Qubits made by advanced semiconductor manufacturing
A.M.J. Zwerver, T. Krähenmann, T.F. Watson, L. Lampert, H.C. George, R. Pillarisetty, S.A. Bojarski, P. Amin, S.V. Amitonov, J.M. Boter, R. Caudillo, D. Corras-Serrano, J.P. Dehollain, G. Droulers, E.M. Henry, R. Kotlyar, M. Lodari, F. Luthi, D.J. Michalak, B.K. Mueller, S. Neyens, J. Roberts, N. Samkharadze, G. Zheng, O.K. Zietz, G. Scappucci, M. Veldhorst, L.M.K. Vandersypen, J.S. Clarke
QQubits made by advanced semiconductor manufacturing
A.M.J. Zwerver , T. Kr¨ahenmann , T.F. Watson , L. Lampert , H.C. George , R. Pillarisetty , S.A. Bojarski , P.Amin , S.V. Amitonov , J.M. Boter , R. Caudillo , D. Corras-Serrano , J.P. Dehollain , G. Droulers , E.M.Henry , R. Kotlyar , M. Lodari , F. Luthi , D.J. Michalak , B.K. Mueller , S. Neyens , J. Roberts , N.Samkharadze , G. Zheng , O.K. Zietz , G. Scappucci , M. Veldhorst , L.M.K. Vandersypen , ∗ , J.S. Clarke , ∗ (Dated: February 1, 2021) QuTech and Kavli Institute of Nanoscience, Delft University of Technology, 2600 GA Delft,The Netherlands Intel Components Research, Intel Corporation, 2501 NW 229th Avenue, Hillsboro, OR, USA ∗ Corresponding authors: [email protected]; [email protected] quantum computers require the integration of millions of quantum bits [1, 2]. Thepromise of leveraging industrial semiconductor manufacturing to meet this requirement has fueledthe pursuit of quantum computing in silicon quantum dots. However, to date, their fabrication hasrelied on electron-beam lithography [3–14] and, with few exceptions [4, 7, 14], on academic stylelift-off processes. Although these fabrication techniques offer process flexibility, they suffer from lowyield and poor uniformity. An important question is whether the processing conditions developed inthe manufacturing fab environment to enable high yield, throughput, and uniformity of transistorsare suitable for quantum dot arrays and do not compromise the delicate qubit properties. Here, wedemonstrate quantum dots hosted at a Si/ SiO interface, fabricated in a 300 mm semiconductormanufacturing facility using all-optical lithography and fully industrial processing [15]. As a result,we achieve nanoscale gate patterns with remarkable homogeneity. The quantum dots are well-behaved in the multi-electron regime, with excellent tunnel barrier control, a crucial feature for fault-tolerant two-qubit gates. Single-spin qubit operation using magnetic resonance reveals relaxationtimes of over 1 s at 1 Tesla and coherence times of over 3 ms, matching the quality of silicon spinqubits reported to date [3–13]. The feasibility of high-quality qubits made with fully-industrialtechniques strongly enhances the prospects of a large-scale quantum computer. The idea of exploiting quantum mechanics to buildcomputers with computational powers beyond the abili-ties of any classical device has gathered momentum sincethe 1980’s [16]. However, in order for full-fledged quan-tum computers to become a reality, they need to befault-tolerant, i.e. errors from unavoidable decoherencemust be reversed faster than they occur [1]. The mostpromising architectures require a scalable qubit system ofindividually-addressable qubits with a gate fidelity over99% and tunable nearest-neighbour couplings [17, 18].Spin qubits in gate-defined quantum dots (QDs) of-fer great potential for quantum computation due to theirsmall size and relatively long coherence times [19–21].Single-qubit gate fidelities over 99.9% [9, 22] as well astwo-qubit gates [8, 11, 23–25], algorithms [10], condi-tional teleportation [26], three-qubit entanglement [27]and four-qubit universal control [28] have already beendemonstrated. Moreover, silicon spin qubits have beenoperated at relatively high temperatures of 1-4 K [6, 29],where the higher cooling power enables scaling strategieswith integration of control electronics [30–34].A major advantage of silicon spin qubits is that theycould leverage decades of technology development in thesemiconductor industry. Today, industry is able to makeuniform transistors with gate lengths of several tens ofnanometers and spaced apart by 34 nm (fins) to 54 nm(gates), feature sizes that are well below the 193 nmwavelength of the light used in the lithography pro-cess [35]. This engineering feat and the high yield that allows integrated circuits containing billions of transis-tors to function, are enabled by adhering to strict de-sign rules and by advanced semiconductor manufactur-ing techniques such as multiple patterning for pitch dou-bling, subtractive processing, chemically-selective plasmaetches, and chemical mechanical polishing (CMP) [36].While these processing conditions are more intrusive thanthe metal lift-off processing conditions typically used onacademic devices, they will be key to achieving the ex-tremely high yield necessary for the fabrication of thou-sands or millions of qubits in a functional array.A quantum dot device bears a strong similarity to atransistor, taken to the limit where the gate above thechannel controls the flow of electrons one at a time [37].In linear qubit arrays, the transistor gate is replaced bymultiple gates, used to shape the potential landscape ofthe channel into multiple potential minima (quantumdots), to control the occupation of each dot down tothe last electron, and to precisely tune the wavefunctionoverlap (tunnel coupling) of electrons in neighbouringdots [38]. In addition, qubit devices commonly rely on in-tegrated nearby charge sensors to enable single-shot spinreadout and high-fidelity initialisation [21, 39]. A firstkey question is then whether the reliable but strict designrules of industrial patterning can produce suitable qubitdevice layouts. A separate consideration is that qubit co-herence is easily affected by microscopic charge fluctua-tions from interface, surface and bulk defects. Therefore,a second key question is whether the coherence proper- a r X i v : . [ c ond - m a t . m e s - h a ll ] J a n ties of the qubits survive the processing conditions neededto achieve high yield and uniformity. Although the firstqubits in quantum dots fabricated on wafer-scale in in-dustrial foundries have been presented [4, 40], they relyon electron-beam lithography and avoid CMP for the ac-tive device area. CMP requires a uniform metal densityacross the wafer, which introduces its own complexitiesfor quantum dot devices, due to the large amount of float-ing metal and added capacitance.In this letter, we demonstrate optically-patternedquantum dots and qubits, made in a state-of-the-art300 mm wafer process line, similar to those used for com-mercial advanced integrated circuits.A dedicated mask set based on 193 nm immersionlithography is created for patterning quantum dot ar-rays of various lengths, as well as a number of test struc-tures, such as transistors of various sizes and Hall bars.These test structures allow us to directly extract impor-tant metrics at both room temperature and low tem-perature such as mobility, threshold voltage, subthresh-old slope and interface trap density. Analysed together,these metrics give us understanding of the gate oxide andcontact quality along with the electrostatics to help trou-bleshoot process targeting [15].As in current complementary metal-oxide-semiconductor (CMOS) transistors, the active region ofthese quantum dot devices consists of a fin etched outof the silicon substrate [35]. Nested top-gates with apitch of 50 nm, separated from the fin by a compositeSiO /high-k dielectric, are used to form and manipulatequantum dots. Figure 1a shows a high-angle annulardark-field scanning transmission electron microscopy(HAADF-STEM) image of the active device area. Across-section transmission electron microscopy (TEM)image along a fin with quantum dot gates is shownin Fig. 1b. N-type ion implants on both ends of thefin, well separated from the active region, serve asOhmic contacts to the fins. We pattern two such linearquantum dot arrays, separated by 100 nm (see Fig. 1cfor a TEM image across both fins). In our experiments,we use a quantum dot in one array as a charge sensorfor the quantum dots and qubits in the other array. A3D device schematic is shown in Fig. 1d.The process flow starts from a conventional transis-tor flow but is adapted to fabricate two sets of gates insuccessive steps using a combination of 300 mm opti-cal lithography, thin film deposition, plasma etch, andchemical mechanical polish processes. The main stepsare illustrated in Fig. 1e. First, the fins are defined in aSi substrate. The area between the fins is filled in with ashallow trench isolation dielectric material and polished.The first gate layer (with even numbers) is defined usingan industry standard replacement metal gate process [41–43] and the second gate layer (with odd numbers) is thenformed adjacent to the first gate layer. Finally, a con-tact layer is patterned to enable routing to bond pads,as well as ohmic and gate contacts. In the devices in-tended for coherent experiments (discussed in Fig. 4), a metal wire that shunts a coplanar stripline (connected toan on-chip coplanar waveguide) is placed parallel to thefins in the contact layer (Extended Data Fig. 2). Thesamples used only for quantum dot formation are fabri-cated on natural silicon substrates, whereas the samplesused for qubit readout and manipulation are fabricatedon an isotopically enriched Si epilayer with a residual Si concentration of 800 ppm [44]. This reduces the hy-perfine interaction of the qubits with nuclear spins in thehost material and thus increases the qubit coherence [20].A single 300 mm wafer contains more than 10 , > . ± . σ from the mean)per dot in the multi-electron regime (see Extended DataFig. 4). This indicates the dot diameter to be on theorder of 45 nm, which is about the width of a singlegate. Next, we measure charge noise in the multi-electronregime, by measuring the current fluctuations at a fixedgate voltage, on the flank of a Coulomb peak. The powerspectral density shows a 1/ f slope that is characteristic ofcharge noise in solid-state devices [45]. The charge noiseamplitude is in the range of 1 − µ eV/ √ Hz at 1 Hz,with some variation between Coulomb peaks (Fig. 2b).These are common charge noise values in Si-MOS QDsamples [5].Figure 2c shows the transport through a double QDas a function of the gate voltages that (mostly) controlthe electrochemical potential of each dot, G3 and G5.Characteristic points of conductance are measured, so-called triple points. At these points, the electrochemicalpotentials of the reservoirs are aligned with the electro-chemical potentials of the left and the right dot, suchthat electrons can tunnel sequentially through the twodots [38]. Increasing the voltage applied to the interme-diate gate G4 is expected to lower the tunnel barrier be-tween the dots, eventually reaching the point where onelarge dot is formed. This behaviour is seen in Figs. 2c-f,as the gradual transition from triple points to single, par-allel and evenly spaced Coulomb peaks. This shows thetunability of the interdot tunnel coupling in this doubledot, which is advantageous for two-qubit control in sucha system [6, 11, 27, 34].In a next step, we use a QD in one fin as a charge sen-sor for the charge occupation of the QDs in the other fin.This allows us to unambiguously map the charge statesof the qubit dots down to the last electron [39]. A char-acteristic charge stability diagram showing the last elec-tron transition is shown in Fig. 3a. The current throughthe sensor is measured as a function of the voltage ontwo gates controlling the qubit dot. In the few-electronregime, we can usually distinguish lines with several dif-ferent slopes, indicating the formation of additional, spu-rious dots next to the intended dot. However, we consis-tently are able to find a clean region in the charge stabil-ity diagram with an isolated addition line correspondingto the last electron. Several iterations of geometry andmaterial changes improved the charge sensing by ordersof magnitude, resulting in a sensing step of about 500 pAfor a source-drain voltage of 500 µ V. This allows single-shot readout of the spin of a single electron by means ofspin-dependent tunneling and real-time charge detection(Fig. 3c) [46].In order to define a qubit via the electron spin states,we apply a magnetic field in the [100]-direction, parallelto the fins, separating the spin-up and spin-down levelsin energy. We apply a three-stage pulse to gate G6 tomeasure the spin relaxation time, T [46]. We find T exceeding 1 s at a magnetic field of 1 Tesla (Fig. 3b).This long T is comparable to those reported previouslyfor silicon quantum dots [5, 6] and indicates that themore complicated processing conditions of the 300 mm-scale fabrication do not degrade the spin relaxation time.Upon measuring T as a function of magnetic field, wefind a striking, non-monotonic dependence, which is welldescribed in the literature and the result of the valleystructure in the conduction band of silicon. Following[5, 47], we fit the magnetic field dependence of the spin relaxation rate (1 /T ) with a model including the effectof Johnson noise and phonons inducing spin transitionsmediated by spin-orbit coupling, and taking into accountthe lowest four valley states (Fig. 3d). The peak in therelaxation rate around 2 .
25 T corresponds to the situ-ation where the Zeeman energy equals the valley split-ting energy, from which we extract a valley splitting of260 ± µ eV, well above the thermal energy and qubitsplitting in this system.To coherently control the spin states, we apply an accurrent to the stripline in order to generate an oscillatingmagnetic field at the QD [48]. Electron spin resonanceoccurs when the driving frequency matches the spin Lar-mor frequency of f = 17 . T ∗ is measured through aRamsey interference measurement (see Extended DataFig. 8). Fitting this Ramsey pattern with a Gaussian-damped oscillation, yields a decay time of T ∗ = 24 ± µs when averaging data over 100 s (the error bar here refersto the statistical variation between 41 post-selected rep-etitions of 100 s segments). As we repeat such Ramseymeasurements, we observe slow jumps in the qubit fre-quency. Averaging the free induction decay over 2 hoursand 40 minutes still gives a T ∗ of 11 ± µs , see Methodssection for more details.To analyse the single-qubit gate fidelity, we employrandomised benchmarking [49] (Extended Fig. 10). Anumber, m , of random Clifford gates is applied to thequbit, followed by a gate that ideally returns the spinto either the spin-up or spin-down state. In reality, theprobability to reach the target state decays with m due toimperfections. The standard analysis gives a single-qubitgate fidelity of 99 .
0% for Q1 and 99 .
1% for Q2. With theRabi decay being dominated by low-frequency noise, thepresent combination of T ∗ and Rabi frequency shouldallow an even higher fidelity [6, 23, 29]. We suspectthe single-qubit gate fidelity to be limited by impropercalibration. Nonetheless, the fidelity is already aroundthe fault-tolerant threshold for the surface code [18].Finally, we study the limits of spin coherence by per-forming dynamical decoupling by means of Carr-Purcell-Meiboom-Gill (CPMG) sequences (see Fig. 4b for thecoherence decay using 50 pulses). These sequences elimi-nate the effect from quasi-static noise sources. Figure 4cshows the normalised amplitude of the CPMG decay asa function of evolution time for different numbers of π -pulses, n . By fitting these curves we extract T CPMG2 ( n ).We use a Gaussian decay envelope which yields distinctlybetter agreement than an exponential decay. The T CPMG2 times are plotted as a function of n in Fig. 4d. We ob-tain a T CPMG2 of over 3 . n = 50 CPMG pulses,more than 100 times larger than T ∗ , with room for fur-ther increases through additional decoupling pulses. TheCPMG data for Q1 is consistent with charge noise asthe limiting mechanism (see Methods). For Q2, an addi-tional noise mechanism is likely present. Again, all thedecay timescales are comparable to the results reportedearlier for Si-MOS devices [3–7, 23].In summary, despite the industrial processing condi-tions used to fabricate the qubit samples, key perfor-mance indicators such as charge noise, the charge sensingsignal, T , T ∗ and T CPMG2 , are already state-of-the-art.The formation of easily tunable double dots bodes wellfor the implementation of two-qubit gates in this sys-tem. Several further improvements are possible. First,the ESR stripline can be redesigned to lower resistanceand dissipation by increasing the trace width up to theshort and using lower resistivity materials. Further-more, bringing the quantum dots on the inside of thestripline [48] will increase the ratio of magnetic field and(unwanted) electric fields and heating. Finally, spuriousdots in the few electron regime and two-level systems canbe removed by reducing the presence of material chargedefects [50, 51]. While growth conditions for high-qualitySi/dielectric interfaces have been identified, performance-limiting defects can be formed through downstream pro-cessing. Further work is ongoing to optimise the processflow and recipes (temperature budget, plasma conditions,chemical exposure, and annealing conditions) to reducedefects at the end of line.These fabrication methods can be adapted to allow for2D quantum dot arrays as well. Moreover, these pro-cessing steps are by default integratable with any otherCMOS technology, which opens up the potential to inte-grate classical circuits next to the qubit chip. Eventually,industrial processing has the potential to achieve the veryhigh quantum dot uniformity that would enable cross-bar addressing schemes [32]. The compatibility of siliconspin qubits with fully-industrial processing demonstratedhere, highlights their potential for scaling and for realis-ing a fault-tolerant full-stack quantum computer. e Spin Qubits Charge SensorsDummificationGate Routing
Silicon Fins a db
ACL G1 G2 G3 G4 G5 G6 G7 ACR c FIG. 1.
Industrially fabricated quantum dot devices.a,
HAADF-STEM image of a typical device. The active re-gion consists of two parallel fins; one hosts the qubits andthe other hosts the sensing dot. The fan-out of the gatesis clearly visible, as are many additional metallic structures(called dummification) needed to maintain a roughly constantdensity of metal on the surface, which ensures homogeneouspolishing on a wafer scale. b, TEM image along a Si fin,showing 7 finger gates to define the quantum dot array andtwo accumulation gates to induce reservoirs connecting to then-type implants that serve as Ohmic contacts (outside the im-age). c, False-coloured TEM image perpendicular to the Sifins, showing the fins and the gates on top. d, Schematic ofthe active region of the device. e, Schematic of the processsteps used to fabricate the devices as explained in the maintext. c d e fa b
FIG. 2.
Tunable single and double quantum dots a,
Charge stability diagram for a single QD measured via elec-tron transport. b, Coulomb blockade peaks in the multi-electron regime (orange line) and the power spectral densityat 1 Hz of the quantum dot potential fluctuations measuredat the flank of each peak (purple dots). c-f,
Charge stabilitydiagrams of a double quantum dot formed under gates G3and G5. The gate voltage on G4 is gradually increased (G4is 1245, 1308, 1353 and 1398 mV from c to f ), showing goodcontrol over the interdot tunnel coupling. e wr c da b FIG. 3.
Charge sensing and single-shot spin readout.a,
Charge stability diagram of the last-electron regime of aQD, measured with a sensing dot in the other fin. The pointsw, r and e refer to the wait, readout and empty stages of thegate voltage pulse. b, Spin-up probability as a function ofload time at a magnetic field of 1 T. The exponential fit yieldsa T of 1 . ± . c, Real-time current through the sensingdot indicating a spin-up (purple line) and spin-down (orangeline) electron, recorded with a measurement bandwidth of3 kHz set by an external low-pass filter. d, Relaxation rate(1 /T ) as a function of the applied magnetic field (purpledots). The relaxation rate is fitted by a model (orange line)that includes the effect of Johnson noise and phonons couplingto the spin via spin-orbit interaction. From this fit, we extracta valley splitting of E v = 260 ± µ eV. c da b FIG. 4.
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Setup and instrumentation
The measurements were performed on two differentsetups on different continents, setup 1 (S1, Delft) andsetup 2 (S2, Hillsboro). The samples were cooleddown in a dilution refrigerator, operated at the basetemperature of around 10 mK (S1: Oxford Triton drydilution refrigerator, S2: Bluefors XLD dry dilution re-frigerator). DC voltages were applied via Delft in-housebuilt, battery-powered voltage sources (S1 and S2).The printed circuit board onto which the sample wasmounted contained bias tees with a cut-off frequency of3 Hz to allow for the application of gate voltage pulses(S1 and S2). The pulses were generated by an arbitrarywaveform generator (AWG, S1: Tektronix AWG5014,S2: Zurich Instruments HDAWG). The baseband cur-rent through the sensing dot was converted to a voltageby means of a home-built amplifier, filtered througha room-temperature low-pass filter (S1: 3 kHz, S2:1.5 kHz) and sampled by a digitiser (S1: M4i spectrum,S2: Zurich Instruments MFLI). Microwave bursts fordriving ESR were generated by a vector source with aninternal IQ mixer (S1 and S2: Keysight PSG8267D),with the I and Q channels controlled by two outputchannels of the AWG.
Charge noise measurements
Each charge noise data point in Fig. 2c is obtained byrecording a 140 second time trace (at 28 Hz samplingrate) of the current through the QD with the plungergate voltage fixed at the steepest point of the Coulombpeak flank. To convert the current signal to energy,we proceed as follows. First, we convert the current togate voltage by multiplying the data by the slope ofthe Coulomb peak at the operating point. Then, wemultiply with the lever arm to convert from plunger gatevoltage to energy. To obtain the power spectral density(PSD), we divide the data in 10 equally long segments,take the single-sided fast Fourier transform (FFT) of thesegments and average these. Fitting the PSD to
A/f α we extract the energy fluctuations at 1 Hz ( √ A ) for eachCoulomb peak. We extract a mean value of α = 1 . ± . Spin readout
In order to read out the spin eigenstate we use energy-selective tunneling to the electron reservoir [46]. Thespin levels are aligned with respect to the Fermi reser-voir, such that a spin-up electron can tunnel out of theQD, while for a spin-down electron it is energeticallyforbidden to leave the QD. Thus, depending on the spinstate, the charge occupation in the QD will change. Tomonitor the charge state, we apply a fixed voltage biasacross the sensing dot and measure the baseband currentsignal through the sensing dot, filtered with a low-passfilter and sampled via the digitiser. In post-analysis wethreshold the sensing dot signal and accordingly assigna spin-up or spin-down to every single shot experiment. After readout, we empty the QD to repeat the sequence.As is commonly seen in spin-dependent tunneling, thereadout errors are not symmetric, which is reflected inthe range of the oscillations in Figs. 4a,b.
Qubit operations
When addressing the qubit, we phenomenologicallyobserve that the qubit resonance frequency shifts de-pending on the burst duration. The precise origin ofthis resonance shift is so-far unclear, but appears tobe caused by heating. Similar observations have beenmade in recent spin qubit experiments [9, 10, 34] thatused electric-dipole spin resonance via micromagnets asthe driving mechanism. To ensure a reproducible qubitfrequency in the experiments, we apply an off-resonantmicrowave burst prior to the intended manipulationphase to saturate this frequency shift. We furtherinvestigate this frequency shift in Extended Data Figs. 6and 7.
Ramsey oscillation
We observe that the qubit resonance frequency inthe devices exhibits jumps of several 100 s of kHz ona timescale of 5-10 minutes. To extract meaningfulresults, we monitor this frequency shift throughout theexperiments and accordingly discard certain data traces,such that we only take into account data acquired withthe qubit in a narrow frequency window. To illustratethe frequency shift, we show the FFT of 100 repeti-tions of a Ramsey interference measurement of qubit1 (measurement time ∼ T ∗ of qubit 1, we fiteach of the 100 repetitions of the Ramsey measurement(measurement time per repetition ∼
100 s) and extracta T ∗ value. Evidently, some of the data quality is ratherpoor due to the previously described frequency jumps inwhich case the extracted T ∗ value is meaningless. Wecalculate the mean square error of each fit and disregardall the measurements with a high error. The average T ∗ of the 41 remaining traces is 24 ± µ s (Extended DataFig. 8b). Averaging the data traces of all 41 traces andthen fitting a decay curve yields a dephasing time of16 ± µ s (Extended Data Fig. 8c); averaging the dataof all 100 traces still gives a dephasing time of 11 ± µ s(Extended Data Fig. 8d). CPMG coherence measurements and powerspectral density
To ensure robust fitting, the CPMG sequences areapplied with artificial detuning. We fit the result-ing curves with a Gaussian damped cosine function: A (cos( ωt + φ ) + B ) exp[ − ( t/T CPMG2 ) ] + C . If, insteadof using a Gaussian decay, we leave the exponent ofthe decay open as a fitting parameter, we obtain valuesfor the exponent between 2.3 and 2.6, but the use ofthe additional parameter results in less robust fits.The offset B is included to compensate for the loss of2readout visibility for long microwave burst duration.We attribute this to heating generated while drivingthe spin rotations. The measurement is divided intosegments, each consisting of 200 single shots. Eachsegment includes a simple calibration part, based onwhich we post-select repetitions for which the spin-upprobability after applying a π -pulse is above 25 percent.In this way, we can exclude repetitions where the qubitresonance frequency has shifted drastically. The remain-ing repetitions are averaged to obtain the characteristicdecay curves for each choice of n , one of which is shownin Fig. 4b. From fitting the decay curves, we extract the T CPMG2 times as a function of n , shown in Fig. 4d. Toextract the CPMG amplitude as a function of evolutiontime from the data, we demodulate the measured valueswith the parameters extracted from the fit, according to A CPMG = ( x − C ) / ( A (cos( ωt + φ ) + B )), with x the mea-sured data. Due to experimental noise, points where thedenominator is small, do not yield meaningful results.Hence, we exclude data points for which the absolutevalue of the expected denominator is smaller than 0.4.The extracted CPMG amplitudes are plotted in Fig. 4c.In a commonly used simplified framework [52, 53], wecan relate the data of Fig. 4d to a noise power spectraldensity of the form S ( ω ) ∝ /ω γ . Specifically, fittingthe data to T CP MG ( n ) ∝ n γ/ ( γ +1) gives γ = 1 . ± . γ by fitting the noisepower spectral density extracted from the individualdata points in the CPMG decays [53] in Extended DataFig. 9(a). This analysis gives γ = 1 . ± .
1. Eitherway, the extracted power spectral density is close to the1 /f dependence that is characteristic of charge noise.Charge noise can affect spin coherence since the spinresonance frequency is sensitive to the gate voltage, asalso reported before for Si-MOS based spin qubits [3].We next estimate how large charge noise would needto be in order to dominate spin decoherence. To doso, we extrapolate the extracted spectral density in therange between 10 and 10 Hz to an amplitude at 1 Hz,which after conversion to units of charge noise gives29 ± µ eV/ √ Hz. With the caveat that this extrapola-tion is not very precise, we note that this value is onlyslightly larger than the charge noise amplitude in themulti-electron regime of 2 − µ eV/ √ Hz. Consideringthat charge noise values are typically higher in thefew-electron regime, this suggests that coherence of Q1may be limited by charge noise [53]. For Q2, which isanother qubit in the same sample, the same proceduregives an extrapolated noise at 1 Hz that is an orderof magnitude larger. Possibly a two-level fluctuator isactive in the vicinity of this qubit in the regime wherethe qubit data was taken.
ACKNOWLEDGEMENTS
We thank Luca Petit and Sander de Snoo for soft-ware support and Raymond Schouten, Raymond Ver-meulen, Marijn Tiggelman, Jason Mensingh, Olaf Ben-ningshof and Matt Sarsby for technical support. More-over, we thank all people from the QuTech spin qubitgroup and from the Intel Components research group fordiscussions. We acknowledge financial support from In-tel Corporation and the QuantERA ERA-NET Cofundin Quantum Technologies implemented within the Euro-pean Union’s Horizon 2020 Program.
AUTHOR CONTRIBUTIONS
A.M.J.Z., T.K., T.F.W., L.L. and F.L. performed thequantum dot and qubit measurements. J.B., D.C.S.,J.P.D., G.D., R.K., D.J.M., R.P., N.S., G.S., M.V.,L.M.K.V. and J.C. designed the devices. S.A.B.,H.C.G., E.M.H. and B.K.M. fabricated the devices.P.A., J.M.B., R.C., T.K., L.L., F.L., D.M., S.N., R.P.,T.F.W., O.K.Z., G.Z. and A.M.J.Z. characterised thetest structures and devices. M.L. characterised theSi-MOS stacks. S.V.A. contributed to the preparationof the experiments. A.M.J.Z., T.K., T.F.W., L.L. andF.L. analysed the data. J.R., L.M.K.V. and J.S.C.conceived and supervised the project. A.M.J.Z., T.K.and L.M.K.V. wrote the manuscript with input from allauthors.
ADDITIONAL INFORMATION
Data availability
Datasets and analysis scripts sup-porting the conclusions of this paper are available athttps://doi.org/10.5281/zenodo.4478855.
Competing interests
The authors declare no compet-ing interests.3
EXTENDED DATA FIG. 1.
Comparison of an academic SiMOS device and a TEM-image of an industrial device.a,
High-angle annular dark-field scanning transmission electron microscopy image (HAADF-STEM) of a SiMOS device from [1],fabricated at Delft University of Technology, using electron beam lithography and lift-off techniques. Below the fine gates, thescreening gates of the device are visible. b, TEM image of a device nominally identical to devices measured here. Comparedto the academic device, the gate profiles and dimensions are much more uniform and well-defined (as stated in the main text,there are two sets of nested gates, so we should compare gates within the same set), and there are no traces of spurious piecesof metal. a r X i v : . [ c ond - m a t . m e s - h a ll ] J a n EXTENDED DATA FIG. 2.
Sample with ESR line.
A scanning electron microscope image of a sample with an ESR line,nominally identical to the samples measured in this letter. The ESR line is false-coloured in red. The active area of the sampleis indicated by the cartoon of a spin. Metal dummification is clearly visible in the image, the quantum dot gates are not visibleas they are covered by dielectric. c da b EXTENDED DATA FIG. 3.
Cross-wafer variation of threshold voltages.
Each 300 mm wafer consists of 82 die and eachdie contains quantum dot arrays with various design skews and array sizes (up to 55 gates), as well as transistor and calibrationtest structures. To analyse cross-wafer sample uniformity, automated probing at room temperature is used to measure oneseven-gate device per die (nominally identical to the devices discussed in the main text). For each device, the threshold voltagefor the seven gates on the qubit fin are analysed. a, Median threshold voltage of the three gates of the first gate layer (even-numbered). For the bulk of the samples, the median threshold voltage is between 0 . . b, Median threshold voltagefor the four gates of the second gate layer (odd-numbered gates). c, Spread (highest minus lowest value) in threshold voltage ofthe three gates in the first gate layer per sample. For the majority of the wafer, the spread in threshold voltage is between 50and 200 mV. d, Spread in threshold voltage for the second gate layer per sample. In general, the spread in threshold voltagefor the majority of the second gate layer is between 100 and 250 mV. The combination of figures a , b , c and d . shows thatthe spread of threshold voltage per sample is low and the median is rather uniform, indicating a good cross-wafer uniformityand thus a precise process control. The cross-wafer variation of the second gate layer is slightly augmented relative to the firstlayer. This arises from small process-induced cross-wafer variability of the final effective oxide thickness of the second gatelayer, which gives rise to an electrostatic change and results in a small shift in threshold voltage. EXTENDED DATA FIG. 4.
Coulomb diamonds
Typical Coulomb blockade diamonds measured in the multi-electron regime,from which the quantum dot charging energy and the gate lever arms are determined. Coulomb diamonds are measured byscanning a gate voltage versus the bias voltage applied between the source and the drain contact. When the electrochemicalpotential of the quantum dot falls outside the bias window, the current through the quantum dot is blocked, i.e. the sample is inCoulomb blockade. Once the electrochemical potential of the quantum dot is aligned within the bias window, Coulomb blockadeis lifted. The bias window increases along the vertical axis, hence Coulomb blockade is lifted over a wider gate voltage range.The regions of blockaded current have the characteristic diamond shape. Here we extract a charging energy of 8 . ± . .
36 meV/mV.
800 900 1000 1100 1200 1300 1400G2 [mV]90010001100120013001400 G [ m V ] dI Sens dV G [a.u.] EXTENDED DATA FIG. 5.
Charge stability diagram for the few-electron regime.
Charge stability diagram for thefew-electron regime of the samples in which qubit 1 and qubit 2 are measured. Smaller charge sensing maps are stitchedtogether to obtain one large map. As pointed out in the main text, the Coulomb peaks become more irregular towards thesingle-electron regime, indicating dots forming under adjacent gates. The approximate gate voltages at which qubit 1 and qubit2 are measured are indicated. a b On resonant (I/Q = 0.05 V, 3 µ s)time Off resonant (-10MHz, 20 µ s) I/ Q EXTENDED DATA FIG. 6.
Frequency shift due to off-resonant microwave pulse amplitude.
Microwave spectroscopyof a, qubit 1 and b, qubit 2 as a function of the I/Q amplitude of an off-resonant microwave pre-burst (orange in schematic)that is applied immediately before the microwave spectroscopy burst (purple in schematic). Both qubit 1 and qubit 2 showsimilar behavior with the qubit frequency shifting to a lower frequency when the I/Q amplitude of the off-resonant microwavepre-burst is above 0 .
05 V. The microwave output power at an I/Q amplitude of 0.2 V is 6 dBm and the LO frequency is13 .
072 GHz for qubit 1 and 13 .
053 GHz for qubit 2 (different tuning than in the main text). The off-resonant burst is 10 MHzaway from the LO frequency with a duration of 20 µ s. The spectroscopic microwave burst has an I/Q amplitude of 0 .
05 V witha duration of 3 µ s. a b On resonant time Off resonant I/ Q time I/ Q EXTENDED DATA FIG. 7.
Time dependent frequency shift of qubit 1.
In these measurements we perform microwavespectroscopy of the qubit at low power to find the qubit resonance frequency. Before the microwave spectroscopy burst (purplein schematic), we apply an off-resonant burst. a, The resonance frequency of the qubit as a function of the duration of theoff-resonant burst applied before spectroscopy (purple dots). An exponential fit gives a time constant of 1 . µ s. b, The resonantfrequency of the qubit as a function of the time between the off-resonant and spectroscopy pulse. An exponential fit gives atime constant of 37 µ s. The time dependence of the resonance frequency of the qubit while turning on and off the microwavesignal indicates that the frequency shift is related to heating. The off-resonant burst is applied 5 MHz away from the LO andhas an I/Q amplitude of 0 . .
05 V and a duration of 3 . µ s. The LOfrequency is 17 . c da b EXTENDED DATA FIG. 8.
Ramsey analysis over time. a,
Fast Fourier transform (FFT) of a Ramsey experiment. Thedata consists of 100 traces, each trace is an average of 200 repetitions. The entire measurement takes three hours in total. TheFFT of the Ramsey fringes shows frequency jumps over the timescale of the measurement. Each trace is fitted individuallywith a decaying Gaussian curve and the fit is analysed by calculating the mean square error. We keep the 41 traces with meansquare error below a given threshold. b, Extracted T ∗ for the selected traces (purple dots). The average of the T ∗ times ofthe selected traces is 24 ± µ s (orange line). c, Ramsey decay curve. The data points are the averaged data of the 41 selectedtraces. The fit gives a T ∗ time of 16 ± µ s. d, Ramsey decay curve. The data points are the average of all 100 traces of thetwo hour and 40 minute measurement. The fit gives a T ∗ time of 11 ± µ s. a b EXTENDED DATA FIG. 9.
Noise analysis for qubit 1. a,
Dynamical decoupling pulses for which the time between twosubsequent π -pulses is fixed, can act as a filter function for Gaussian noise [2, 3]. The filter function peaks at a frequencyof f = n t wait , with n the number of π -pulses and t wait the evolution time. When the filter is sufficiently narrow aroundthe frequency f , the noise within the bandwidth can be regarded as constant. We can use this to relate the amplitudeof the CPMG decay, A CPMG , for each wait time and number of π pulses, to the dominant noise spectrum for the qubit: S ( f ) = − ln( A CPMG ) / π t wait [4, 5]. Here, we plot the noise spectrum, S ( f ) as a function of frequency for the data shown inFig. 4c of the main text. We only take data points into account for which 0 . < A CPMG < .
85. We assume that the noisespectrum dominating spin decoherence is described by a power law and use the fit function Bf − γ . We obtain γ = 1 . ± . B = 820 ±
750 Hz/ √ Hz. This agrees well with the fitting of the individual CPMG curves and the T ,CPMG scaling in Fig. 4 of the main text. Using the susceptibility of the qubit resonance frequency to a voltage change ofa nearby gate and the energy lever arm of that gate we can compare the noise value B to charge noise. With this conversionwe obtain B = 29 ± µ eV/ √ Hz. b, Following [2], we now use the filter function as described in a to estimate the noisedecay and the noise level at 1 Hz that gives the given T ,CPMG for the number of π -pulses and fit this to our data. We obtain γ = 1 . ± .
13 and B = 30 ± µ eV/ √ Hz. This is comparable to the results obtained in a . ce fda b EXTENDED DATA FIG. 10.
AllXY and Randomised benchmarking. a, d,
To check the calibration of the single-qubitgates we perform an AllXY sequence [6] on a, qubit 1 and d, qubit 2. Each data point corresponds to the outcome aftersequentially applying two gates from the set I, X, X , Y, Y , where X and Y indicate 90 ◦ rotations. The data points shouldideally follow a staircase pattern (solid line) and deviations from this indicate calibration errors. b, c, e, f, To determinethe single-qubit gate fidelity, we perform randomised benchmarking [7, 8]. In randomised benchmarking, we randomly select p gates from a set of 24 gates that form the Clifford group and apply them to the qubit. At the end of the sequence we apply aninverting gate from the Clifford group that ideally takes the qubit state back to either | i or | i . In this experiment, the Cliffordgates are decomposed to the set of primitive gates I, ± X, ± X , ± Y, ± Y . On average a Clifford gate contains 1.875 primitivegates. b, e, Normalised spin-up probability as a function of the number of Clifford operations applied for b, qubit 1 and e, qubit 2. The orange and purple data points correspond to sequences producing a net Clifford of X or I, respectively, ideallytaking the spin to either spin up or spin down. Each data point corresponds to 40 randomisations of the Clifford sequenceand the normalisation is done by additional calibration experiments where we apply either just I or X to the qubit. c, f, Thedifference between the purple and orange data points in b, e, is fitted with and exponential of the form
V P pc . From this wederive an average Clifford-gate fidelity of F C = 1 − (1 − P c ) / . ± .
7% and 98 . ± .
4% for qubit 1 and qubit 2 respectively.This translates to a primitive gate fidelity of 99 . ± .
4% and 99 . ± .
7% for Q1 and Q2. ceda b EXTENDED DATA FIG. 11.
Rabi oscillations for qubit 2. a-d,
Rabi oscillation of qubit 2 measured for different outputpowers of the microwave source. e, The extracted Rabi frequency is plotted versus the square root of the applied power,showing the expected linear dependence. ca b
EXTENDED DATA FIG. 12.
Coherence of qubit 2. a,
CPMG-curve for qubit 2 for n = 20. Fitting this curve, as describedin the methods, gives T ,CPMG = 1 . ± . b, Analogously to the case of qubit 1 (see main text), we demodulate andnormalise the CPMG amplitude as a function of evolution time for different numbers of π pulses, giving the CPMG amplitude. c, The measured CPMG decay time as a function of the number of π -pulses. The orange line represents a fit through the data(excluding n = 1) following T CPMG2 ∝ n ( γ/ ( γ +1)) . We extract γ = 1 . ± .
15. Performing a similar analysis as has been donefor qubit 1 (see Extended Data Fig. 9) gives unreliable results. a b EXTENDED DATA FIG. 13.
Rabi oscillations for qubit 3 . a, Rabi oscillation for a third qubit measured on a differentdevice than qubits 1 and 2. The qubit was measured at an external magnetic field of B = 0 .
675 T, giving a Larmor frequencyof 18 .
757 GHz. From fitting the curve, we extract a Rabi frequency of 1.4 MHz. b, Spin-up probability versus burst durationand microwave frequency in a slightly different tuning regime. The expected Chevron pattern is visible. We observed a second(spurious) quantum dot in the vicinity of qubit 3 and expect that hybridisation with this extra quantum dot is limiting the T ∗ and also the T ,Rabi of qubit 3.[1] Eenink, H. G. J. et al. Tunable coupling and isolation of single electrons in silicon metal-oxide-semiconductor quantumdots.
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Nat. Phys. ,565–570 (2011).[3] Cywinski, L., Lutchyn, R. M., Nave, C. P. & Das Sarma, S. How to enhance dephasing time in superconducting qubits. Phys. Rev. B , 11 (2008).[4] Kawakami, E. et al. Gate fidelity and coherence of an electron spin in an Si/SiGe quantum dot with micromagnet.
Proc.Natl. Acad. Sci. USA , 11738–11743 (2016).[5] Yoneda, J. et al.
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