. 6(a). The noise is quasi-static, white Gaussian Stark shift noise of amplitude σ = 20 kHz to imitate g-factor variability in a qubit ensemble. For the dressed scheme, the average Clifford gate fidelity and the noise coherence are found to be $98.6(14)%$ and $99.1(14)%$ without added noise, and $95.2(48)%$ and $98.3(48)%$ with added noise, respectively. For the SMART protocol, we measure $99.1(9)%$ and $99.4(6)%$ without added noise, and $98.2(18)%$ and $99.0(11)%$ with added noise, respectively. The SMART qubit protocol is more robust against detuning noise, dropping by less than $1%$ in both average Clifford gate fidelity and noise coherence. From the randomized benchmarking data, coherence times are extracted and recorded in the figure legend. These are found to be in the millisecond range for both the dressed and SMART qubit protocol. This tells us that we have similar coherence times during gate control sequences and during the global drive alone [Fig. 3(a)]. Hence, if we can reduce the gate duration, we expect to achieve higher fidelities. For fitting of the Clifford gate fidelity and the coherence, we use $A(1−2B)x+0.5$ and $A(1−2B)2x+C$⁠, respectively. Here, A and C absorb state preparation and measurement errors, respectively, and B is the error per Clifford gate. The number of Clifford gates is given by x. The extracted fidelity numbers are, therefore, calculated according to $1−B$⁠. The fitting parameters for Fig. 6 can be found in Table II in Sec. IV.

In summary, we have presented a proof-of-principle demonstration of a scalable protocol that relaxes the engineering constraints for full-scale quantum processors. We have also shown that there exist optimal modulation parameters for a global dressing field to make spin qubits more robust against detuning and microwave amplitude noise, while the qubits are individually addressable via electrical Stark shift control. We demonstrate universal control, investigated with process tomography as well as randomized benchmarking, with fidelities exceeding 99%. To account for higher order noise, we suggest more advanced modulation protocols.

We stress that pure Stark shift control is applied in this work for universal control as opposed to conventional and more widely used microwave control. The fidelities measured here are most likely limited by the Rabi frequency. To work at higher Rabi frequencies, a microwave antenna or a microwave cavity with lower electric field is required.^15,16 The Rabi frequency limits the gate speed in the SMART qubit protocol, since one gate lasts for at least one period of the global field modulation (⁠$Tmod∝1/ΩR$⁠). The linearity of the Stark shift³⁸ is another factor affecting the calibration of the SMART gates. From Fig. 2, we know that the Stark shift is not perfectly linear; hence, it might be necessary to implement more complex frequency modulations. With a proper cavity design, one can potentially reduce thermal and electrical noise that the current on-chip ESR antenna design suffers from at higher power, to provide higher Rabi frequency and shorter gate durations. In previous randomized benchmarking data from the same device, an average Clifford gate fidelity of 99.96% has been achieved using pulse engineering of the microwave and gate durations of $∼8$ μs.³⁵ This approach is, however, not compatible with global control. In this work, we present an agnostic approach to provide noise robustness to arrays of qubits, taking into account the Larmor frequency variability and spatial microwave amplitude variations.

The SMART protocol demonstrated here can be implemented with any qubit that allows dressing, providing robustness to qubit variability and improving the prospects for scaling up to full-scale quantum processors. The natural next step is to demonstrate local electrical addressing and synchronous control/echoing/idling for arrays of qubits using a single global microwave field and perform two-qubit gates. To demonstrate the full potential of the SMART protocol, implementation with a microwave resonator will also be important.

The qubit device used in this work is fabricated on an isotopically enriched ²⁸Si substrate (residual ²⁹Si concentration of 800 ppm). The details of fabrication and operation of an identical device can be found in Refs. 28, 35, and 39. The experiment is conducted in a dilution refrigerator with an electron temperature of $100–150$ mK. Stanford Research System SIM928 rechargeable isolated voltage sources were used to supply all the d.c. voltages, and a LeCroy ArbStudio 1104 arbitrary waveform generator (AWG) was combined with the d.c. voltages through a resistive voltage divider. The shaped microwave pulses were delivered by an Agilent E8267D vector signal generator, using the external AWG for IQ modulation. The SET current signals were detected by a FEMTO transimpedance amplifier DLPCA-200 and finally sampled by an oscilloscope (pico Technology PicoScope 4824).

An overview containing the optimal rotation power, the leading noise term, and the corresponding T₁-axis for different qubit protocols is given in Table I.

The fidelities extracted from the randomized benchmarking data in Fig. 6 are found using the fitting parameters in Table II.

Simulation of Figs. 4(c) and 4(f) is performed similarly to what is reported in Ref. 27, except here we assume quasi-static Gaussian detuning noise estimated from the experimental data.

See the supplementary material for additional information about the basis transformation, Hamiltonian deduction, feedback protocol and data processing.

We acknowledge support from the Australian Research Council (Nos. FL190100167 and CE170100012), the U.S. Army Research Office (No. W911NF-17–1-0198), and the New South Wales Node of the Australian National Fabrication Facility. 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 or the U.S. Government. The U.S. Government is authorized to reproduce and distribute reprints for Government purposes notwithstanding any copyright notation herein. I.H. and A.E.S. acknowledge support from Sydney Quantum Academy.

I.H., A.E.S., A.L., A.S., C.H.Y., and A.S.D. are inventors on a patent related to this work (Australia Provisional Application No. 2021902356) field by the University of New South Wales with a priority date of 30th July 2021. All other authors declare they have no competing interest.

I.H. and C.H.Y. performed the experiments and analyzed the data. I.H., C.H.Y., A.E.S., A.L., and A.S. discussed the results. K.W.C. and F.E.H. fabricated the device. K.M.I. prepared and supplied the ²⁸Si wafer. I.H. and C.H.Y. wrote the manuscript with input from all coauthors. A.S.D. and C.H.Y. supervised the project.

Ingvild Hansen: Formal analysis (equal); Investigation (lead); Methodology (equal); Visualization (lead); Writing – original draft (lead). Amanda E. Seedhouse: Formal analysis (supporting); Writing – review & editing (supporting). Kok Wai Chan: Resources (equal). Fay E. Hudson: Resources (equal). Kohei M. Itoh: Resources (supporting). Arne Laucht: Formal analysis (equal); Methodology (supporting); Supervision (supporting); Writing – review & editing (equal). Andre Saraiva: Formal analysis (equal); Methodology (equal); Supervision (supporting); Writing – review & editing (equal). Chih-Hwan Yang: Conceptualization (lead); Formal analysis (equal); Investigation (equal); Methodology (equal); Software (equal); Supervision (lead); Visualization (supporting); Writing – review & editing (equal). Andrew S. Dzurak: Funding acquisition (lead); Resources (lead); Supervision (lead); Writing – review & editing (supporting).

The data that support the findings of this study are available from the corresponding author upon reasonable request.