A single hole spin with enhanced coherence in natural silicon

Semiconductor spin qubits based on spin–orbit states are responsive to electric field excitations, allowing for practical, fast and potentially scalable qubit control. Spin electric susceptibility, however, renders these qubits generally vulnerable to electrical noise, which limits their coherence time. Here we report on a spin–orbit qubit consisting of a single hole electrostatically confined in a natural silicon metal-oxide-semiconductor device. By varying the magnetic field orientation, we reveal the existence of operation sweet spots where the impact of charge noise is minimized while preserving an efficient electric-dipole spin control. We correspondingly observe an extension of the Hahn-echo coherence time up to 88 μs, exceeding by an order of magnitude existing values reported for hole spin qubits, and approaching the state-of-the-art for electron spin qubits with synthetic spin–orbit coupling in isotopically purified silicon. Our finding enhances the prospects of silicon-based hole spin qubits for scalable quantum information processing.

I n the global effort to build scalable quantum processors, spin qubits in semiconductor quantum dots 1 are progressively making their mark 2 . We highlight, in particular, the achievement of single- 3,4 and two-qubit 5-8 gate fidelities well above 99%, the first realizations of multi-qubit arrays 9,10 and a demonstrated compatibility with industrial-grade semiconductor manufacturing technologies 11-13 . Due to their long coherence time, electron-spin qubits in silicon quantum dots have so far attracted the most attention 2 . That said, their control requires add-ons such as metal microstrips 3 , micromagnets 4 or dielectric resonators 14 , the large-scale integration of which is technically challenging 13 . 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 silicon 11,15 and germanium [16][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 works [20][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 processors 23 .

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).
In our device geometry, the first holes primarily accumulate in the upper corners of the silicon nanowire 25 . Figure 1c displays the expected single-hole wave function in QD2, computed with a finite-differences k ⋅ p model including the six topmost valence bands 26 (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 mixing 27,28 . 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 components [29][30][31] . To verify this, we measure the hole spin resonance frequency f L while varying the orientation of the magnetic field B in the xz and yz planes. The effective g-factor g = hf L /(μ B |B|) (with μ B the Bohr magneton and h the Planck constant) is plotted in Fig. 1d,e as a function of the magnetic field angles θ zx and θ zy , respectively. These maps highlight the Semiconductor spin qubits based on spin-orbit states are responsive to electric field excitations, allowing for practical, fast and potentially scalable qubit control. Spin electric susceptibility, however, renders these qubits generally vulnerable to electrical noise, which limits their coherence time. Here we report on a spin-orbit qubit consisting of a single hole electrostatically confined in a natural silicon metal-oxide-semiconductor device. By varying the magnetic field orientation, we reveal the existence of operation sweet spots where the impact of charge noise is minimized while preserving an efficient electric-dipole spin control. We correspondingly observe an extension of the Hahn-echo coherence time up to 88 μs, exceeding by an order of magnitude existing values reported for hole spin qubits, and approaching the state-of-the-art for electron spin qubits with synthetic spin-orbit coupling in isotopically purified silicon. Our finding enhances the prospects of silicon-based hole spin qubits for scalable quantum information processing.
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 temperature 32 .

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 LSES G1 = ∂f L ∂V G1 and LSES G2 = ∂f L ∂V G2 , respectively. In essence, LSES G1 and LSES G2 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 f L at different V G2 (Extended Data Fig. 4). The resulting LSES G2 is plotted as a function of the magnetic field angle θ zx in Fig. 2a. The observed angular dependence is in good agreement with the theoretical expectation.
Noticeably, LSES G2 is positive along x and negative along z. Indeed, when increasing V G2 , the hole wave function extends proportionally more in the yz plane than in the vertical x direction, which increases g x and decreases g y and g z (Extended Data Fig. 2b and Supplementary Information, section 1). As a result of the sign change, LSES G2 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 V G1 in a Hahn-echo sequence 4 as outlined in Fig. 2b. This defines a phase gate, controlled by the amplitude δV G1 and duration τ z of the pulse. Figure 2b displays the coherent oscillations recorded as a function of τ z for three different pulse amplitudes. The frequency of these oscillations is expected to increase linearly with δV G1 , with a slope LSES G1 = ∂f L ∂V G1 . This is shown in Fig. 2c for different magnetic field orientations. LSES G1 , plotted in Fig. 2d as a function of θ zx , ranges from −0.5 MHz mV −1 to −0.1 MHz mV −1 . Its magnitude is much smaller than that of LSES G2 because G1 is further from QD2 than G2 and its field effect is partly screened by the hole gas beneath. The numerically calculated LSES G1 (solid line) reproduces reasonably well the order of magnitude but not the angular dependence of the measured LSES G1 . 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 ). 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 θ zy (dots). θ zy = 90 ∘ corresponds to a magnetic field applied along the y axis. e, Same as d but in the xz plane. θ zx = 90 ∘ corresponds to a magnetic field applied along the x axis. BOX, buried oxide.
Information, section 1). We also notice that LSES G1 never vanishes and that the minimum of |LSES G1 | happens to be almost at the same θ zx as a zero of LSES G2 .

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 susceptibility 33 . To get rid of low-frequency noise sources, we measure the coherence time using a conventional Hahn-echo protocol 2 . The control sequence, applied to G1 (see upper inset of Fig. 3a), consists of π x /2, π y and π ϕ /2 pulses separated by a time delay τ wait /2. For each τ wait , 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. A representative Hahn-echo plot is shown in Fig. 3a. We fit the echo amplitude to an exponential decay exp(−(τ wait /T E 2 ) β ), 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 hf (f 0 /f) α , where f 0 = 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 germanium 10 ).
To explore the angular dependence of T E 2 in the xz plane, we measure the decay of the Hahn-echo amplitude for different values of θ zx . The results, shown in Fig. 3b, reveal a strong anisotropy, with T E 2 ranging from 15 μs to 88 μs. Strikingly, the spin coherence time peaks at θ zx = 99°, an angle between the minimum of |LSES G1 | and a zero of LSES G2 , 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 θ zx = 99 ∘ , where we could readily achieve Rabi frequencies F Rabi as large as 5 MHz limited by the attenuation on the microwave line. The echo quality factor Q E = F Rabi × T E 2 also peaks at θ zx = 99°, reaching Q E ≈ 440 with further room for improvement (Supplementary Information, section 2 and Extended Data Fig. 5).
The observed angular dependence of T E 2 can be understood by assuming that the electrical noise is the sum of uncorrelated voltage fluctuations on the different gates Gi with respective spectral den- Given the Hahn-echo noise filter function, the decoherence rate can then be expressed as (Supplementary Information, section 3): Using the longitudinal spin-electric susceptibilities from Fig. 2a-d and leaving the weights S hf Gi as adjustable parameters, we achieve a remarkable agreement with the experimental T E 2 (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, LSES G1 and LSES G2 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 hf G1 = (1.7 μV/ √ Hz) 2 and S hf G2 = (66 nV/ √ Hz) 2 . We speculate that the large S hf G1 /S hf G2 ratio results from an artificial enhancement of S hf G1 accounting for hidden sources of electric field fluctuations along the silicon nanowire. Control Measure Initialize Spin-electric susceptibility with respect to V G2 (LSES G2 ) as a function of magnetic field angle θ zx (symbols), at constant f L = 19 GHz. The LSES vanishes at θ zx = 41 ∘ and 106 ∘ , as indicated by the two arrows. The solid line corresponds to the numerically calculated LSES G2 . b, Top: pulse sequence used to measure LSES G1 , a voltage pulse of amplitude δV G1 and duration τ z is applied to G1 during the first free evolution time of a Hahn-echo sequence. Bottom: spin-up fraction P ↑ as a function of τ z for δV G1 = 2.16 mV (diamonds), 3.12 mV (stars) and 4.80 mV (squares), at θ zx = 90 ∘ . The oscillation frequency varies with δV G1 . c, δV G1 dependence of the frequency shift extracted from the Hahn-echo measurements at θ zx = 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 |LSES G1 |. d, Measured (symbols) and calculated (solid line) LSES G1 as a function of θ zx , at constant f L = 17 GHz. The negative sign of LSES G1 is inferred from the shift of f L under a change in V G1 .
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 LSES G3 to be comparable to LSES G1 . 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 hf G1 when lumped into ∝LSES G1 terms. To further investigate the hole spin coherence, we implement Carr-Purcell-Meiboom-Gill (CPMG) sequences at the most favourable field orientation θ zx = 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 τ wait for series of N π = 2 n π pulses, where n is an integer ranging from 1 to 8. The CPMG decay times T CPMG 2 extracted from Fig. 3c (see caption) are plotted against N π in Fig. 3d. As expected, the data points follow a power law T CPMG 2 ∝ N γ π , where γ = α α+1 for a ∝1/f α noise spectrum 4 . The best-fit value γ = 0.34 yields again α ≈ 0.5. For the largest sequence of 256 π pulses, we find T CPMG 2 = 0.4 ms, an exceptionally long coherence for a hole spin 34 . 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 τ wait . 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 ↑ (τ wait ) on a subset of consecutive sequences measured within a total time t meas . This way, an averaged Ramsey oscillation is obtained for each t meas , the amplitude of which is fitted to a Gaussian-decay function yielding T * 2 (tmeas). Representative Ramsey data sets and corresponding fits are shown in Fig. 4b for three values of t meas . The inhomogeneous dephasing time decreases with increasing t meas 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 t meas . The overall anisotropy of the Hahn-echo decay time of Fig. 3b can still be identified, although it reduces at large t meas starting from t meas > 50 s.
However, if the 1/f 0.5 charge noise prevailed over the whole mHz to MHz range, T * 2 would be ∼50 μs when T E 2 ≈ 88 μs ( 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/f α 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/f 0.5 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 ≈ 1-2 μs measured at long t meas is below but fairly close to the expected hole spin dephasing T E 2 versus magnetic field angle θ zx (symbols). The solid line is a fit to equation (1), using the experimental LSES G1 and LSES G2 from Fig. 2a-d. c, Normalized CPMG amplitude as a function of free evolution time τ wait for different numbers N π 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 CPMG 2 as a function of N π . The dashed line is a linear fit with slope γ = 0.34. The inset sketches the CPMG pulse sequence: N π equally spaced π y pulses between two π x /2 pulses. For the Hahn-echo, we detune the phase of the last pulse. time due to hyperfine interactions with the naturally present 29 Si nuclear spins 25 (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/f 0.5 charge noise at high frequencies (10 4 −10 6 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 28 Si 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 devices 21 , 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 systems 35 .

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Any methods, additional references, Nature Research reporting summaries, source data, extended data, supplementary information, acknowledgements, peer review information; details of author contributions and competing interests; and statements of data and code availability are available at https://doi.org/10.1038/ s41565-022-01196-z.

Methods
Device. The device is a four-gate silicon-on-insulator nanowire transistor fabricated in an industry-standard 300 mm CMOS platform 11 . 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 (Si 3 N 4 ) spacers. The gate stack consists of a 6-nm-thick SiO 2 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 devices 37,38 , 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 C p = 0.54 pF, leading to resonance frequency f = 449.81 MHz. The high value of the loaded quality factor Q ≈ 10 3 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 devices 39 . 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 ϕ drain 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 V G2 and V G3 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 α G2 ≈ 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, α G1 ≈ 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 E Z = gμ B B where g is the g-factor, μ B 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 conversion 40 . 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 V G3 and V G4 . 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 F Rabi , 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 t tot = 10.26 h. For each realization, we varied the free evolution time τ wait 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 = |f MW1 − f L | between the MW1 frequency f MW1 and the spin resonance frequency f L . To track low-frequency noise on f L , we make a Fourier transform of each fringe and extract its fundamental frequency Δf reported in Extended Data Fig. 6a (bottom). Throughout the experiment, f MW1 is set to 17 GHz. The low-frequency spectral noise on the Larmor frequency (in units of Hz 2 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) as 4 : 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 S L (f) = S lf (f 0 /f) with S lf = 10 9 Hz 2 Hz −1 , which is comparable to what has been measured for a hole spin in natural germanium 41 .
To further characterize the noise spectrum, we add the CPMG measurements as coloured dots in Extended Data Fig. 6b 4 : where A CPMG is the normalized CPMG amplitude. As discussed in the main text, the resulting high-frequency noise scales as S hf (f0/f) 0.5 , where S hf = 8 × 10 4 Hz 2 Hz −1 is four orders of magnitude lower than S lf . This high-frequency noise appears to be dominated by electrical fluctuations, as supported by the correlations between the Hahn-echo/CPMG T 2 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 model 26 . 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

Data availability
All of the data used to produce the figures in this paper and to support our analysis and conclusions are available at https://zenodo.org/search?page=1&size=2 0&q=6638442. This repository includes the original data, jupyter notebooks for data analysis and figure plotting. Additional data are available upon reasonable request to the corresponding author.

Code availability
The code is part of the Commisariat à l'Energie Atomique et aux Energies Alternatives strategy and could not be made public. However, the authors are ready to collaborate with anyone interested in the modelling tools used in this work, as they already do with several international teams.  Fig. 3 | Comparison between the experimental and calculated g-factors. (a)g-factors for a magnetic field in the xz (red) and yz (blue) planes, as a function of the angles θ zx and θ zy , respectively. The symbols are the experimental data, and the dotted lines are calculated in the pristine device at the experimental bias point (with the hole gases below G1, G3 and G4). The lateral electric field is however too weak at this bias point to match the experimental anisotropy g y > g x . The solid lines are calculated in a particular realization of a disordered device with roughness and positive charge traps at the Si/SiO 2 interface and in Si 3 N 4 (see Methods and Supp. Info S1). These traps tend to strengthen confinement on the side facets (because they are much better screened near the corners of the wrap gate), which increases g y and decreases g x , as shown in Extended Data Fig. 2. Moreover, θ zx is shifted by ≈ − 25 ∘ and θ zy by ≈ 10 ∘ to account for the experimental rotations of the principal axes of the g-tensor (resulting from residual strains, see Supp. Info S1). The polar plots of the g-factors and the LSES of this disordered device are shown in Figs Extended Data Fig. 7 | experimental set-up. Dilution fridge with all electrical connections to the sample. We operate in a dilution refrigerator system equipped with a three-axis vector superconducting magnet. The main solenoid magnet produces a magnetic field of up to 6 T in the z direction, while both transverse Helmholtz coils ramp up to 1 T in the x and y directions. However, one of the axis was broken during the experiment. Therefore, after recording Fig. 1d of the main text, the sample was warmed up, physically rotated by 90 ∘ , and cooled down again to record Fig. 1e. 24 twisted pairs are filtered at the mixing chamber by 6 low pass filters. The DC gate voltages are generated by Itest high stability voltage sources (BE2141). To perform charge and spin manipulation, semi-rigid coaxial lines with 20 GHz bandwidth are routed to G1, G2 and G3 using on-PCB bias tees. Microwave frequency signals are supplied by a vector signal generator (R&S SMW200A) with IQ modulating signals originating from two channels of an arbitrary waveform generator (AWG) Tektronix AWG5200. Other channels of the AWG are used to generate the pulse sequences. The homodyne readout of the resonator connected to the drain electrode is performed with a Zurich Instrument UHFLI lock-in with an excitation power of − 105 dBm at the PCB stage. The reflected signal from the resonator is amplified at 4 K with an ultra-low noise cryogenic amplifier LNF-LNC0.2-3A.