Sweet-spot operation of a germanium hole spin qubit with highly anisotropic noise sensitivity

Spin qubits defined by valence band hole states are attractive for quantum information processing due to their inherent coupling to electric fields, enabling fast and scalable qubit control. Heavy holes in germanium are particularly promising, with recent demonstrations of fast and high-fidelity qubit operations. However, the mechanisms and anisotropies that underlie qubit driving and decoherence remain mostly unclear. Here we report the highly anisotropic heavy-hole g-tensor and its dependence on electric fields, revealing how qubit driving and decoherence originate from electric modulations of the g-tensor. Furthermore, we confirm the predicted Ising-type hyperfine interaction and show that qubit coherence is ultimately limited by 1/f charge noise, where f is the frequency. Finally, operating the qubit at low magnetic field, we measure a dephasing time of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${T}_{2}^{* }$$\end{document}T2* = 17.6 μs, maintaining single-qubit gate fidelities well above 99% even at elevated temperatures of T > 1 K. This understanding of qubit driving and decoherence mechanisms is key towards realizing scalable and highly coherent hole qubit arrays.


INTRODUCTION
The development of a fault-tolerant quantum computer 1 able to solve relevant problems 2 requires the integration of many highly coherent qubits.Spin qubits based on quantum dots 3 hold excellent promise for scaling towards large-scale quantum processors, due to their small footprint and long coherence.Recently, great progress has been made, with demonstrations of single-qubit gate 4,5 , two-qubit gate [6][7][8] , and readout 9 fidelities well above the fault-tolerant threshold.Furthermore, rudimentary quantum algorithms and simulations have been executed on multi-qubit arrays 10,11 including minimal error correction schemes 12,13 and compatibility with semiconductor manufacturing has been demonstrated 14 .
In particular, hole qubits in strained germanium quantum wells have gained a strong interest over recent years 15 , with demonstrations of single 16,17 and multi-qubit 10,12 operations and first steps towards the operation of large, multiplexed qubit registers 18,19 .This surge of interest is rooted in the combination of favourable properties that holes in germanium possess: a low-effective mass that relaxes the constraints on device fabrication 20 and a lowdisorder quantum well that provides a low-noise qubit environment 21 and enables excellent quantum dot control 22 , without the complication of low-energy valley states that have hindered progress for electrons in silicon.
The spin properties of valence band holes can be highly anisotropic [23][24][25][26][27] , which results in a field-dependent coupling to the two dominant sources of decoherence in spin qubits: nuclear spin fluctuations 4 and charge noise 28 .These anisotropies present both opportunities and challenges for building a scalable qubit platform.For example, the anisotropic heavy hole g-tensor can amplify small variations in quantum dot confinement, leading to site-dependent qubit properties 10,26 and increasing requirements on material uniformity.However, when well controlled, the anisotropy enables operational sweet spots where qubit control is maximized while decoherence is minimized 24,[29][30][31] , overcoming the general trade-off between qubit control and coherence.Theoretical considerations predict the operating point of such sweet spots to depend on the exact material and device parameters like strain 32 or device geometry 33 , but an experimental demonstration of the heavy hole anisotropies and their implications on qubit performance is lacking.

GERMANIUM TWO-QUBIT DEVICE
Here, we unveil the mechanisms that enable qubit driving and mediate decoherence in germanium hole qubits.We fully characterize the heavy-hole g-tensors of a twoqubit system and their sensitivity to electric fields.A comparison with the dependence of qubit coherence and Rabi frequency on the orientation and magnitude of the external magnetic field demonstrates that both qubit driving and charge-noise induced qubit decoherence are explained by the distortion of the g-tensor through electric fields.Furthermore, we confirm the predicted Ising character of the hyperfine interaction between the heavy-hole spin and the 73 Ge nuclear spin bath, leading to a strong suppression of hyperfine coupling when the magnetic field is oriented in the plane of the qubit g-tensor.This understanding enables us to find an optimal operation regime that yields an improvement in spin coherence times of more than an order of magnitude compared to the state-of-the-art.
We define a two-qubit system based on confined hole spins in a strained Ge/SiGe heterostructure quantum well 34 .The spins are confined in gate-defined quantum dots, formed respectively underneath plunger gates P1 and P2, with an additional gate B12 controlling the interdot coupling (see Fig. 1a).Additionally, we form a large quantum dot underneath gate SP to act as a charge sensor of which the tunnel rates to in-diffused PtSiGe ohmic leads can be controlled by gates SB1 and SB2.We measure the charge sensor conductance to detect nearby tunnelling events.Using two virtual gates P1 and P2 (see Methods), we measure the charge stability diagram as plotted in Fig. 1b.Well-defined charge occupancy regions can be observed, with the top right corner corresponding to both dots being fully depleted.We operate the device in the (1,1) charge region and perform latched Pauli spin blockade readout 9,10,35 , as shown in Fig. 1c, where a distinct difference in the differential charge sensor current can be observed for the preparation of a |↓↓⟩ and |↓↑⟩ state.

HEAVY HOLE g-TENSOR
The confinement of holes in a two-dimensional (2D) strained Ge quantum well splits heavy hole (HH) and light hole (LH) bands, with the former defining the ground state 20 .As the electrical confinement in the plane of the quantum well is notably weaker than the confinement in growth direction, the hole wave function is expected to contain mostly HH components 20,30 .The degree of HH-LH mixing will affect the observed hole g-tensor, which is predicted to be highly anisotropic for the heavy hole states and much more isotropic for the light hole states 36 .The general g-tensor can be modelled as a rotated diagonal 3x3 matrix g , where ϕ, θ, and ζ are Euler angles corresponding to successive intrinsic rotations around axes zyz and g x ′ , g y ′ , and g z ′ define the effective g-factors along the principle axes x ′ y ′ z ′ of the g-tensor (Fig. 2d).We reconstruct the g-tensor for both Q1 and Q2 by measuring the effective g-factor g * = hf Q /(µ B B), with h the Planck constant, f Q = |f Q | the qubit Larmor frequency, and µ B the Bohr magneton, for different magnetic field orientations B = B b.The measured data and fit of g ↔ are plotted in Fig. 2a-c,e-g for cuts through the xy, xz, and yz planes respectively.The observed g-tensor is found to be extremely anisotropic for both qubits, with g z ′ ≈ 30g y ′ ≈ 180g x ′ , and the largest principle axis almost aligned to the sample growth direction z.The g-tensors of the two qubits are remarkably identical, with their principle axes lengths differing by < 10%, the azimuth rotations ϕ and ζ by < 15°, and the elevation θ by less than 2°(see Fig. 2h).
Due to the strong anisotropy, the qubit quantisation axis hf Q = µ B g ↔ B is not necessarily aligned with the applied magnetic field B as is the case in isotropic systems.In particular, any small deviation of B from the plane spanned by the two minor principal axes x ′ y ′ of the g-tensor will result in a strong rotation of the qubit quantisation axis towards ±ẑ (see Supp.Fig. 2b,c).Therefore, small variations between qubit g-tensors can still lead to a sizeable difference between the qubit quantisation axes, in particular for in-plane magnetic field orientations.Because Pauli spin blockade readout measures the relative spin projection of two qubits, we observe readout to be affected for these magnetic field orientations and being completely suppressed when the angle between f Q1 and f Q2 equals π/2 (Supp.Fig. 2f,g).While the anisotropy between g z ′ and .147(2)°2.541(8)°9 .0(4)°-5.5(13)°0 . g x ′ ,y ′ is expected from the quantum well confinement, the in-plane anisotropy points to a non-circular confinement of the quantum dot.This can be explained by the device layout, as the finite potential on the interdot barrier breaks the symmetry of the individual quantum dots and gives rise to an non-circular confinement potential.We suspect that the small tilt of the g-tensor with respect to the sample axes is caused by localized strain gradients as imposed by the nanostructured gate electrodes 32,37 .

CHARGE NOISE
The connection between confinement potential of the hole and LH-HH mixing gives rise to a sensitivity of the g-tensor to local electric fields 31,36 .An electric field modulation will thus induce a variation of the g-tensor δ g ↔ , which leads to a modulation of the Larmor vector hδf Q = µ B δ g ↔ • B. These modulations can be separated into changes parallel (longitudinal) or perpendicular (transverse) to the qubit quantisation axis.The former will change the qubit energy splitting and provide a channel for dephasing due to e.g.charge noise 24 , while the latter enables driving the qubit through g-tensor magnetic resonance (g-TMR) 24,38,39 .The dependency of g ↔ on electric field fluctuations will depend on the direction of the electric field, which we study by considering potential modulations on differently oriented gates.
We first focus on the longitudinal electric field sensitivity ∂f Q2 /∂V P2 of Q2 with respect to its plunger gate, as we expect charge noise to mostly originate from the interfaces and oxides directly above the qubit.We determine the change in qubit frequency from an acquired phase when applying a small voltage pulse δV i to different gate electrodes i during a Hahn echo measurement 24,39 (see Fig. 3a-d and Methods).Fig. 3e shows ∂f Q2 /∂V P2 for different magnetic field orientations, for f Q2 = 1.36 (7) GHz.We observe the qubit energy splitting to be most sensitive to electric field fluctuations when B is in the plane x ′ y ′ spanned by the g-tensor minor principal axes (indicated by the red dashed line) with ∂f Q2 /∂V P2 > 2 GHz/V.
If qubit decoherence is limited by fluctuations of the gtensor caused by charge noise, we expect the qubit frequency fluctuations δf Q to linearly increase with B and to strongly depend on the orientation of B as governed by the corresponding longitudinal electric field sensitivity.To this end, we perform a Hahn echo experiment and extract the echo coherence times T H 2 by fitting the data to an envelope exponential decay disregarding nuclear spin effects (see Methods).We plot T H 2 as a function of the qubit frequency (obtained by varying B) in Fig. 3f for different orientations of B indicated by the coloured markers in Fig. 3e.For large enough B, we observe a power law dependence of T H 2 ∝ f −1 Q , consistent with a 1/f noise spectrum acting on the qubit 24,40 (see Methods).We note that for small B, the finite spread of the precession frequencies of the nuclear spin ensemble limits qubit coherence, resulting in a sharp decrease 41 of the extracted T H 2 .Next, we corre- FFT(P(t Z ))  II).Solid lines correspond to T H 2 as extracted from a pure decay, while dashed lines correspond to T H 2 as extracted from the envelope of the nuclear spin induced collapse-and-revival. Data indicated by opaque markers are used to fit the power law dependence of T H 2 .g, Expected T H 2,f Q2 =1 GHz as extracted from a power law fit to the opaque data markers in f as a function of the gate voltage sensitivity (∂fQ2/∂VP2)/fQ2 from e. Coloured markers correspond to the different magnetic field orientations as indicated in e. Solid black line is a fit of T H 2 = ax β to the data, yielding an exponent of β = −1.04(8).
late the observed charge-noise limited echo coherence time to the electric field sensitivity of g ↔ for different orientations B. The extracted charge-noise limited T H 2 obtained at different (θ B , ϕ B ) and extrapolated to f Q2 = 1 GHz is plotted as a function of the measured relative voltage sensitivity (∂f Q2 /∂V P2 )/f Q2 (Fig. 3g).The good fit to a power law with exponent −1 confirms the dominant source of charge noise originates from the oxide interfaces directly above the qubit (see Supp.Fig. 4 for the noise sensitivity of other gates).
To get a full understanding of the mechanism underlying the electric modulation of the g-tensor, we reconstruct ∂ g ↔ /∂V i for Q2 as a function of the voltage applied to plunger gate P2, and two neighbouring barrier gates B2 and B12, oriented at a 90°angle to each other (Fig. 4d).We measure (∂f Q2 /∂V i )/f Q2 for selected magnetic field orientations, that together with the previously extracted g ↔ allows fitting ∂ g ↔ /∂V i (see Methods).All measurements are performed at constant f Q2 = 225 MHz and we show the measured relative electric potential sensitivity and corre-sponding fits in Fig. 4a-c.The extracted fit parameters are detailed in Supp.Table I.To illustrate what happens to the g-tensor as the gates are pulsed, we sketch the crosssections of g ↔ and g ↔ + δ g ↔ i (δV i ) in the xz, yz, and xy planes of the magnet frame for δV i = 100 mV in Fig. 4h-j.We observe that the plunger gate directly above the qubit mostly scales the g-tensor principle axes ('breathing'), while the neighbouring barrier gates also induce a rotation of the g-tensor (Supp.Table I).A true sweet spot to noise originating near gate i exists when ∂f Q /∂V i = 0. We only find such a zero crossing for potentials applied to side gate B2, as visible in Fig. 4b (see Supp.Fig. 8 for the full θ B ϕ Bprojections).For voltage fluctuations applied to gates P2 and B12, we find that an improvement in the electric field sensitivity is possible, but no true sweet spot exists for any (θ B , ϕ B ).These effects are dominated by the dynamic tilting of g ↔ , which we believe to be caused by hole wave function moving in a local strain gradient 32,37,42 , not taken into account by previous models 30 .
While the longitudinal component of the g-tensor mod- ulation leads to decoherence, the transverse part enables an electric drive of the qubit through g-TMR.Therefore, our reconstruction of ∂ g ↔ /∂V i allows us to compare the expected Rabi frequency from g-TMR with the observed Rabi frequency.We measure the angular dependence of the Rabi frequency of the qubit, for a resonant electric drive with amplitude V i applied to either gate P2, B2, or B12 and extract (∂f Rabi /∂V i )/f Q2 .The results, shown in Fig. 4eg, reveal a striking agreement between the measured Rabi frequency and the expected drive due to the g-TMR (see Methods for details).The agreement between the data and the projection of ∂ g ↔ /∂V i , both in absolute size and magnetic field dependence, confirms that the main driving mechanism of planar germanium hole qubits is in fact g-TMR.

HYPERFINE INTERACTION
Our qubits are defined in a natural germanium quantum well, where with a concentration of 7.7 %, 73 Ge is the only isotope with non-zero nuclear spin.As a result, the hole wave function overlaps with ∼10 6 nuclear spins (see Methods), leading to a fluctuating Overhauser field acting on the hole spin.One can separate the contributions of the Overhauser field into longitudinal and transverse components with respect to the quantisation axis of the nuclear spins 41 .While temporal fluctuations of both components can lead to qubit dephasing, longitudinal field fluctuations are mainly caused by the quasi-static dipole-dipole interaction between nuclear spins 43,44 and can easily be echoed out.However, the transverse part contains a spectral component at the Larmor frequency of the nuclear spins, that leads to a collapse-and-revival of coherence when performing spin echo experiments, as predicted in Refs. 43,45and observed in GaAs 41 and germanium 46 .
The hyperfine interaction between heavy hole states and S 0,hf = a cos 2 (θ ) nuclear spins is expected to be highly anisotropic 25 , unlike the isotropic contact hyperfine interaction observed for conduction band electrons.In fact, for the 73 Ge isotope, the Ising term (out-of-plane, ∝ s z I z ) is numerically estimated to be ∼50 times larger than in-plane (∝ s x I x , s y I y ) components 47 .As a result, hyperfine interaction between the heavy hole and the surrounding nuclear spin bath is expected to be negligible for an in-plane magnetic field 25,47 .
To study the hyperfine anisotropy for planar germanium qubits, we perform a Carr-Purcell-Meiboom-Gill (CPMG) experiment, which constitutes an effective band pass filter for the noise acting on the qubit with a frequency f = 1/τ (Fig. 5a).We apply CPMG sequences with N = 1, 2, 4, and 8 decoupling pulses to Q2 and measure the spin state as a function of the free evolution time τ between the Y π -pulses, as shown in Fig. 5c,d for N = 1 and N = 4 (data for N = 2, N = 8 in Supp.Fig. 10c,d).We observe the expected collapse-and-revival of the coherence and find f revival = γ|B| with γ = 1.485(2)MHz/T (Fig. 5b), in good agreement with the gyromagnetic ratio of the 73 Ge nuclear spin γ Ge-73 = 1.48 MHz/T, confirming its origin.
Following Refs. 40,48, we fit the data using the formalism developed to describe decoherence of dynamically decoupled qubits suffering from dephasing noise with a given noise spectrum.We assume a noise spectrum S fq acting on the qubit, consisting of a 1/f part caused by charge noise, as well as a sharp spectral component at the precession frequency of the 73 Ge nuclear spins (see Methods).
We extract the strength of the nuclear noise S 0,hf and plot this as a function of the elevation of the Larmor vector (Fig. 5e).We find that the data closely follows a relation S 0,hf ∝ cos 2 θ f Q2 , providing strong experimental evidence of the predicted Ising coupling 25,47,49 .As a result, there exists a sweet plane approximately spanned by the x ′ y ′ axes of the g-tensor, where the qubit is mostly insensitive to nuclear spin noise.The finite width of the hyperfine distribution of σ Ge-73 = 9 − 16 kHz, results in a loss of qubit coherence at small B, as seen in Fig. 3f.This line width is several orders of magnitude larger than expected for a single 73 Ge spin 50 , but is in good agreement with values previously observed in Ge 46 and could be caused either by interactions between the nuclei or by the quadrupolar splitting present in the 73 Ge isotope.
Finally, assuming all charge noise to originate near P2, we fit the extracted S 0,E (Fig. 5f) to ∂f Q2 /∂V P2 as measured in Fig. 4 and find an effective electric noise power spectral density of S V = 610 µV 2 /Hz at 1 Hz, corresponding to an effective voltage noise of 25 µV / √ Hz on P2.Using the estimated plunger gate lever arm α P = 7.4% (Supp.Fig. 7) we extract a charge noise level of 1.9 µeV / √ Hz, in good agreement with charge noise measurements on similar devices 21 .

SWEET-SPOT OPERATION
The detailed understanding of the hole qubit coherence for different magnetic field orientations, allows to select an optimal operation regime.For any magnetic field orientation away from the hyperfine sweet plane, nuclear spin noise limits qubit coherence in natural germanium samples.However, the slight but significant tilt between the two qubit g-tensors limits this further to a single spot where the two circles intersect: ϕ B = 97.5 °and θ B = 89.7 °for this device (see Supp.Fig. 10).The existence of such common hyperfine sweet spots is not guaranteed for larger qubit systems when the individual qubit g-tensors slightly differ.Furthermore, we observe that this hyperfine sweet plane coincides with the hot spots for charge-induced decoherence (Supp.Fig. 8), preventing full employment of charge noise sweet spots.In fact, we estimate charge-noise limited coherence times and quality factors to be improved by about an order of magnitude for the optimal magnetic field orientation.This showcases the need for isotopically purified materials, despite the Ising type hyperfine interaction of the heavy hole.
For our device, we aim to optimize the coherence of Q2 by lowering the magnetic field strength and operate along the hyperfine sweet plane of Q2, with ϕ B = 0°to strike a balance between low charge noise sensitivity and high operation speed.We first assess the free induction decay coherence time by performing a Ramsey experiment (Fig. 6a).We set B = 20 mT, such that f Q2 ≈ 21 MHz and f Rabi = 1 MHz and find T * 2 = 17.6 µs, which is about an order of magnitude larger than shown previously for germanium hole qubits 46 .We can further extend the coherence by using dynamical decoupling and find coherence times beyond 1 ms (Fig. 6b).Operation at low magnetic field also has implications on the speed of single qubit operations, as these are expected and observed to scale with B (Eq. 13 in Methods).Single qubit gate performance is ultimately governed by the ratio of the operation time and the coherence time, and should thus in principle be preserved even at low magnetic fields.To test this, we perform randomized benchmarking, with a Clifford group based on X π and X π/2 pulses and virtual Z updates (see Supp.Table III).We find an optimal average single qubit gate fidelity (with 0.875 physical gates per Clifford) of 99.94 % at B = 12 mT (Fig. 6c), well above 99 %.Furthermore, we find that the fidelity remains significantly higher than 99 % when operating our qubits at an elevated temperature of T = 1.1 K, where more cooling power is available (Fig. 6d).Lowering the qubit frequency thus opens a path to increase qubit coherence, while maintaining high single-qubit gate performance.This can provide a potential avenue to improve two-qubit gate performance, which has typically been limited by the comparatively short coherence time of the germanium hole qubit 10,12 .

CONCLUSIONS
In summary, we report on a fully electrically controlled two-qubit system defined by single hole spins in a strained germanium quantum well.The hole g-tensor of both qubits is characterized, revealing a strong anisotropy with respect to the heterostructure growth direction.The two qubit gtensors are remarkably similar and vary by less than 10 %, indicative of a high degree of uniformity of the electrostatic confinement.However, the small tilt (δθ ≈ 1 °) combined with the large anisotropy of g ↔ leads to measurable effects, in particular for magnetic field orientations in proximity to the g-tensor minor principle axes.The slight tilt of the gtensor is likely the result of local strain gradients and could thus be controlled through material and gate stack optimization, or by modifying the LH-HH mixing, defined by material stoichiometry 51 and quantum dot confinement 29 .
The g-tensor anisotropy is also reflected in the qubit sensitivity to electric field fluctuations.We find that g ↔ breathes and tilts under electric field fluctuations, leading to charge-noise induced decoherence, but also enabling qubit control through g-TMR, both strongly anisotropic in strength with respect to the magnetic field orientation.Furthermore, also the hyperfine interaction between the qubit and the 73 Ge nuclear spin bath is extremely anisotropic and only suppressed when the qubit quantisation axis aligns with the quantum well plane.As a result, the hyperfine interaction is detrimental to qubit coherence for any B ∦ x ′ y ′ .When the nuclear spin noise can be mitigated, we find qubit coherence to be limited by charge noise with a 1/f power spectrum, such that coherence times are inversely correlated to the qubit energy splitting and its electric field sensitivity.The hyperfine interaction hinders leveraging of the electric field sensitivity sweet spots that would enable a significant further improvement to qubit coherence, underpinning the need for isotopic purification of the germanium quantum well 52 .Finally, we find that qubit coherence can be substantially increased by operating in the low-field regime, while maintaining high-fidelity single qubit control with a gate fidelity well above the fault tolerant threshold, even at operation temperatures above 1 K.This understanding of the dominant decoherence mechanisms and sweet spots for hole spins is key for the future design and operation of large-scale, high-fidelity spin qubit arrays.
= 100 mT ϕ B = 97.5°Figure 6. Coherence figures at low magnetic field in the hyperfine sweet spot.a, Free induction decay coherence as measured through a Ramsey experiment.The data constitute of an average of 10 traces, for a total integration time of approximately 5 minutes (12-hour dataset Supp.Fig. 11) and we find a coherence time of T * 2 = 17.6 µs.b, CPMG dynamical decoupling coherence, as measured for a sequence with 250 refocusing pulses.We find a coherence time of T DD 2 = 1.3 ms.c, Randomized benchmarking of the performance of Q2.The solid line is a fit of the data to P = a exp(−(1 − 2Fc)NC), from which we extract a single qubit gate fidelity of Fg = 99.94%.The reduced visibility for larger NC is caused by the readout being affected by the large number of pulses applied to the gate, but does not affect the extracted fidelity (see Methods), as indicated by the dashed line where we fix a = 1.d, Randomized benchmarking at a fridge temperature of T = 1.1 K for Q2.We now operate in the joint Q1-Q2 hyperfine sweet spot at ϕ B = 97.5 °.We extract a single qubit gate fidelity of Fg = 99.7 %.

Device fabrication
The quantum dot device is fabricated on a Ge/SiGe heterostructure consisting of a 20-nm-thick quantum well buried 48 nm below the wafer surface, grown in an industrial reduced-pressure chemical vapour deposition reactor 34 .The virtual substrate consists of a strain-relaxed germanium layer on a silicon wafer and multiple layers with increasing silicon content to reach the Si 0.2 Ge 0.8 stoichiometry used for the quantum well barriers.Ohmic contacts to the quantum well are defined by in-diffusion of Pt at a temperature of 300°C.We note that in the device used for this work, the Pt-silicide did not diffuse in deep enough to reach the quantum well, resulting in a larger contact resistance (∼MΩ).Electrostatic gates are defined using electron beam lithography and lift-off of Ti/Pd (20 nm), separated by thin (7 nm) layers of atomic layer deposited SiO 2 .

Experimental setup
All measurements are performed in a Bluefors LD400 dilution refrigerator with a base temperature of T mc = 10 mK.The sample is mounted on a QDevil QBoard circuit board, and static biases are applied to the gates using a QDevil QDAC through dc looms filtered using a QDevil QFilter at the millikelvin stage of our fridge.In addition, all plunger and barrier gates are also connected to coaxial lines through on-PCB bias-tees.All rf lines are attenuated by 10 dB at the 4K stage and an additional 3 dB at the still.We use Tektronix AWG5204 arbitrary waveform generators (AWGs) to deliver fast voltage excitation pulses to the quantum dot gates.Furthermore, we use the AWGs to drive the vector input of a Rohde & Schwarz SGS100A source to generate microwave control signals when f Q > 500 MHz.For experiments when f Q < 500 MHz, we directly synthesize the qubit drive pulses using the AWG.Unfortunately, the coaxial line connected to gate P1 was defective at the time of the experiments.To enable fast pulsing throughout the charge stability diagram of the double quantum dot, we applied pulses to the coaxial line connected to RB1 the reservoir side gate of Q1 (see Fig. 1) instead and account for the difference in dot-gate capacitance between P1 and RB1.The independent control over the dc voltage on RB1 and P1 still allows to select a reservoir tunnel rate suitable for the experiments.
The qubits are read out using a charge sensor defined in the lower channel of the four quantum dot device.We tune the device to form a single quantum dot underneath the central gate SP, with the tunnel rates being controlled by SB1 and SB2 as defined in Fig. 1 of the main text.We measure the sensor conductance using a pair of Basel Precision Instruments (BasPI) SP983c IV-converters with a gain of 10 6 and a low-pass output filter with a cut-off frequency of 30 kHz and applying a source-drain bias excitation of V SD = 300 − 800 µV.We directly extract the differential current using a BasPI SP1004 differential amplifier with a gain of 10 3 and record the signal using an Alazar ATS9440 digitizer card.
An external magnetic field is applied through an American Magnetics three-axis magnet with a maximum field of 1/1/6 Tesla in the xyz direction and a high-stability option on all coils.We note that due to an offset z = 2.78 cm of the sample with respect to the xy coil centres, a correction of −11.2% is applied to B x and B y as following from a simulation of the magnet coil fields.As the sample is correctly centred with respect to the z solenoid, no off-diagonal components of the applied magnetic field are present (i.e.B x−coil ∥ x, B y−coil ∥ y, and B z−coil ∥ z).The correctly observed gyromagnetic ratio of the 73 Ge nuclear spin confirms the accuracy of this correction.Small common rotations of the Q1 and Q2 g-tensor rotations may occur due to imperfect planar mounting of the sample.Finally, we note that our magnet coils typically show a few mT of hysteresis, which becomes significant at very low fields.To ensure operation in a hyperfine sweet spot, we sweep θ B before every measurement in Fig. 6 and locate the sweet plane by minimizing the qubit frequency as a function of θ B .

Virtual gate matrices
To compensate for the cross capacitance between the different electrostatic gates and the quantum dots, we define a set of virtual gates 51 : with G i the real gate voltage, and G i the virtual gate voltage, which leaves the chemical potential of the other quan-tum dots unchanged.Furthermore, we define a second pair of axes detuning ϵ and on-site energy U, as illustrated in Fig. 1b of the main text:

Pauli spin blockade readout
To overcome rapid spin relaxation as mediated by the spin-orbit interaction 52 , we make use of charge latching, where we tune the tunnel rates between each dot and its respective reservoir to be asymmetric t Q2 ≪ t Q1 .By pulsing across the extended (1,1)-(0,1) charge transition line, we can latch the blocking (1,1) states into a (0,1) charge state 9,10 , with a characteristic decay time to the (0,2) ground state governed by t Q2 .Furthermore, the spin-orbit interaction introduces a coupling between the |T (1, 1)⟩ and |S(0, 2)⟩ states, resulting in the presence of an anticrossing between the |↓↓⟩ and the |S(0, 2)⟩ states.As a result, depending on the sweep rate across the interdot transition line, as well as the orientation of the external magnetic field B, we observe either parity or single-state readout 10,53 .We typically operate the device in single-state readout by sweeping fast across the anti-crossing, unless this was prohibited due to the finite bandwidth of our setup with respect to the different tunnel rates.

Fitting procedure of the g-tensor
The g-tensor of the device can be described as a rotated diagonal matrix: where Euler angles ϕ, θ, and ζ define the successive intrinsic rotations around the zyz axes.The rotation matrix R is thus defined as: The g-tensor can thus be reconstructed by measuring the qubit energy splitting hf Q for different orientations of the magnetic field B. We measure f Q for various magnetic field orientations (θ B , ϕ B ) and fit the data to: using g ↔ as defined in Eqs.1-2 and g x ′ , g y ′ , g z ′ , ϕ, θ, and ζ as fitting parameters.The data used for the fitting include but are not limited to the data presented in Fig. 2 of the main text.All magnetic field orientations at which f Q is measured are shown in Supp.Fig. 1c.These field orientations (θ B , ϕ B ) are selected to enable a reliable fit of the g-tensor, with the error on the different parameters indicated in Fig. 2h of the main text.
Fitting procedure of the charge-noise limited coherence We measure the qubit coherence by extracting the Hahn echo coherence time, which is insensitive to quasi-static noise and experimental parameters such as the integration time.We measure the normalized charge sensor current as a function of the total free evolution time 2τ and observe two different regimes (see Supp.Fig. 3).In the first (Supp.Fig. 3a), the echo decay follows an exponential decay and we fit the data to I SD = exp −(2τ /T H 2 ) α , with the exponent α left free as a fitting parameter (Supp.Fig. 3a).However, for magnetic field orientations where the echo decay is dominated by the nuclear spin induced decoherence (B ∦ x ′ y ′ ), we extract the envelope coherence T H 2 by fitting the envelope of the nuclear spin induced collapse-and-revival (Supp.Fig. 3a) 41 to , with a 0 and α free fitting parameters and f Ge-73 = γ Ge-73 B, as discussed further in the main text.
The exponent of the dependence of the Hahn echo coherence time on both ∂f Q /∂V i /f Q and f Q (Fig. 3f,g of the main text), is related to the colour of the electric noise spectrum.Assuming charge noise with a power law noise spectrum S ∝ f α acting on a qubit and following the filter formalism from Refs. 24,40, we find: Therefore, both the dependence of T H 2 on the qubit frequency (by varying B, Fig. 3f) and on the electric field sensitivity (by varying θ B and ϕ B , Fig. 3g) should obey a power law with exponent β = 2 α−1 .From this we can derive the noise exponent α To obtain the expected charge-noise limited T H 2 at f Q2 = 1 GHz, we fit a power law T H 2 = T H 2 [1 GHz] • 1 GHz/f Q2 to the data in Fig. 3f where B > B hyperfine (opaque markers).Here, B hyperfine indicates the magnetic field strength below which the finite spread of the nuclear spin precession frequencies limits qubit coherence 41 .
Because of the limited maximum field strength we can apply along the x and y axis B max,x = B max,y = 1 T, the electric field sensitivity for the pink data point is obtained at a lower qubit frequency f Q2 = 785 MHz and extrapolated to f Q2 = 1.36 GHz.

Fitting procedure of the hyperfine noise
We follow the method presented in Refs. 40,48,54and assume a noise spectrum acting on the qubit consisting of a 1/f noise spectrum caused by a large number of charge fluctuators and a Gaussian line caused by the hyperfine interaction with the precession of the 73 Ge nuclear spins: Here, S 0,hf (B) defines the effective strength of the nuclear spin noise acting on the qubit, which can be related to the hyperfine coupling constants as detailed below.Furthermore, γ Ge-73 = 1.48 MHz/T is the 73 Ge gyromagnetic ratio and σ Ge-73 represents the finite spread of the 73 Ge precession frequencies.The charge noise acting on the qubit is most likely originating from charge traps in the interfaces and oxides directly above the qubit, so we model its coupling as coming from the qubit plunger gate, in agreement with what we find in Fig. 3 of the main text.S 0,V is the effective voltage noise power spectral density and ∂f Q2 ∂V P2 (B) is the sensitivity of the qubit frequency to electric potential fluctuations from the plunger gate P2.The qubit will undergo dephasing as a result of the energy splitting noise, which will lead to a decay as defined by: with P the measured spin-up probability and The unitless filter function F N for the CPMG experiment is defined as follows 40 : F N (ω, τ ) = 8 sin 4 (ωτ /4) sin 2 (N ωτ /2) cos 2 (ωτ /2) , N is even 8 sin 4 (ωτ /4) cos 2 (N ωτ /2) cos 2 (ωτ /2) , N is odd (8)   As both the strength of the nuclear spin noise and charge noise are expected to depend on B, we fit the data for each θ B independently, fixing γ Ge-73 = 1.48 MHz/T and keeping σ Ge-73 , S 0,V , and S 0,hf as fit parameters.
We note that we find that σ Ge-73 is independent of θ B within the experimental range, with an average σ Ge-73 = 9 kHz (see Supp.Fig. 10e,f).The finite width of the hyperfine line is mostly reflected in the loss of the coherence for low magnetic fields, when f Ge-73 ≈ σ Ge-73 .This can be observed in the data presented in Fig. 3, as well as when performing the CPMG experiment as a function of the magnetic field strength (see Supp.Fig. 9).However, we observe this line width to be dependent on the azimuth orientation of the external magnetic field ϕ B (see Supp.Fig. 10), potentially indicative of a quadrupolar origin, which would depend on strain and electric fields and thus be magnetic field orientation dependent.
Increasing the number of refocusing pulses also sharpens the effective band pass filter of the CPMG 40, 55 , thus enhancing the sensitivity to the nuclear spin precession frequency.As a result, a higher accuracy of θ B is required to align exactly to the hyperfine sweet spot and avoid loss of coherence due to hyperfine interaction with the 73 Ge nuclear spins.This is illustrated in Supp.Fig. 12, where we measure the CPMG decay as a function of the number of refocusing pulses N .

Estimation of the hyperfine coupling constant
The reconstruction of the hyperfine noise spectrum allows for an estimation of the hyperfine coupling constants for a heavy hole in germanium.From the fit to the data in Fig. 5 in the main text, we have S 0,hf = 2.52(4)kHz 2 /Hz for an out-of-plane field and σ Ge-73 = 9.9 (11) kHz.This equates to an integrated detuning noise of: Assuming a Gaussian noise distribution, this corresponds to an expected phase coherence time 56 of T * 2 = 1/(π √ 2σ f ) = 900 ns.We can estimate the out-of-plane hyperfine coupling A ∥ using Eq.2.65 from Ref. 57 : Such that: with g Ge-73 = 0.0776 the natural abundance of the 73 Ge isotope, I = 9/2 the 73 Ge nuclear spin and N the number of nuclei the quantum dot wave function overlaps.To estimate N, we consider a cylindrical quantum dot, such that N = πr 2 w/v 0 , with r the radius and w the height of the dot, and v 0 = 2.3 • 10 −29 m 3 the atomic volume of germanium.We can estimate r from the single particle level splitting ∆E ≈ 1.2 meV as can be obtained from the extend of the PSB readout window, and find r ≈ 35 nm.This is in good agreement with r ≈ 50 nm as expected from the charging energy E C ≈ 2.8 meV and the capacitance of a disk: r = e 2 /(8ϵ r E c ).Assuming r = 35 nm and w = 10 nm (half of the quantum well width), we then find N ≈ 1.7 • 10 6 .Using Eq. 11, we estimate the hyperfine coupling constant to be |A ∥ | ≈ 1.9 µeV, which is in good agreement with the theoretical prediction A ∥ = −1.1 µeV from Ref. 47 .Similarly from the extracted S 0,hf for an inplane B, we estimate an upper bound for the in-plane hyperfine coupling constant A ⊥ < 0.1µeV compatible with the predicted A ⊥ = 0.02 µeV.

Randomized benchmarking
To extract the single qubit gate fidelity, we perform randomized benchmarking of the Clifford gate set presented in Supp.Table III.For every randomization, we measure both the projection to |↑⟩ and |↓⟩ and fit the difference to avoid inaccuracies due to the offset of the charge sensor current.The measured current is normalized to the signal obtained from a separate measurement of our |↑⟩ and |↓⟩ states.We fit the data to P = a exp(−(1 − 2F C )N C ), with F C the Clifford gate fidelity and N the number of applied Clifford gates.a is an additional scaling parameter we include to account for the reduced visibility we observe when applying a large number of rf pulses.Fixing a = 1 does not significantly alter the fit as shown by the dashed line in Fig. 6.In fact, we find F g = 99.92% for T mc = 20 mK and F g = 99.7 % for T mc = 1.1 K when fixing a = 1.The primitive gate fidelity F g can be calculated by accounting for the number of physical gates per Clifford: 0.875 for this gate set.

Extraction of the g-tensor sensitivity
We measure the modulation of the qubit energy splitting δf Q as the result of a small voltage pulse δV on one of the quantum dot gates.The voltage pulse will temporarily shift the qubit resonance frequency, thus inducing an effective phase gate, controlled by the length of the pulse t Z .By incorporating this phase gate within the free evolution of a Hahn echo experiment, we can observe the phase oscillations as a function of t Z , as shown in Fig. 3c of the main text.From the frequency of these oscillations, we obtain |δf Q |.We confirm that for a small δV , |δf Q | is linear in δV , allowing us to extract the sensitivity |∂f Q /∂V i | from a single data point of δV (see Supp.Fig. 5).To exclude effects caused by the exchange interaction J between the qubits, we tune J < 1 MHz using the interdot barrier B12.Furthermore, we tune the device to the point of symmetric exchange in the (1,1) region 58,59 and apply symmetric pulses in the first and second free evolution period of the Hahn sequence, echoing out effects caused by changes of the double dot detuning.To extract the sign of ∂f Q /∂V i , we measure the qubit resonance frequency for three different gate voltage settings (see Fig. 3c) for a few selected magnetic field orientations.
Given a g-tensor g ↔ and a g-tensor sensitivity ∂ g ↔ /∂V i , ∂f Q /∂V i only depends on the magnetic field direction b and on f Q ∝ B: We extract ∂ g ↔ /∂V i by fitting Eq. 12 to the data presented in Fig. 4 of the main text, using g ↔ as extracted previously and displayed in Fig. 2h.We then calculate the expected g-TMR mediated Rabi frequency using ) with f Rabi the Rabi frequency and µ the signal attenuation for a microwave signal at a frequency of f Q .
We fit the data to the Eq. 13, with µ as the only fit parameter.We find a line attenuation of µ P2 = 2.1, µ B2 = 2.1, and µ B12 = 2.0.These value are in good agreement with the attenuation of our experimental setup at f = 225 MHz as extracted from the broadening of the charge sensor Coulomb peak (µ = 2.1 − 2.5) (see Supp.Fig. 6).Expected ratio of the transverse and longitudinal components (∂f Rabi /∂Vi)/(∂fQ2/∂VP2) as a result of a drive amplitude on gate P2 (g), B2 (h), and B12 (i) for different magnetic field orientations.We assume the noise to couple in predominantly as if it is applied to the plunger gate P2.

Figure 1 .
Figure1.A germanium hole two-qubit system.a, Schematic drawing of the three quantum dot device.We define qubits Q1 and Q2 underneath plunger gates P1 and P2 respectively, that can be read out using the nearby charge sensor (CS) defined by gates SP, SB1, and SB2.The coupling between the qubits is controlled by B12 while the coupling of Q1 (Q2) to its reservoir is controlled by RB1 (RB2).We record the response of the charge sensor by measuring the differential current flowing into and out of the source (S) and drain (D) contacts.b, Two-quantum-dot charge stability diagram as a function of two virtualized plunger gate voltages V P 1 and V P 2 .The different charge configurations are indicated by the numbers in parentheses (N1, N2).The direction of the virtual detuning ϵ and on-site energy U axes are indicated.c, Spin-to-charge conversion is performed by latched Pauli spin blockade readout.The pulses applied to the ϵ and U axes, as well as the qubit drive pulses V rf are shown in the top panels.The spins are initialized in the |↓↑⟩ state by adiabatically sweeping across the interdot transition (1→2).Next we apply either no pulse (left panel) or a Xπ pulse (right panel) to Q2 (2) and sweep (2→3) to the readout point (Vϵ,3, VU,3), which is rasterized to compose the entire map.Red lines indicate (extended) lead transition lines, while the white lines corresponds to the interdot transition lines of the quantum dot ground (solid) and excited (dashed) states.

Figure 3 .
Figure 3. Electric field sensitivity and coherence dependence on magnetic field orientation.a, Pulse sequences used to measure the voltage sensitivity of the energy splitting ∂fQ/∂Vi.A positive (negative) voltage pulse δVi of varying length tZ is applied to the test gate electrode i in the first (second) free evolution of a Hahn echo to extract |∂fQ/∂Vi|.b, Pulse sequences used to infer the sign of ∂fQ/∂Vi by assessing the shift of the qubit resonance frequency as a result of a voltage pulse δVi.c, Left: charge sensor current Isensor as a function of tZ , solid line is a fit to the data.Right: fast Fourier transform of Isensor, allowing us to extract |∂fQ/∂Vi|.d, Isensor as a function of the drive frequency fX and δVi.The shift of the resonance frequency allows us to extract the sign of ∂fQ/∂Vi.e, The qubit energy splitting sensitivity to a voltage change on the plunger gate ∂fQ2/∂VP2, as a function of different magnetic field orientations ϕ B and θ B .B is adapted to keep fQ2 constant at fQ2 = 1.36(7)GHz.Data acquisition is hindered for the white areas as a result of limited qubit readout or addressability for these magnetic field orientations.f, Hahn coherence time T H 2 as a function of the qubit frequency fQ2, for different magnetic field orientations, indicated by the coloured markers in e (exact field orientation in Supp.TableII).Solid lines correspond to T H 2 as extracted from a pure decay, while dashed lines correspond to T H 2 as extracted from the envelope of the nuclear spin induced collapse-and-revival. Data indicated by opaque markers are used to fit the power law dependence of T H 2 .g, Expected T H 2,f Q2 =1 GHz as extracted from a power law fit to the opaque data markers in f as a function of the gate voltage sensitivity (∂fQ2/∂VP2)/fQ2 from e. Coloured markers correspond to the different magnetic field orientations as indicated in e. Solid black line is a fit of T H 2 = ax β to the data, yielding an exponent of β = −1.04(8).

Figure 4 .
Figure 4. Reconstruction of ∂ g ↔ /∂V i for differently oriented electrostatic gates.a-c, Relative voltage sensitivity of the energy splitting (∂fQ2/∂Vi)/fQ2 of Q2 for a voltage excitation on gates P2 (a), B2 (b), and B12 (c).Top panels correspond to sweeps of the magnetic field elevation θ B , while bottom panels correspond to sweeps of the in-plane angle ϕ B .The solid lines correspond to projections of the ∂ g ↔ /∂Vi fitted to the data.d, Schematic illustration of the qubit layout indicating the different electrostatic gates.e-g, Relative Rabi frequency of (∂f Rabi /∂Vi)/fQ2 of Q2 for a drive voltage excitation Vi on gates P2 (e), B2 (f ), and B12 (g).Solid lines correspond to the projection of the ∂ g ↔ /∂Vi as fitted to the data in panels a-c.h-j, Cross-section of the change of the g-tensor in the xy, xz, and yz-planes of the magnet frame.Dashed lines correspond to cross-sections of g ↔ , while solid lines represent g ↔ + δ g ↔ i(0.1V ), for gates P2 (h), B2 (i), and B12 (j).

Figure 5 .
Figure 5. Collapse-and-revival of qubit coherence due to hyperfine interaction.a, Filter function of the CPMG pulse sequence, for N = 1 and 4 decoupling pulses, illustrating full suppression of noise with a characteristic frequency f = n/(2τ ), with n any integer.b, Extracted revival frequency as a function of the magnetic field strength B, full data shown in Supp.Fig. 9.We extract a gyromagnetic ratio of the 73 Ge nuclear spin of γ = 1.485(2)MHz/T.c, d, Normalized charge sensor signal for a CPMG sequence with respectively 1 (c), and 4 (d) decoupling pulses, as a function of the spacing between two subsequent decoupling pulses τ and θ B .N τ is the total evolution time.ϕ B = 97.5°andB = 133 mT.The inset displays the fit to the data from which we extract S 0,hf (θ B ) and S0,E(θ B ). e, The extracted strength of the hyperfine interaction as a function of θ f Q .The black line is a fit of the data to a cos 2 (θ f Q ), with a = 2.5 • 10 6 Hz 2 /Hz.Error bars indicate 1σ of the fit.f, The extracted strength of the 1/f noise at 1 Hz.The black line is a fit of the data to S0,E = S0,V • (∂fQ2/∂VP2(θ B )) 2 , with ∂fQ2/∂VP2(θ B ) the electric field sensitivity of the qubit frequency to the top gate voltage as extracted from Fig. 4 and S0,V = 6.1 • 10 −10 V 2 /Hz the only fit parameter.Error bars indicate 1σ.

Supplementary Figure 8 .
Longitudinal and transverse components of the fitted ∂ g ↔ /∂V i .a-c.Using the fitted ∂ g ↔ /∂Vi, as detailed in Fig. 4 of the main text and Supp.Table I , we plot the expected normalized resonance frequency fluctuation of Q2 as a result of a voltage fluctuation on gate P2 (a), B2 (b), and B12 (c) for different magnetic field orientations.Zero crossings are marked in green, to indicate the presence of a true sweet spot.d-f.Expected normalized Rabi frequency fluctuation of Q2 as a result of a drive excitation with amplitude Vi on gate P2 (d), B2 (e), and B12 (f ) for different magnetic field orientations.g-i.
xz-plane (b, e), and the yz-plane (c, f ) of the magnet frame.Dots indicate measurements of g * and the solid line corresponds to the fit of the g-tensor.Exemplary EDSR spectra used to extract g are plotted in Supp.Fig.1.d, Diagram indicating the zyz-Euler rotation angles ϕ, θ, ζ of the principle g-tensor axes g x ′ , g y ′ , and g z ′ .The approximate crystal directions are indicated in brackets.h, Overview of the three zyz Euler angles ϕ, θ, and ζ of the rotation of a g-tensor with principle components g x ′ , g y ′ , and g z ′ , for Q1 and Q2.