Main

In the global effort to build scalable quantum processors, spin qubits in semiconductor quantum dots1 are progressively making their mark2. We highlight, in particular, the achievement of single-3,4 and two-qubit5,6,7,8 gate fidelities well above 99%, the first realizations of multi-qubit arrays9,10 and a demonstrated compatibility with industrial-grade semiconductor manufacturing technologies11,12,13.

Due to their long coherence time, electron-spin qubits in silicon quantum dots have so far attracted the most attention2. That said, their control requires add-ons such as metal microstrips3, micromagnets4 or dielectric resonators14, the large-scale integration of which is technically challenging13. Hole spin qubits, on the other hand, can circumvent this difficulty due to their intrinsically large spin–orbit coupling, which enables electric-dipole spin manipulation. Over the last five years a variety of hole spin qubits have been reported in both silicon11,15 and germanium16,17,18,19 quantum dots. In all these qubits, quantum operations are performed using high-frequency gate voltage excitations.

The downside of all-electrical spin control is that the required spin–orbit coupling exposes the qubit to charge noise, leading to a reduced hole spin coherence. Recent theoretical works20,21,22, however, have shown that, for properly chosen structural geometries and magnetic field orientations, careful tuning of the electrostatic confinement can bring the hole qubit to an optimal operation point where the effects of charge noise vanish to first order while enabling efficient electric-dipole spin resonance. Here, using a single hole spin confined in natural silicon, we pinpoint the existence of operation sweet spots where the longitudinal spin-electric susceptibility is minimized, resulting in a large enhancement of the spin coherence time.

Numerical simulations are found in remarkable agreement with the experimental observations, and predict that such sweet spots are resilient to realistic amounts of disorder. This advocates the use of such sweet spots as a reliable way to decouple hole spin qubits from charge noise, thereby reinforcing the promises of emergent hole-based quantum processors23.

Device design and g-factor anisotropy

Our device consists of an undoped silicon nanowire of rectangular cross section in which the electrostatics is controlled by four gates (G1–G4) as shown in Fig. 1a,b. We define a large hole island below G3 and G4 to be used simultaneously as a reservoir and as a charge sensor for a single hole trapped in a quantum dot, QD2, under G2. Single-shot readout of this hole spin is performed by means of a spin-to-charge conversion technique based on the real-time detection of spin-selective tunnelling to the reservoir, a widely used method often referred to as ‘Elzerman readout’24. Tunnelling events are detected by dispersive radiofrequency reflectometry on the charge sensor (see Methods and Extended Data Fig. 1 for technical details).

Fig. 1: Device, measurement scheme and properties of the first confined hole.
figure 1

a, Simplified three-dimensional representation of a silicon (yellow)-on-insulator (green) nanowire device with four gates (light blue) labelled G1, G2, G3 and G4. Gate G2 defines a quantum dot (QD2) hosting a single hole; G3 and G4 define a hole island used as reservoir and sensor for hole spin readout; G1 defines a hole island screening QD2 from dopant disorder and fluctuations in the source. Using bias tees, both static voltages (VG1, VG2) and time-dependent, high-frequency voltages (MW1, MW2) can be applied to G1 and G2, respectively. The drain contact is connected to an off-chip, surface-mount inductor to enable radiofrequency reflectometry readout. The coordinate system used for the magnetic field is shown on the left side (in the crystal frame, x = [001], \(y=[1\bar{1}0]\) and z = [110]). Each axis is given a different colour, which is used throughout the manuscript to indicate the magnetic field orientation. b, Colourized scanning electron micrograph showing a tilted view of a device similar to the measured one. Image taken just after the etching of the spacer layers. Scale bar, 100 nm. c, Rendering of the calculated wave function of the first hole accumulated under G2. d, Measured (dots) and calculated (solid line) hole g-factor as a function of the in-plane magnetic field angle θzy (dots). θzy = 90∘ corresponds to a magnetic field applied along the y axis. e, Same as d but in the xz plane. θzx = 90∘ corresponds to a magnetic field applied along the x axis. BOX, buried oxide.

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In our device geometry, the first holes primarily accumulate in the upper corners of the silicon nanowire25. Figure 1c displays the expected single-hole wave function in QD2, computed with a finite-differences k ⋅ p model including the six topmost valence bands26 (see Methods and Supplementary Information, section 1). At low energy, that is, close to the valence-band edge, the hole wave function primarily contains heavy-hole (HH) and light-hole (LH) components. The strong two-axes confinement readily seen in Fig. 1c favours HH–LH mixing27,28. This mixing is expected to manifest in the anisotropy of the hole g-tensor, which carries information on the relative weight of the HH and LH components29,30,31. To verify this, we measure the hole spin resonance frequency fL while varying the orientation of the magnetic field B in the xz and yz planes. The effective g-factor g = hfL/(μB∣B∣) (with μB the Bohr magneton and h the Planck constant) is plotted in Fig. 1d,e as a function of the magnetic field angles θzx and θzy, respectively. These maps highlight the strong anisotropy of the Zeeman splitting, with a maximal g = 2.7 close to the y axis (in-plane, perpendicular to the wire) and a minimal g = 1.4 close the z axis (in-plane, along the wire). The calculated g-factors are also plotted in the same figures as coloured solid lines. The agreement with the experimental data is remarkable. From the numerical simulation, we conclude that the measured g-factor anisotropy results from a strong electrical confinement against the side facet of the channel (along y), which prevails over the mostly structural vertical confinement (along x). The experimental g-factors and the small misalignment between the principal axes of the g-tensor and the device symmetry axes are best reproduced by introducing a moderate amount of charge disorder in combination with small (∼0.1%) shear strains in the silicon channel (Extended Data Figs. 2 and 3, and Supplementary Information, section 1). The latter probably originate from device processing and thermal contraction at the measurement temperature32.

Longitudinal spin-electric susceptibility

Given that the g-factor anisotropy is intimately related to the HH–LH mixing, which is controlled by the electrostatic confinement potential, the Larmor frequency is expected to be gate-voltage dependent. As a consequence, the hole spin coherence must be generally susceptible to charge noise. We thus measure the longitudinal spin-electric susceptibility (LSES) with respect to the voltages applied to the lateral gate G1 and to the accumulation gate G2, which we define as \({{{{\rm{LSES}}}}}_{{{{\rm{G1}}}}}=\frac{\partial {f}_{\mathrm{L}}}{\partial {V}_{{{{\rm{G1}}}}}}\) and \({{{{\rm{LSES}}}}}_{{{{\rm{G2}}}}}=\frac{\partial {f}_{\mathrm{L}}}{\partial {V}_{{{{\rm{G2}}}}}}\), respectively. In essence, LSESG1 and LSESG2 characterize the response of the Larmor frequency to the electric-field components parallel (z) and perpendicular (x,y) to the channel direction, respectively.

To probe the response to G2, we directly measure the spin resonance frequency fL at different VG2 (Extended Data Fig. 4). The resulting LSESG2 is plotted as a function of the magnetic field angle θzx in Fig. 2a. The observed angular dependence is in good agreement with the theoretical expectation.

Fig. 2: LSES.
figure 2

a, Spin-electric susceptibility with respect to VG2 (LSESG2) as a function of magnetic field angle θzx (symbols), at constant fL = 19 GHz. The LSES vanishes at θzx = 41∘ and 106∘, as indicated by the two arrows. The solid line corresponds to the numerically calculated LSESG2. b, Top: pulse sequence used to measure LSESG1, a voltage pulse of amplitude δVG1 and duration τz is applied to G1 during the first free evolution time of a Hahn-echo sequence. Bottom: spin-up fraction P↑ as a function of τz for δVG1 = 2.16 mV (diamonds), 3.12 mV (stars) and 4.80 mV (squares), at θzx = 90∘. The oscillation frequency varies with δVG1. c, δVG1 dependence of the frequency shift extracted from the Hahn-echo measurements at θzx = 0∘, 42∘ and 90∘. Symbols in the latter data set correspond to the P↑ oscillations shown in b. The solid lines are linear fits to the experimental data whose slope directly yields ∣LSESG1∣. d, Measured (symbols) and calculated (solid line) LSESG1 as a function of θzx, at constant fL = 17 GHz. The negative sign of LSESG1 is inferred from the shift of fL under a change in VG1.

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Noticeably, LSESG2 is positive along x and negative along z. Indeed, when increasing VG2, the hole wave function extends proportionally more in the yz plane than in the vertical x direction, which increases gx and decreases gy and gz (Extended Data Fig. 2b and Supplementary Information, section 1). As a result of the sign change, LSESG2 vanishes at two magnetic field orientations in the xz plane (marked by arrows in Fig. 2a), which are sweet spots for electric-field fluctuations perpendicular to the silicon channel.

To probe the response to G1, we introduce a pulse on VG1 in a Hahn-echo sequence4 as outlined in Fig. 2b. This defines a phase gate, controlled by the amplitude δVG1 and duration τz of the pulse. Figure 2b displays the coherent oscillations recorded as a function of τz for three different pulse amplitudes. The frequency of these oscillations is expected to increase linearly with δVG1, with a slope \({{{{\rm{LSES}}}}}_{{{{\rm{G1}}}}}=\frac{\partial {f}_{\mathrm{L}}}{\partial {V}_{{{{\rm{G1}}}}}}\). This is shown in Fig. 2c for different magnetic field orientations. LSESG1, plotted in Fig. 2d as a function of θzx, ranges from −0.5 MHz mV−1 to −0.1 MHz mV−1. Its magnitude is much smaller than that of LSESG2 because G1 is further from QD2 than G2 and its field effect is partly screened by the hole gas beneath. The numerically calculated LSESG1 (solid line) reproduces reasonably well the order of magnitude but not the angular dependence of the measured LSESG1. This discrepancy may be due to inaccuracies in the description of the hole gases near QD2 and to unaccounted charge disorder and strains (see discussion in Supplementary Information, section 1). We also notice that LSESG1 never vanishes and that the minimum of ∣LSESG1∣ happens to be almost at the same θzx as a zero of LSESG2.

Coherence times and frequency-dependent noise contributions

We now turn to the angular dependence of the hole spin coherence time and investigate its correlation with the longitudinal spin-electric susceptibility33. To get rid of low-frequency noise sources, we measure the coherence time using a conventional Hahn-echo protocol2. The control sequence, applied to G1 (see upper inset of Fig. 3a), consists of πx/2, πy and πϕ/2 pulses separated by a time delay τwait/2. For each τwait, we extract the averaged amplitude of the P↑ oscillation obtained by varying the phase ϕ of the last π/2 pulse, and normalize it to the P↑ oscillation amplitude in the zero-delay limit.

Fig. 3: Anisotropy of the hole spin coherence and sweet-spot operation.
figure 3

a, Normalized Hahn-echo amplitude versus free evolution time τwait at fL = 17 GHz. The top-right inset sketches the pulse sequence. The bottom-left inset displays P↑ (τwait = 31.4 μs) versus the phase ϕ of the last π/2 pulse for 100 repetitions. For each τwait, we extract the average amplitude of the P↑(ϕ) oscillations and normalize it to the average amplitude in the zero-delay limit. The resulting normalized echo amplitudes are reported on the main plot. The dashed curve is a fit to \(\exp (-{({\tau }_{{{{\rm{wait}}}}}/{T}_{2}^{\mathrm{E}})}^{\beta })\) with β = 1.5 ± 0.1. b, Measured \({T}_{2}^{{{{\rm{E}}}}}\) versus magnetic field angle θzx (symbols). The solid line is a fit to equation (1), using the experimental LSESG1 and LSESG2 from Fig. 2a–d. c, Normalized CPMG amplitude as a function of free evolution time τwait for different numbers Nπ of π pulses (curves are offset for clarity). The solid lines are fits to the same exponential decay function as in a with β = 1.5. d, Extracted \({T}_{2}^{{{{\rm{CPMG}}}}}\) as a function of Nπ. The dashed line is a linear fit with slope γ = 0.34. The inset sketches the CPMG pulse sequence: Nπ equally spaced πy pulses between two πx/2 pulses. For the Hahn-echo, we detune the phase of the last pulse.

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A representative Hahn-echo plot is shown in Fig. 3a. We fit the echo amplitude to an exponential decay \(\exp (-{({\tau }_{{{{\rm{wait}}}}}/{T}_{2}^{{{{\rm{E}}}}})}^{\beta })\), where the exponent β is left as a free parameter. The best fit is obtained for β = 1.5 ± 0.1, which implies a high-frequency noise with a characteristic spectrum \(S(f)={S}_{{{{\rm{hf}}}}}{({f}_{0}/f)}^{\alpha }\), where f0 = 1 Hz is a reference frequency and α = β − 1 ≈ 0.5 (we note that the same α value was reported for hole spin qubits in germanium10).

To explore the angular dependence of \({T}_{2}^{{{{\rm{E}}}}}\) in the xz plane, we measure the decay of the Hahn-echo amplitude for different values of θzx. The results, shown in Fig. 3b, reveal a strong anisotropy, with \({T}_{2}^{{{{\rm{E}}}}}\) ranging from 15 μs to 88 μs. Strikingly, the spin coherence time peaks at θzx = 99°, an angle between the minimum of ∣LSESG1∣ and a zero of LSESG2, highlighting a correlation with the correspondingly suppressed electrical noise. The extended coherence time is much longer than previously reported for hole spin qubits in both silicon (1.5 μs (ref. 15)) and germanium (3.8 μs (ref. 23))34. In addition, we notice that spin control remains efficient at all angles including θzx = 99∘, where we could readily achieve Rabi frequencies FRabi as large as 5 MHz limited by the attenuation on the microwave line. The echo quality factor \({Q}^{{{{\rm{E}}}}}={F}_{{{{\rm{Rabi}}}}}\times {T}_{2}^{{{{\rm{E}}}}}\) also peaks at θzx = 99°, reaching QE ≈ 440 with further room for improvement (Supplementary Information, section 2 and Extended Data Fig. 5).

The observed angular dependence of \({T}_{2}^{{{{\rm{E}}}}}\) can be understood by assuming that the electrical noise is the sum of uncorrelated voltage fluctuations on the different gates Gi with respective spectral densities \({S}_{{{{\rm{G}}}}i}(f)={S}_{{{{\rm{G}}}}i}^{{{{\rm{hf}}}}}{({f}_{0}/f)}^{0.5}\). Given the Hahn-echo noise filter function, the decoherence rate can then be expressed as (Supplementary Information, section 3):

$$\frac{1}{{T}_{2}^{{{{\rm{E}}}}}}\approx 7.8{f}_{0}^{1/3}{\left(\mathop{\sum}\limits_{i}{\left(\frac{\partial {f}_{\mathrm{L}}}{\partial {V}_{{{{\rm{G}}}}i}}\right)}^{2}{S}_{{{{\rm{G}}}}i}^{{{{\rm{hf}}}}}\right)}^{2/3}.$$
(1)

Using the longitudinal spin-electric susceptibilities from Fig. 2a–d and leaving the weights \({S}_{{{{\rm{G}}}}i}^{{{{\rm{hf}}}}}\) as adjustable parameters, we achieve a remarkable agreement with the experimental \({T}_{2}^{{{{\rm{E}}}}}\) (coloured solid line in Fig. 3b). This strongly supports the hypothesis that the Hahn-echo coherence time is limited by electrical noise. As already argued before, LSESG1 and LSESG2 indeed quantify the susceptibility of the hole spin to electric field fluctuations parallel and perpendicular to the channel, respectively.

The best fit in Fig. 3b is obtained with \({S}_{{{{\rm{G}}}}1}^{{{{\rm{hf}}}}}={(1.7\,\upmu {{{\rm{V}}}}/\sqrt{{{{\rm{Hz}}}}})}^{2}\) and \({S}_{{{{\rm{G}}}}2}^{{{{\rm{hf}}}}}={(66\,{{{\rm{nV}}}}/\sqrt{{{{\rm{Hz}}}}})}^{2}\). We speculate that the large \({S}_{{{{\rm{G}}}}1}^{{{{\rm{hf}}}}}/{S}_{{{{\rm{G}}}}2}^{{{{\rm{hf}}}}}\) ratio results from an artificial enhancement of \({S}_{{{{\rm{G}}}}1}^{{{{\rm{hf}}}}}\) accounting for hidden sources of electric field fluctuations along the silicon nanowire. Certainly, equation (1) misses the contribution from the electrical noise on G3, the LSES of which could not be measured. For reasons of symmetry, we expect LSESG3 to be comparable to LSESG1. A possible additional source of longitudinal electric field fluctuations are the randomly oscillating charges and dipoles in the silicon nitride spacers between the gates. Because these noise sources are closer to QD2 than is gate G1, and because they are much less screened by the hole gas beneath, they presumably make a large contribution to the apparent \({S}_{{{{\rm{G}}}}1}^{{{{\rm{hf}}}}}\) when lumped into ∝LSESG1 terms.

To further investigate the hole spin coherence, we implement Carr–Purcell–Meiboom–Gill (CPMG) sequences at the most favourable field orientation θzx = 99∘. These consist in increasing the number of π pulses cancelling faster and faster dephasing mechanisms. Figure 3c displays the CPMG echo amplitudes as a function of the total waiting time τwait for series of Nπ = 2nπ pulses, where n is an integer ranging from 1 to 8. The CPMG decay times \({T}_{2}^{{{{\rm{CPMG}}}}}\) extracted from Fig. 3c (see caption) are plotted against Nπ in Fig. 3d. As expected, the data points follow a power law \({T}_{2}^{{{{\rm{CPMG}}}}}\propto {N}_{\uppi }^{\gamma }\), where \(\gamma =\frac{\alpha }{\alpha +1}\) for a ∝1/fα noise spectrum4. The best-fit value γ = 0.34 yields again α ≈ 0.5. For the largest sequence of 256 π pulses, we find \({T}_{2}^{{{{\rm{CPMG}}}}}=0.4\,{\mathrm{ms}}\), an exceptionally long coherence for a hole spin34.

Finally, to gain insight into the low-frequency noise acting on the hole spin, we perform systematic measurements of the inhomogeneous dephasing time \({T}_{2}^{* }\). To this aim, we apply Ramsey control sequences consisting of two π/2 pulses separated by a variable delay τwait. Contrary to Hahn-echo, the dephasing induced by low-frequency noise sources is not cancelled due to the absence of the refocusing π pulse. Figure 4a displays P↑ for a series of identical Ramsey sequences recorded on an overall time frame of 1 h, with each sequence lasting approximately 5.5 s. The next step is to average P↑(τwait) on a subset of consecutive sequences measured within a total time tmeas. This way, an averaged Ramsey oscillation is obtained for each tmeas, the amplitude of which is fitted to a Gaussian-decay function yielding \({T}_{2}^{* }({t}_{{{{\rm{meas}}}}})\). Representative Ramsey data sets and corresponding fits are shown in Fig. 4b for three values of tmeas. The inhomogeneous dephasing time decreases with increasing tmeas due to the contribution of noise components with lower and lower frequency. To unveil the angular dependence of \({T}_{2}^{* }\), we repeat the same measurement for different magnetic field orientations. The results are plotted in Fig. 4c for the same three values of tmeas. The overall anisotropy of the Hahn-echo decay time of Fig. 3b can still be identified, although it reduces at large tmeas starting from tmeas > 50 s.

Fig. 4: Free induction decay.
figure 4

a, Collection of 600 Ramsey oscillations as a function of τwait, the free evolution time between two πx/2 pulses, at θzx = 118∘. The applied microwave frequency is detuned by ~700 kHz from the Larmor frequency. Each Ramsey oscillation is measured in ∼5.5 s. The locations of the representative traces shown in b are indicated by a diamond and a dot. b, Selected averages of Ramsey oscillations taken over different measurement times: tmeas = 5.5 s corresponding to a single trace (diamonds); tmeas = 27.5 s, corresponding to five consecutive traces (circles); tmeas ≈ 1 h, corresponding to the full set of 600 traces (squares). The solid lines are fits to Gaussian decaying oscillations. Note that the decay time \({T}_{2}^{* }\) depends on the chosen subset of consecutive traces (except for tmeas ≈ 1 h), which is a signature of non-ergodicity36 at small tmeas (Supplementary Information, section 4). Hence we observe a distribution of \({T}_{2}^{* }\) values with mean \({\overline{T}}_{2}^{* }\). c, Mean \({\overline{T}}_{2}^{* }\) for different tmeas (same symbols as in b) as a function of the magnetic field angle θzx. The solid lines are guides to the eye. The dashed black line is the calculated dephasing time due to hyperfine interactions (Supplementary Information, section 5).

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However, if the 1/f0.5 charge noise prevailed over the whole mHz to MHz range, \({T}_{2}^{* }\) would be ∼50 μs when \({T}_{2}^{{{{\rm{E}}}}}\approx 88\,\upmu{\mathrm{s}}\) (Supplementary Information, section 3), well above the 7 μs seen in Fig. 4c. The power spectrum S(f) at low frequency can be extracted from the data of Fig. 4a (Extended Data Fig. 6). This reveals a 1/fα noise with α closer to 1, and a power (at 1 Hz) four orders of magnitude larger than the one expected by extrapolating the high-frequency 1/f0.5 noise inferred from CPMG. The change of exponent α and amplitude of S(f) when going from the mHz to the MHz points to the presence of different mechanisms dominating the dephasing at low and high frequencies. We note that the \({T}_{2}^{* }\approx 1{\text{--}}2\,\upmu{\mathrm{s}}\) measured at long tmeas is below but fairly close to the expected hole spin dephasing time due to hyperfine interactions with the naturally present 29Si nuclear spins25 (see the dashed line in Fig. 4c, and Supplementary Information, section 5 for details). This suggests that low-frequency dephasing may be partially due to such hyperfine interactions.

Conclusions

We report on a spin qubit with electrical control and single-shot readout based on a single hole in a silicon nanowire device issued from an industrial-grade fabrication line. The hole wave function and corresponding g-factors could be modelled with an excellent level of accuracy in these types of devices, denoting a relatively low level of structural and charge disorder. The hole-spin coherence was found to be limited by a 1/f0.5 charge noise at high frequencies (104−106 Hz), with a strong dependence on the magnetic-field orientation that could be faithfully accounted for by the spin-electric susceptibilities. A largely enhanced spin coherence was measured at the sweet-spot angle, far beyond the current state-of-the-art for hole-spin qubits and close to the best figures reported for 28Si electron-spin qubits electrically driven via a micromagnet. Our study of the inhomogeneous dephasing time revealed a much stronger noise at low frequencies (10−4−10−2 Hz) that could be partially ascribed to the expected hyperfine interaction. In this scenario, the possible introduction of isotopically purified silicon devices would lead to a significant improvement of hole–spin coherence in the low-frequency range. Finally, we would like to emphasize that such sweet spots should be ubiquitous in hole spin qubit devices21, and that a careful design and choice of operation point can make them usefully robust to disorder (see example in Supplementary Information, section 1). The engineering of sweet spots should therefore open new opportunities for an efficient realization of multi-qubit or coupled spin-photon systems35.

Methods

Device

The device is a four-gate silicon-on-insulator nanowire transistor fabricated in an industry-standard 300 mm CMOS platform11. The undoped [110]-oriented silicon nanowire channel is 17 nm thick and 100 nm wide. It is connected to wider boron-doped source and drain pads used as reservoirs of holes. The four wrapping gates (G1–G4) are 40 nm long and are spaced by 40 nm. The gaps between adjacent gates and between the outer gates and the doped contacts are filled with silicon nitride (Si3N4) spacers. The gate stack consists of a 6-nm-thick SiO2 dielectric layer followed by a metallic bilayer with 6 nm of TiN and 50 nm of heavily doped polysilicon. The yield of the four-gate devices across the full 300 mm wafer reaches 90% and their room temperature characteristics exhibit excellent uniformity (see Supplementary Information, section 6 for details).

Dispersive readout

Similar to charge detection methods recently applied to silicon-on-insulator nanowire devices37,38, we accumulate a large hole island under gates G3 and G4, as sketched in Fig. 1a. The island acts both as a charge reservoir and electrometer for the quantum dot QD2 located under G2. However, unlike the aformentioned earlier implementations, the electrometer is sensed by radiofrequency dispersive reflectometry on a lumped element resonator connected to the drain rather than to a gate electrode. To this aim, a commercial surface-mount inductor (L = 240 nH) is wire bonded to the drain pad (see Extended Data Fig. 7 for the measurement set-up). This configuration involves a parasitic capacitance to ground Cp = 0.54 pF, leading to resonance frequency f = 449.81 MHz. The high value of the loaded quality factor Q ≈ 103 enables fast, high-fidelity charge sensing. We estimate a charge readout fidelity of 99.6% in 5 μs, which is close to the state-of-the-art for silicon MOS devices39. The resonator characteristic frequency experiences a shift at each Coulomb resonance of the hole island, that is, when the electrochemical potential of the island lines up with the drain Fermi energy. This leads to a dispersive shift in the phase ϕdrain of the reflected radiofrequency signal, which is measured through homodyne detection.

Energy-selective single-shot readout of the spin state of the first hole in QD2

Extended Data Fig. 1a displays the stability diagram of the device as a function of VG2 and VG3 when a large quantum dot (acting as a charge sensor) is accumulated under gates G3 and G4. The dashed grey lines outline the charging events in the quantum dot QD2 under G2, detected as discontinuities in the Coulomb peak stripes of the sensor dot. The lever-arm parameter of gate G2 is αG2 ≈ 0.37 eV V−1, as inferred from temperature-dependence measurements. Comparatively, the lever-arm parameter of gate G1 with respect to the first hole under G2, αG1 ≈ 0.03 eV V−1, is much smaller. The charging energy, measured as the splitting between the first two charges, is U = 22 meV. Extended Data Fig. 1b shows a zoom on the stability diagram around the working point used for single-shot spin readout in the main text. The three points labelled Empty (E), Load (L) and Measure (M) are the successive stages of the readout sequence sketched in Extended Data Fig. 1c. The quantum dot is initially emptied (E) before loading (L) a hole with a random spin. Both spin states are separated by the Zeeman energy EZ = gμBB where g is the g-factor, μB is the Bohr magneton and B is the amplitude of the magnetic field. This opens a narrow window for energy-selective readout using spin to charge conversion40. Namely, we align at stage M the centre of the Zeeman split energy levels in QD2 with the chemical potential of the sensor. In this configuration, only the excited spin-up hole can tunnel out of QD2 while only spin-down holes from the sensor can tunnel in. These tunnelling events are detected by thresholding the phase of the reflectometry signal of the sensor to achieve single-shot readout of the spin state. Typical time traces of the reflected signal phase at stage M, representative of a spin up (spin down) in QD2, are shown in Extended Data Fig. 1d. We used this three-stage pulse sequence to optimize the readout. For that purpose, the tunnel rates between QD2 and the charge sensor were adjusted by fine tuning VG3 and VG4. For the spin-manipulation experiment discussed in the main text, we use a simplified two-stage sequence for readout by removing the empty stage. The measure stage duration is set to 200 μs for all experiments, while the load stage duration (seen as a manipulation stage duration) ranges from 50 μs to 1 ms. To obtain the spin-up probability P↑ after a given spin manipulation sequence, we repeat the single-shot readout a large number of times, typically 100–1,000 times.

Pulse sequences

For Ramsey, Hahn-echo, phase-gate and CPMG pulse sequences, we set a π/2 rotation time of 50 ns. Given the angular dependence of FRabi, we calibrate the microwave power required for this operation time for each magnetic field orientation. We also calibrate the amplitude of the π pulses to achieve a π rotation in 150 ns. In extracting the noise exponent γ from CPMG measurements, we do not include the time spent in the π pulses (this time amounts to about 10% of the duration of each pulse sequence).

Noise spectrum

We measured 3,700 Ramsey fringes over ttot = 10.26 h. For each realization, we varied the free evolution time τwait up to 7 μs, and averaged 200 single-shot spin measurements to obtain P↑ (Extended Data Fig. 6a, top). The fringes oscillate at the detuning Δf = ∣fMW1 − fL∣ between the MW1 frequency fMW1 and the spin resonance frequency fL. To track low-frequency noise on fL, we make a Fourier transform of each fringe and extract its fundamental frequency Δf reported in Extended Data Fig. 6a (bottom). Throughout the experiment, fMW1 is set to 17 GHz. The low-frequency spectral noise on the Larmor frequency (in units of Hz2 Hz−1) is calculated (here we make use of two-sided power spectral densities, which are even with respect to the frequency) from Δf(t) as4:

$${S}_{\mathrm{L}}=\frac{{t}_{{{{\rm{tot}}}}}{\left|{{{\rm{FFT}}}}[{{\Delta }}f]\right|}^{2}}{{N}^{2}}\,,$$
(2)

where FFT[Δf] is the fast Fourier transform (FFT) of Δf(t) and N is the number of sampling points. We observe that the low-frequency noise, plotted in Extended Data Fig. 6b, behaves approximately as SL(f) = Slf(f0/f) with Slf = 109 Hz2 Hz−1, which is comparable to what has been measured for a hole spin in natural germanium41. To further characterize the noise spectrum, we add the CPMG measurements as coloured dots in Extended Data Fig. 6b4:

$${S}_{\mathrm{L}}\left({N}_{\uppi }/(2{\tau }_{{{{\rm{wait}}}}})\right)=-\frac{\ln ({A}_{{{{\rm{CPMG}}}}})}{2{\uppi }^{2}{\tau }_{{{{\rm{wait}}}}}},$$
(3)

where ACPMG is the normalized CPMG amplitude. As discussed in the main text, the resulting high-frequency noise scales as \({S}^{{{{\rm{hf}}}}}{({f}_{0}/f)}^{0.5}\), where Shf = 8 × 104 Hz2 Hz−1 is four orders of magnitude lower than Slf. This high-frequency noise appears to be dominated by electrical fluctuations, as supported by the correlations between the Hahn-echo/CPMG T2 and the LSESs. Additional quasi-static contributions thus emerge at low frequency, and may include hyperfine interactions (Supplementary Information, section 5).

Modelling

The hole wave functions and g-factors are calculated with a six-band k ⋅ p model26. The screening by the hole gases under gates G1, G3 and G4 is accounted for in the Thomas–Fermi approximation. As discussed extensively in Supplementary Information, section 1, the best agreement with the experimental data is achieved by introducing a moderate amount of charge disorder. The theoretical data displayed in Figs. 1, 2 and Extended Data Fig. 3 correspond to a particular realization of this charge disorder (point-like positive charges with density σ = 5 × 1010 cm−2 at the Si/SiO2 interface and ρ = 5 × 1017 cm−3 in bulk Si3N4). The resulting variability, and the robustness of the operation sweet spots with respect to disorder, are discussed in Supplementary Information, section 1. The rotation of the principal axes of the g-tensor visible in Fig. 1d,e are most probably due to small inhomogeneous strains (<0.1%); however, in the absence of quantitative strain measurements, we have simply shifted θzx by ∼−25° and θzy by ∼10° in the calculations of Figs. 1, 2 and Extended Data Fig. 3.