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

N.W. Hendrickx<EMAIL_ADDRESS>L. Massai  M. Mergenthaler  F. Schupp  S.
Paredes  IBM Research Europe - Zurich, Säumerstrasse 4, 8803 Rüschlikon,
Switzerland  S.W. Bedell  IBM Quantum, T.J. Watson Research Center, 1101
Kitchawan Road, Yorktown Heights, New York 10598, USA  G. Salis  A. Fuhrer
<EMAIL_ADDRESS>IBM Research Europe - Zurich, Säumerstrasse 4, 8803
Rüschlikon, Switzerland

(August 28, 2024)

###### Abstract

Spin qubits defined by valence band hole states comprise an attractive
candidate for quantum information processing due to their inherent coupling to
electric fields enabling fast and scalable qubit control. In particular, heavy
holes in germanium have shown great promise, with recent demonstrations of
fast and high-fidelity qubit operations. However, the mechanisms and
anisotropies that underlie qubit driving and decoherence are still mostly
unclear. Here, we report on the highly anisotropic heavy-hole $g$-tensor and
its dependence on electric fields, allowing us to relate both qubit driving
and decoherence to an electric modulation of the $g$-tensor. We also confirm
the predicted Ising-type hyperfine interaction but show that qubit coherence
is ultimately limited by $1/f$ charge noise. Finally, we operate the qubit at
low magnetic field and measure a dephasing time of
$T_{2}^{*}=9.2\leavevmode\nobreak\ \mu$s, while maintaining a single-qubit
gate fidelity of 99.94 %, that remains well above 99 % at an operation
temperature T>1 K. This understanding of qubit driving and decoherence
mechanisms are key for the design and operation of scalable and highly
coherent hole qubit arrays.

## Introduction

The development of a fault-tolerant quantum computer knill_resilient_1998
able to solve relevant problems reiher_elucidating_2017  requires the
integration of many highly coherent qubits. Spin qubits based on quantum dots
loss_quantum_1998  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
veldhorst_addressable_2014 ; lawrie_simultaneous_2021 , two-qubit gate
mills_two-qubit_2022 ; xue_quantum_2022 ; tanttu_stability_2023 , and readout
harvey-collard_high-fidelity_2018  fidelities well above the fault-tolerant
threshold. Furthermore, rudimentary quantum algorithms and simulations have
been executed on multi-qubit arrays hendrickx_four-qubit_2021 ;
philips_universal_2022  including minimal error correction schemes
van_riggelen_phase_2022 ; takeda_quantum_2022  and compatibility with
semiconductor manufacturing has been demonstrated zwerver_qubits_2022 .

In particular, hole qubits in strained germanium quantum wells have gained a
strong interest over recent years scappucci_germanium_2020 , with
demonstrations of single hendrickx_fast_2020 ; jirovec_singlet-triplet_2021
and multi-qubit hendrickx_four-qubit_2021 ; van_riggelen_phase_2022
operations and first steps towards the operation of large, multiplexed qubit
registers li_crossbar_2018 ; borsoi_shared_2022 . 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 sammak_shallow_2019  and a low-disorder quantum well that provides
a low-noise qubit environment lodari_low_2021  and enables excellent quantum
dot control lawrie_quantum_2020 , 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
winkler_spin-orbit_2003 ; piot_single_2022 ; fischer_spin_2008 ;
jirovec_dynamics_2022 ; zhang_anisotropic_2021 , which results in a field-
dependent coupling to the two dominant sources of decoherence in spin qubits:
nuclear spin fluctuations veldhorst_addressable_2014  and charge noise
yoneda_quantum-dot_2018 . 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 jirovec_dynamics_2022
; hendrickx_four-qubit_2021  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
bosco_squeezed_2021 ; wang_modelling_2022 ; wang_optimal_2021 ;
piot_single_2022 , 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
abadillo-uriel_hole_2022  or device geometry martinez_hole_2022 , but an
experimental demonstration of the heavy hole anisotropies and their
implications on qubit performance is lacking.

Figure 1: 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 $V_{\overline{\mathrm{P}_{1}}}$ and
$V_{\overline{\mathrm{P}_{2}}}$. The different charge configurations are
indicated by the numbers in parentheses ($N_{1}$, $N_{2}$). The direction of
the virtual detuning $\epsilon$ 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 $\epsilon$ and $U$ axes, as well as the qubit drive
pulses $V_{\text{rf}}$ are shown in the top panels. The spins are initialized
in the $\ket{\downarrow\uparrow}$ state by adiabatically sweeping across the
interdot transition (1$\rightarrow$2). Next we apply either no pulse (left
panel) or a $X_{\pi}$ pulse (right panel) to Q2 (2) and sweep
(2$\rightarrow$3) to the readout point ($V_{\epsilon,3}$, $V_{U,3}$), 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.

## 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 two-qubit 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 73Ge 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 bedell_low-temperature_2020 . 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 $\overline{\text{P1}}$ and $\overline{\text{P2}}$ (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
ono_current_2002 ; harvey-collard_high-fidelity_2018 ; hendrickx_four-
qubit_2021 , as shown in Fig. 1c, where a distinct difference in the
differential charge sensor current can be observed for the preparation of a
$\ket{\downarrow\downarrow}$ and $\ket{\downarrow\uparrow}$ state.

Figure 2: Measurement of the hole $\bm{g}$-tensor. a-c, e-g, Cross section of
the $g$-tensor of Q1 (a-c) and Q2 (e-g) in the $xy$-plane (a, d), $xz$-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 $\phi$,
$\theta$, $\zeta$ of the principle $g$-tensor axes $\mathbf{g}_{x^{\prime}}$,
$\mathbf{g}_{y^{\prime}}$, and $\mathbf{g}_{z^{\prime}}$. The approximate
crystal directions are indicated in brackets. h, Overview of the three $zyz$
Euler angles $\phi$, $\theta$, and $\zeta$ of the rotation of a $g$-tensor
with principle components $g_{x^{\prime}}$, $g_{y^{\prime}}$, and
$g_{z^{\prime}}$, for Q1 and Q2.

## Heavy hole $\bm{g}$-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 sammak_shallow_2019 . 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
sammak_shallow_2019 ; wang_modelling_2022 . 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 winkler_spin_2008 . The general $g$-tensor can be modelled as a
rotated diagonal 3x3 matrix $\tensor{g}=R(\phi,\leavevmode\nobreak\
\theta,\leavevmode\nobreak\
\zeta)\operatorname{diag}{(g_{x^{\prime}},g_{y^{\prime}},g_{z^{\prime}})}R^{-1}(\phi,\leavevmode\nobreak\
\theta,\leavevmode\nobreak\ \zeta)$, where $\phi$, $\theta$, and $\zeta$ are
Euler angles corresponding to successive intrinsic rotations around axes $zyz$
and $g_{x^{\prime}}$, $g_{y^{\prime}}$, and $g_{z^{\prime}}$ define the
effective $g$-factors along the principle axes
$x^{\prime}y^{\prime}z^{\prime}$ of the $g$-tensor (Fig. 2d). We reconstruct
the $g$-tensor for both Q1 and Q2 by measuring the effective $g$-factor
$g^{*}=hf_{\text{Q}}/(\mu_{\text{B}}B)$, with $h$ the Planck constant,
$f_{\text{Q}}=\absolutevalue{\mathbf{f}_{\text{Q}}}$ the qubit Larmor
frequency, and $\mu_{\text{B}}$ the Bohr magneton, for different magnetic
field orientations $\mathbf{B}=B\hat{\bm{b}}$. The measured data and fit of
$\tensor{g}$ 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 $g_{z^{\prime}}\approx
30g_{y^{\prime}}\approx 180g_{x^{\prime}}$, 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 $\phi$ and $\zeta$ by $<15\degree$, and the
elevation $\theta$ by less than $2\degree$ (see Fig. 2h).

Due to the strong anisotropy, the qubit quantisation axis
$h\mathbf{f}_{\text{Q}}=\mu_{\text{B}}\tensor{g}\mathbf{B}$ is not necessarily
aligned with the applied magnetic field $\mathbf{B}$ as is the case in
isotropic systems. In particular, any small deviation of $\mathbf{B}$ from the
plane spanned by the two minor principal axes $x^{\prime}y^{\prime}$ of the
$g$-tensor will result in a strong rotation of the qubit quantisation axis
towards $\pm\hat{z}$ (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
$\mathbf{f_{\text{Q1}}}$ and $\mathbf{f_{\text{Q2}}}$ equals $\pi/2$ (Supp.
Fig. 2f,g). While the anisotropy between $g_{z^{\prime}}$ and
$g_{x^{\prime},y^{\prime}}$ 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 abadillo-
uriel_hole_2022 ; corley-wiciak_nanoscale_2023 .

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 $\partial f_{\text{Q}}/\partial V_{i}$. A positive
(negative) voltage pulse $\delta V_{i}$ of varying length $t_{Z}$ is applied
to the test gate electrode $i$ in the first (second) free evolution of a Hahn
echo to extract $|\partial f_{\text{Q}}/\partial V_{i}|$. b, Pulse sequences
used to infer the sign of $\partial f_{\text{Q}}/\partial V_{i}$ by assessing
the shift of the qubit resonance frequency as a result of a voltage pulse
$\delta V_{i}$. c, Left: charge sensor current $I_{\text{sensor}}$ as a
function of $t_{Z}$, solid line is a fit to the data. Right: fast Fourier
transform of $I_{\text{sensor}}$, allowing us to extract $|\partial
f_{\text{Q}}/\partial V_{i}|$. d, $I_{\text{sensor}}$ as a function of the
drive frequency $f_{X}$ and $\delta V_{i}$. The shift of the resonance
frequency allows us to extract the sign of $\partial f_{\text{Q}}/\partial
V_{i}$. e, The qubit energy splitting sensitivity to a voltage change on the
plunger gate $\partial f_{\text{Q2}}/\partial V_{\text{P2}}$, as a function of
different magnetic field orientations $\phi_{\mathbf{B}}$ and
$\theta_{\mathbf{B}}$. $B$ is adapted to keep $f_{\text{Q2}}$ constant at
$f_{\text{Q2}}=1.36(7)$ 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 $T_{2}^{\text{H}}$ as a function of
the qubit frequency $f_{\text{Q2}}$, for different magnetic field
orientations, indicated by the coloured markers in e (exact field orientation
in Supp. Table 2). Solid lines correspond to $T_{2}^{\text{H}}$ as extracted
from a pure decay, while dashed lines correspond to $T_{2}^{\text{H}}$ 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
$T_{2}^{\text{H}}$. g, Expected
$T_{\mathrm{2,f_{\text{Q2}}=1\leavevmode\nobreak\ GHz}}^{H}$ as extracted from
a power law fit to the opaque data markers in f as a function of the gate
voltage sensitivity $(\partial f_{\text{Q2}}/\partial
V_{\text{P2}})/f_{\text{Q2}}$ from e. Coloured markers correspond to the
different magnetic field orientations as indicated in e. Solid black line is a
fit of $T_{2}^{H}=ax^{\beta}$ to the data, yielding an exponent of
$\beta=-1.04(8)$.

## 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
winkler_spin_2008 ; wang_optimal_2021 . An electric field modulation will thus
induce a variation of the $g$-tensor $\delta\tensor{g}$, which leads to a
modulation of the Larmor vector
$h\delta\mathbf{f}_{\text{Q}}=\mu_{\text{B}}\delta\tensor{g}\cdot\mathbf{B}$.
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 piot_single_2022 , while the latter enables driving the
qubit through $g$-tensor magnetic resonance ($g$-TMR) kato_gigahertz_2003 ;
crippa_electrical_2018 ; piot_single_2022 . The dependency of $\tensor{g}$ 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 $\partial
f_{\text{Q2}}/\partial V_{\text{P2}}$ 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 $\delta V_{i}$ to different
gate electrodes $i$ during a Hahn echo measurement crippa_electrical_2018 ;
piot_single_2022  (see Fig. 3a-d and Methods). Fig. 3e shows $\partial
f_{\text{Q2}}/\partial V_{\text{P2}}$ for different magnetic field
orientations, for $f_{\text{Q2}}=1.36(7)$ GHz. We observe the qubit energy
splitting to be most sensitive to electric field fluctuations when
$\mathbf{B}$ is in the plane $x^{\prime}y^{\prime}$ spanned by the $g$-tensor
minor principal axes (indicated by the red dashed line) with $\partial
f_{\text{Q2}}/\partial V_{\text{P2}}>2$ GHz/V.

If qubit decoherence is limited by fluctuations of the $g$-tensor caused by
charge noise, we expect the qubit frequency fluctuations $\delta f_{\text{Q}}$
to linearly increase with $B$ and to strongly depend on the orientation of
$\mathbf{B}$ as governed by the corresponding longitudinal electric field
sensitivity. To this end, we perform a Hahn echo experiment and extract the
echo coherence times $T_{2}^{\text{H}}$ by fitting the data to an envelope
exponential decay disregarding nuclear spin effects (see Methods). We plot
$T_{2}^{\text{H}}$ as a function of the qubit frequency (obtained by varying
$B$) in Fig. 3f for different orientations of $\mathbf{B}$ indicated by the
coloured markers in Fig. 3e. For large enough $B$, we observe a power law
dependence of $T_{2}^{\text{H}}\propto f_{\text{Q}}^{-1}$, consistent with a
$1/f$ noise spectrum acting on the qubit piot_single_2022 ; cywinski_how_2008
(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 bluhm_dephasing_2011  of the extracted $T_{2}^{\text{H}}$.
Next, we correlate the observed charge-noise limited echo coherence time to
the electric field sensitivity of $\tensor{g}$ for different orientations
$\mathbf{B}$. The extracted charge-noise limited $T_{2}^{\text{H}}$ obtained
at different $(\theta_{\mathbf{B}}$, $\phi_{\mathbf{B}})$ and extrapolated to
$f_{\text{Q2}}=1$ GHz is plotted as a function of the measured relative
voltage sensitivity $(\partial f_{\text{Q2}}/\partial
V_{\text{P2}})/f_{\text{Q2}}$ (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).

Figure 4: Reconstruction of $\bm{\partial\tensor{g}/\partial V_{i}}$ for
differently oriented electrostatic gates. a-c, Relative voltage sensitivity of
the energy splitting $(\partial f_{\text{Q2}}/\partial V_{i})/f_{\text{Q2}}$
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
$\theta_{\mathbf{B}}$, while bottom panels correspond to sweeps of the in-
plane angle $\phi_{\mathbf{B}}$. The solid lines correspond to projections of
the $\partial\tensor{g}/\partial V_{i}$ fitted to the data. d, Schematic
illustration of the qubit layout indicating the different electrostatic gates.
e-g, Relative Rabi frequency of $(\partial f_{\text{Rabi}}/\partial
V_{i})/f_{\text{Q2}}$ of Q2 for a drive voltage excitation $V_{i}$ on gates P2
(e), B2 (f), and B12 (g). Solid lines correspond to the projection of the
$\partial\tensor{g}/\partial V_{i}$ 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
$\tensor{g}$, while solid lines represent
$\tensor{g}+\delta\tensor{g}_{i}(0.1V)$, for gates P2 (h), B2 (i), and B12
(j).

To get a full understanding of the mechanism underlying the electric
modulation of the $g$-tensor, we reconstruct $\partial\tensor{g}/\partial
V_{i}$ for Q2 as a function of the voltage applied to plunger gate P2, and two
neighbouring barrier gates B2 and B12, oriented at a $90\degree$ angle to each
other (Fig. 4d). We measure $(\partial f_{\text{Q2}}/\partial
V_{i})/f_{\text{Q2}}$ for selected magnetic field orientations, that together
with the previously extracted $\tensor{g}$ allows fitting
$\partial\tensor{g}/\partial V_{i}$ (see Methods). All measurements are
performed at constant $f_{\text{Q2}}=225$ MHz and we show the measured
relative electric potential sensitivity and corresponding fits in Fig. 4a-c.
The extracted fit parameters are detailed in Supp. Table 1. To illustrate what
happens to the $g$-tensor as the gates are pulsed, we sketch the cross-
sections of $\tensor{g}$ and $\tensor{g}+\delta\tensor{g}_{i}(\delta V_{i})$
in the $xz$, $yz$, and $xy$ planes of the magnet frame for $\delta V_{i}=100$
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 1). A true sweet spot to noise originating near gate $i$ exists when
$\partial f_{\text{Q}}/\partial V_{i}=0$. 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 $\theta_{\mathbf{B}}\phi_{\mathbf{B}}$-projections). 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 $(\theta_{\mathbf{B}}$, $\phi_{\mathbf{B}})$. These effects are
dominated by the dynamic tilting of $\tensor{g}$, which we believe to be
caused by hole wave function moving in a local strain gradient corley-
wiciak_nanoscale_2023 ; liles_electrical_2020 ; abadillo-uriel_hole_2022 , not
taken into account by previous models wang_modelling_2022 .

While the longitudinal component of the $g$-tensor modulation leads to
decoherence, the transverse part enables an electric drive of the qubit
through $g$-TMR. Therefore, our reconstruction of $\partial\tensor{g}/\partial
V_{i}$ 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 $(\partial
f_{\text{Rabi}}/\partial V_{i})/f_{\text{Q2}}$. The results, shown in Fig.
4e-g, 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 $\partial\tensor{g}/\partial V_{i}$,
both in absolute size and magnetic field dependence, confirms that the main
driving mechanism of planar germanium hole qubits is in fact $g$-TMR.

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 $f=n/(2\tau)$, 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 73Ge nuclear
spin of $\gamma=1.485(2)$ 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 $\tau$ and
$\theta_{\mathbf{B}}$. $N\tau$ is the total evolution time.
$\phi_{\mathbf{B}}=97.5\degree$ and $B=133$ mT. The inset displays the fit to
the data from which we extract $S_{0,\text{hf}}(\theta_{\mathbf{B}})$ and
$S_{0,\text{E}}(\theta_{\mathbf{B}})$. e, The extracted strength of the
hyperfine interaction as a function of $\theta_{\mathbf{f_{\text{Q}}}}$. The
black line is a fit of the data to
$a\cos^{2}(\theta_{\mathbf{f_{\text{Q}}}})$, with $a=2.5\cdot
10^{6}\leavevmode\nobreak\ \text{Hz}^{2}/\text{Hz}$. Error bars indicate
$1\sigma$ 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 $S_{0,\text{E}}=S_{0,V}\cdot(\partial
f_{\text{Q2}}/\partial V_{\text{P2}}(\theta_{\mathbf{B}}))^{2}$, with
$\partial f_{\text{Q2}}/\partial V_{\text{P2}}(\theta_{\mathbf{B}})$ the
electric field sensitivity of the qubit frequency to the top gate voltage as
extracted from Fig. 4 and $S_{0,V}=6.1\cdot 10^{-10}\leavevmode\nobreak\
V^{2}/\text{Hz}$ the only fit parameter. Error bars indicate $1\sigma$.

## Hyperfine interaction

Our qubits are defined in a natural germanium quantum well, where with a
concentration of 7.7 %, 73Ge is the only isotope with non-zero nuclear spin.
As a result, the hole wave function overlaps with ${\sim}10^{6}$ 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 bluhm_dephasing_2011 . 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 cywinski_electron_2009 ; chekhovich_nuclear_2013  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. cywinski_electron_2009 ; cywinski_pure_2009  and observed in GaAs
bluhm_dephasing_2011  and germanium lawrie_spin_2022 .

The hyperfine interaction between heavy hole states and nuclear spins is
expected to be highly anisotropic fischer_spin_2008 , unlike the isotropic
contact hyperfine interaction observed for conduction band electrons. In fact,
for the 73Ge isotope, the Ising term (out-of-plane, $\propto s_{z}I_{z}$) is
numerically estimated to be ${\sim}50$ times larger than in-plane ($\propto
s_{x}I_{x},s_{y}I_{y}$) components philippopoulos_hyperfine_2020 . 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
fischer_spin_2008 ; philippopoulos_hyperfine_2020 . 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 $f=1/\tau$ (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 $\tau$ between the
$\text{Y}_{\pi}$-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 $f_{\text{revival}}=\gamma|\mathbf{B}|$ with
$\gamma=1.485(2)$ MHz/T (Fig. 5b), in good agreement with the gyromagnetic
ratio of the 73Ge nuclear spin $\gamma_{\text{Ge-73}}=1.48$ MHz/T, confirming
its origin.

Following Refs. uhrig_keeping_2007 ; cywinski_how_2008 , 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 $S_{f_{q}}$ 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 73Ge nuclear spins (see Methods). We extract the
strength of the nuclear noise $S_{0,\text{hf}}$ and plot this as a function of
the elevation of the Larmor vector
$\theta_{\mathbf{f}_{\text{Q2}}}=\arccos\left(\frac{\mathbf{f_{\text{Q2}}}\cdot\hat{z}}{f_{\text{Q2}}}\right)$
(Fig. 5e). We find that the data closely follows a relation
$S_{0,\text{hf}}\propto\cos^{2}\left(\theta_{\mathbf{f}_{\text{Q2}}}\right)$,
providing strong experimental evidence of the predicted Ising coupling
fischer_spin_2008 ; prechtel_decoupling_2016 ; philippopoulos_hyperfine_2020 .
As a result, there exists a sweet plane approximately spanned by the
$x^{\prime}y^{\prime}$ axes of the $g$-tensor, where the qubit is mostly
insensitive to nuclear spin noise. The finite width of the hyperfine
distribution of $\sigma_{\text{Ge-73}}=9-16$ 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 73Ge spin kaufmann_73ge_1971 ,
but is in good agreement with values previously observed in Ge
lawrie_spin_2022  and could be caused either by interactions between the
nuclei or by the quadrupolar splitting present in the 73Ge isotope.

Finally, assuming all charge noise to originate near P2, we fit the extracted
$S_{0,E}$ (Fig. 5f) to $\partial f_{\text{Q2}}/\partial V_{\text{P2}}$ as
measured in Fig. 4 and find an effective electric noise power spectral density
of $S_{V}=610\leavevmode\nobreak\ \mu V^{2}/\mathrm{Hz}$ at 1 Hz,
corresponding to an effective voltage noise of $25\leavevmode\nobreak\ \mu
V/\sqrt{\mathrm{Hz}}$ on P2. Using the estimated plunger gate lever arm
$\alpha_{\mathrm{P}}=7.4\%$ (Supp. Fig. 7) we extract a charge noise level of
$1.9\leavevmode\nobreak\ \mu eV/\sqrt{\mathrm{Hz}}$, in good agreement with
charge noise measurements on similar devices lodari_low_2021 .

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 $T_{2}^{*}=17.6\leavevmode\nobreak\ \mu$s. b, CPMG dynamical
decoupling coherence, as measured for a sequence with 250 refocusing pulses.
We find a coherence time of $T_{2}^{\text{DD}}=1.3$ ms. c, Randomized
benchmarking of the performance of Q2. The solid line is a fit of the data to
$P=a\exp(-(1-2F_{\text{c}})N_{\text{C}})$, from which we extract a single
qubit gate fidelity of $F_{\text{g}}=99.94$ %. The reduced visibility for
larger $N_{\text{C}}$ 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
$\phi_{\mathbf{B}}=97.5\leavevmode\nobreak\ \degree$. We extract a single
qubit gate fidelity of $F_{\text{g}}=99.7$ %.

## 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:
$\phi_{\mathbf{B}}=97.5\leavevmode\nobreak\ \degree$ and
$\theta_{\mathbf{B}}=89.7\leavevmode\nobreak\ \degree$ 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 $\phi_{\mathbf{B}}=0\degree$ 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 $f_{\text{Q2}}\approx 21$ MHz and $f_{\text{Rabi}}=1$ MHz and find
$T_{2}^{*}=17.6\leavevmode\nobreak\ \mu$s, which is about an order of
magnitude larger than shown previously for germanium hole qubits
lawrie_spin_2022 . 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 $X_{\pi}$ and $X_{\pi/2}$ pulses
and virtual $Z$ updates (see Supp. Table 3). 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 hendrickx_four-qubit_2021 ; van_riggelen_phase_2022 .

## 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 $g$-tensors are
remarkably similar and vary by less than $10$ %, indicative of a high degree
of uniformity of the electrostatic confinement. However, the small tilt
($\delta\theta\approx 1\leavevmode\nobreak\ \degree$) combined with the large
anisotropy of $\tensor{g}$ leads to measurable effects, in particular for
magnetic field orientations in proximity to the $g$-tensor minor principle
axes. The slight tilt of the $g$-tensor 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 lodari_lightly_2022  and quantum dot confinement
bosco_squeezed_2021 .

The $g$-tensor anisotropy is also reflected in the qubit sensitivity to
electric field fluctuations. We find that $\tensor{g}$ 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 73Ge
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
$\mathbf{B}\nparallel x^{\prime}y^{\prime}$. 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
itoh_high_1993 . 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.

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## Methods

### 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 bedell_low-
temperature_2020 . The virtual substrate consists of a strain-relaxed
germanium layer on a silicon wafer and multiple layers with increasing silicon
content to reach the Si0.2Ge0.8 stoichiometry used for the quantum well
barriers. Ohmic contacts to the quantum well are defined by in-diffusion of Pt
at a temperature of $300\degree$ 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 ($\sim$M$\Omega$).
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
SiO2.

### Experimental setup

All measurements are performed in a Bluefors LD400 dilution refrigerator with
a base temperature of $T_{\text{mc}}=10$ 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 $f_{\text{Q}}>500$ MHz. For experiments when
$f_{\text{Q}}<500$ 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
$V_{\text{SD}}=300-800\leavevmode\nobreak\ \mu$V. 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. $B_{x-\text{coil}}\parallel x$, $B_{y-\text{coil}}\parallel y$, and
$B_{z-\text{coil}}\parallel z$). The correctly observed gyromagnetic ratio of
the 73Ge 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
$\theta_{\mathbf{B}}$ before every measurement in Fig. 6 and locate the sweet
plane by minimizing the qubit frequency as a function of
$\theta_{\mathbf{B}}$.

### 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
hensgens_quantum_2017 :

$\begin{pmatrix}V_{\text{P1}}\\\ V_{\text{P2}}\\\ V_{\text{P3}}\\\
V_{\text{P4}}\\\
V_{\text{B12}}\end{pmatrix}=\begin{pmatrix}1&-0.28&0&0&-1.65\\\
-0.18&1&0&0&-1.30\\\ 0&-0.11&1&0&0.10\\\ -0.11&0&0&1&0.10\\\ 0&0&0&0&1\\\
\end{pmatrix}\begin{pmatrix}V_{\overline{\text{P1}}}\\\
V_{\overline{\text{P2}}}\\\ V_{\overline{\text{SB2}}}\\\
V_{\overline{\text{SB1}}}\\\ V_{\overline{\text{B12}}}\\\ \end{pmatrix}$

with Gi the real gate voltage, and $\overline{\text{G}_{i}}$ the virtual gate
voltage, which leaves the chemical potential of the other quantum dots
unchanged. Furthermore, we define a second pair of axes detuning $\epsilon$
and on-site energy U, as illustrated in Fig. 1b of the main text:

$\begin{pmatrix}V_{\overline{\text{P1}}}\\\
V_{\overline{\text{P2}}}\end{pmatrix}=\begin{pmatrix}-0.5&0.5\\\ 0.5&0.5\\\
\end{pmatrix}\begin{pmatrix}V_{\epsilon}\\\ V_{\text{U}}\\\ \end{pmatrix}$

### Pauli spin blockade readout

To overcome rapid spin relaxation as mediated by the spin-orbit interaction
danon_pauli_2009 , we make use of charge latching, where we tune the tunnel
rates between each dot and its respective reservoir to be asymmetric
$t_{\text{Q2}}\ll t_{\text{Q1}}$. 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 harvey-collard_high-fidelity_2018 ; hendrickx_four-qubit_2021 ,
with a characteristic decay time to the (0,2) ground state governed by
$t_{\text{Q2}}$. Furthermore, the spin-orbit interaction introduces a coupling
between the $\ket{T(1,1)}$ and $\ket{S(0,2)}$ states, resulting in the
presence of an anticrossing between the $\ket{\downarrow\downarrow}$ and the
$\ket{S(0,2)}$ 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
seedhouse_pauli_2021 ; hendrickx_four-qubit_2021 . 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 $\bm{g}$-tensor

The $g$-tensor of the device can be described as a rotated diagonal matrix:

$\tensor{g}=R(\phi,\leavevmode\nobreak\ \theta,\leavevmode\nobreak\
\zeta)\operatorname{diag}{(g_{x^{\prime}},g_{y^{\prime}},g_{z^{\prime}})}R^{-1}(\phi,\leavevmode\nobreak\
\theta,\leavevmode\nobreak\ \zeta)$ (1)

where Euler angles $\phi$, $\theta$, and $\zeta$ define the successive
intrinsic rotations around the $zyz$ axes. The rotation matrix $R$ is thus
defined as:

$R(\phi,\leavevmode\nobreak\ \theta,\leavevmode\nobreak\
\zeta)=\left(\begin{array}[]{ccc}\cos(\phi)\cos(\theta)\cos(\zeta)-\sin(\phi)\sin(\zeta)&-\sin(\phi)\cos(\zeta)-\cos(\phi)\cos(\theta)\sin(\zeta)&\cos(\phi)\sin(\theta)\\\
\sin(\phi)\cos(\theta)\cos(\zeta)+\cos(\phi)\sin(\zeta)&\cos(\phi)\cos(\zeta)-\sin(\phi)\cos(\theta)\sin(\zeta)&\sin(\phi)\sin(\theta)\\\
-\sin(\theta)\cos(\zeta)&\sin(\theta)\sin(\zeta)&\cos(\theta)\\\
\end{array}\right)$ (2)

The $g$-tensor can thus be reconstructed by measuring the qubit energy
splitting $hf_{\text{Q}}$ for different orientations of the magnetic field
$\mathbf{B}$. We measure $f_{\text{Q}}$ for various magnetic field
orientations $(\theta_{\mathbf{B}},\phi_{\mathbf{B}})$ and fit the data to:

$hf_{\text{Q}}=\absolutevalue{\mu_{B}\tensor{g}\cdot\mathbf{B}}$ (3)

using $\tensor{g}$ as defined in Eqs. 1-2 and $g_{x^{\prime}}$,
$g_{y^{\prime}}$, $g_{z^{\prime}}$, $\phi$, $\theta$, and $\zeta$ 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 $f_{\text{Q}}$ is measured are shown in Supp. Fig. 1c. These field
orientations $(\theta_{\mathbf{B}},\phi_{\mathbf{B}})$ 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 $2\tau$ 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
$I_{\text{SD}}=\exp(-(2\tau/T_{2}^{\text{H}})^{\alpha})$, with the exponent
$\alpha$ 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 ($\mathbf{B}\nparallel x^{\prime}y^{\prime}$), we
extract the envelope coherence $T_{2}^{\text{H}}$ by fitting the envelope of
the nuclear spin induced collapse-and-revival (Supp. Fig. 3a)
bluhm_dephasing_2011  to
$I_{\text{SD}}=\exp(-(2\tau/T_{2}^{\text{H}})^{\alpha})/|1-a_{0}\cos(2\pi
f_{\text{Ge-73}}\tau)|^{2}$, with $a_{0}$ and $\alpha$ free fitting parameters
and $f_{\text{Ge-73}}=\gamma_{\text{Ge-73}}B$, as discussed further in the
main text.

The exponent of the dependence of the Hahn echo coherence time on both
$\partial f_{\text{Q}}/\partial V_{i}/f_{\text{Q}}$ and $f_{\text{Q}}$ (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 $S\propto
f^{\alpha}$ acting on a qubit and following the filter formalism from Refs.
piot_single_2022 ; cywinski_how_2008 , we find:

$T_{2}^{H}\propto\left(\frac{\frac{\partial f_{\text{Q}}}{\partial
V_{i}}}{f_{\text{Q}}}(\theta_{\mathbf{B}},\phi_{\mathbf{B}})\cdot
f_{\text{Q}}(B)\right)^{\frac{2}{\alpha-1}}$ (4)

Therefore, both the dependence of $T_{2}^{H}$ on the qubit frequency (by
varying $B$, Fig. 3f) and on the electric field sensitivity (by varying
$\theta_{\mathbf{B}}$ and $\phi_{\mathbf{B}}$, Fig. 3g) should obey a power
law with exponent $\beta=\frac{2}{\alpha-1}$. From this we can derive the
noise exponent $\alpha=\frac{2}{\beta}+1$, such that $\alpha=-1$ if
$\beta=-1$.

To obtain the expected charge-noise limited $T_{2}^{H}$ at $f_{\text{Q2}}=1$
GHz, we fit a power law $T_{2}^{H}=T_{2}^{H}[1\leavevmode\nobreak\
\text{GHz}]\cdot 1\leavevmode\nobreak\ \text{GHz}/f_{\text{Q2}}$ to the data
in Fig. 3f where $B>B_{\text{hyperfine}}$ (opaque markers). Here,
$B_{\text{hyperfine}}$ indicates the magnetic field strength below which the
finite spread of the nuclear spin precession frequencies limits qubit
coherence bluhm_dephasing_2011 .

Because of the limited maximum field strength we can apply along the $x$ and
$y$ axis $B_{\text{max},x}=B_{\text{max},y}=1$ T, the electric field
sensitivity for the pink data point is obtained at a lower qubit frequency
$f_{\text{Q2}}=785$ MHz and extrapolated to $f_{\text{Q2}}=1.36$ GHz.

### Fitting procedure of the hyperfine noise

We follow the method presented in Refs. uhrig_keeping_2007 ; uhrig_exact_2008
; cywinski_how_2008  and 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 73Ge nuclear spins:

$\begin{split}S_{f_{q}}(f,\mathbf{B})&=S_{\text{hf}}(f,\mathbf{B})+S_{E}(f,\mathbf{B})\\\
S_{\text{hf}}(f,\mathbf{B})&=S_{0,\text{hf}}(\mathbf{B})\exp\left(-\frac{f-\gamma_{\text{Ge-73}}B}{2\sigma_{\text{Ge-73}}^{2}}\right)\\\
S_{E}(f,\mathbf{B})&=\frac{S_{0,E}}{f}(\mathbf{B})=\frac{S_{0,V}\left(\frac{\partial
f_{\text{Q}}}{\partial V_{\text{P2}}}(\mathbf{B})\right)^{2}}{f}\end{split}$
(5)

Here, $S_{0,\text{hf}}(\mathbf{B})$ 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,
$\gamma_{\text{Ge-73}}=1.48$ MHz/T is the 73Ge gyromagnetic ratio and
$\sigma_{\text{Ge-73}}$ represents the finite spread of the 73Ge 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 $\frac{\partial f_{\text{Q2}}}{\partial
V_{\text{P2}}}(\mathbf{B})$ 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:

$P(\tau)\propto\exp\left(-\mathcal{X}(\tau)\right)\\\ $ (6)

with $P$ the measured spin-up probability and

$\mathcal{X}(\tau)=\int_{0}^{\infty}S_{\omega_{q}}(\omega)\cdot\frac{F_{N}(\omega,\tau)}{\pi\omega^{2}}d\omega$
(7)

The unitless filter function $F_{N}$ for the CPMG experiment is defined as
follows cywinski_how_2008 :

$F_{N}(\omega,\tau)=\begin{cases}8\sin^{4}\left(\omega\tau/4\right)\frac{\sin^{2}\left(N\omega\tau/2\right)}{\cos^{2}\left(\omega\tau/2\right)},&N\text{
is even}\\\
8\sin^{4}\left(\omega\tau/4\right)\frac{\cos^{2}\left(N\omega\tau/2\right)}{\cos^{2}\left(\omega\tau/2\right)},&N\text{
is odd}\end{cases}$ (8)

As both the strength of the nuclear spin noise and charge noise are expected
to depend on $\mathbf{B}$, we fit the data for each $\theta_{\mathbf{B}}$
independently, fixing $\gamma_{\text{Ge-73}}=1.48$ MHz/T and keeping
$\sigma_{\text{Ge-73}}$, $S_{0,V}$, and $S_{0,\text{hf}}$ as fit parameters.

We note that we find that $\sigma_{\text{Ge-73}}$ is independent of
$\theta_{\mathbf{B}}$ within the experimental range, with an average
$\overline{\sigma}_{\text{Ge-73}}=9$ 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 $f_{\text{Ge-73}}\approx\sigma_{\text{Ge-73}}$.
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 $\phi_{\mathbf{B}}$ (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 cywinski_how_2008 ; biercuk_dynamical_2011 , thus
enhancing the sensitivity to the nuclear spin precession frequency. As a
result, a higher accuracy of $\theta_{\mathbf{B}}$ is required to align
exactly to the hyperfine sweet spot and avoid loss of coherence due to
hyperfine interaction with the 73Ge 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
$S_{\text{0,hf}}=2.52(4)\text{kHz}^{2}/\text{Hz}$ for an out-of-plane field
and $\sigma_{\text{Ge-73}}=9.9(11)$ kHz. This equates to an integrated
detuning noise of:

$\sigma_{f}=\sqrt{\sqrt{2\pi}S_{\text{0,hf}}\sigma_{\text{Ge-73}}}=250\leavevmode\nobreak\
\text{kHz}$ (9)

Assuming a Gaussian noise distribution, this corresponds to an expected phase
coherence time nakajima_coherence_2020  of
$T_{2}^{*}=1/(\pi\sqrt{2}\sigma_{f})=900\leavevmode\nobreak\ \text{ns}$. We
can estimate the out-of-plane hyperfine coupling $A_{\parallel}$ using Eq.
2.65 from Ref. philippopoulos_first-principles_2020 :

$h^{2}\sigma_{f}^{2}\approx\frac{1}{4N}g_{\text{Ge-73}}I(I+1)A_{\parallel}^{2}$
(10)

Such that:

$\displaystyle A_{\parallel}$
$\displaystyle\approx\sqrt{\frac{4N}{g_{\text{Ge-73}}I(I+1)}}h\sigma_{f}$ (11)

with $g_{\text{Ge-73}}=0.0776$ the natural abundance of the 73Ge isotope,
$I=9/2$ the 73Ge 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 $N=\pi r^{2}w/v_{0}$, with $r$ the radius and $w$ the height of the
dot, and $v_{0}=2.3\cdot 10^{-29}\leavevmode\nobreak\ \text{m}^{3}$ the atomic
volume of germanium. We can estimate $r$ from the single particle level
splitting $\Delta E\approx 1.2$ meV as can be obtained from the extend of the
PSB readout window, and find $r\approx 35$ nm. This is in good agreement with
$r\approx 50$ nm as expected from the charging energy $E_{\text{C}}\approx
2.8$ meV and the capacitance of a disk: $r=e^{2}/(8\epsilon_{r}E_{c})$.
Assuming $r=35$ nm and $w=10$ nm (half of the quantum well width), we then
find $N\approx 1.7\cdot 10^{6}$. Using Eq. 11, we estimate the hyperfine
coupling constant to be $|A_{\parallel}|\approx 1.9\leavevmode\nobreak\
\mu$eV, which is in good agreement with the theoretical prediction
$A_{\parallel}=-1.1\leavevmode\nobreak\ \mu$eV from Ref.
philippopoulos_hyperfine_2020 . Similarly from the extracted $S_{\text{0,hf}}$
for an in-plane $\mathbf{B}$, we estimate an upper bound for the in-plane
hyperfine coupling constant $A_{\perp}<0.1\mu$eV compatible with the predicted
$A_{\perp}=0.02\leavevmode\nobreak\ \mu$eV.

### Randomized benchmarking

To extract the single qubit gate fidelity, we perform randomized benchmarking
of the Clifford gate set presented in Supp. Table 3. For every randomization,
we measure both the projection to $\ket{\uparrow}$ and $\ket{\downarrow}$ 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 $\ket{\uparrow}$ and $\ket{\downarrow}$ states.
We fit the data to $P=a\exp(-(1-2F_{\text{C}})N_{\text{C}})$, with
$F_{\text{C}}$ 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 $F_{\text{g}}=99.92$ % for
$T_{\text{mc}}=20$ mK and $F_{\text{g}}=99.7$ % for $T_{\text{mc}}=1.1$ K when
fixing $a=1$. The primitive gate fidelity $F_{\text{g}}$ can be calculated by
accounting for the number of physical gates per Clifford: $0.875$ for this
gate set.

### Extraction of the $\bm{g}$-tensor sensitivity

We measure the modulation of the qubit energy splitting $\delta f_{\text{Q}}$
as the result of a small voltage pulse $\delta V$ 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 $|\delta f_{\text{Q}}|$. We confirm that for a small
$\delta V$, $|\delta f_{\text{Q}}|$ is linear in $\delta V$, allowing us to
extract the sensitivity $|\partial f_{\text{Q}}/\partial V_{i}|$ from a single
data point of $\delta V$ (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 reed_reduced_2016 ; martins_noise_2016
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 $\partial f_{\text{Q}}/\partial V_{i}$, 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 $\tensor{g}$ and a g-tensor sensitivity
$\partial\tensor{g}/\partial V_{i}$, $\partial f_{\text{Q}}/\partial V_{i}$
only depends on the magnetic field direction $\mathbf{b}$ and on
$f_{\text{Q}}\propto B$:

$\frac{\delta f_{\text{Q}}}{\delta
V}(\theta_{\mathbf{B}},\phi_{\mathbf{B}},B)=\frac{\left(\partial\tensor{g}/\partial
V_{i}\cdot\mathbf{b}\right)\cdot\left(\tensor{g}\cdot\mathbf{b}\right)}{\left(\tensor{g}\cdot\mathbf{b}\right)^{2}}f_{\text{Q}}(B)$
(12)

We extract $\partial\tensor{g}/\partial V_{i}$ by fitting Eq. 12 to the data
presented in Fig. 4 of the main text, using $\tensor{g}$ as extracted
previously and displayed in Fig. 2h. We then calculate the expected $g$-TMR
mediated Rabi frequency using

$\frac{\delta f_{\text{Rabi}}}{\delta
V}(\theta_{\mathbf{B}},\phi_{\mathbf{B}},B)=\mu\frac{\left|\left(\partial\tensor{g}/\partial
V_{i}\cdot\mathbf{b}\right)\crossproduct\left(\tensor{g}\cdot\mathbf{b}\right)\right|}{2|\tensor{g}\cdot\mathbf{b}|^{2}}f_{\text{Q}}(B)$
(13)

with $f_{\text{Rabi}}$ the Rabi frequency and $\mu$ the signal attenuation for
a microwave signal at a frequency of $f_{\text{Q}}$.

We fit the data to the Eq. 13, with $\mu$ as the only fit parameter. We find a
line attenuation of $\mu_{\text{P2}}=2.1$, $\mu_{\text{B2}}=2.1$, and
$\mu_{\text{B12}}=2.0$. 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 ($\mu=2.1-2.5$) (see Supp. Fig. 6).

## References

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  * (2) Danon, J. & Nazarov, Y. V.  Pauli spin blockade in the presence of strong spin-orbit coupling.  _Physical Review B_ 80, 041301 (2009).
  * (3) Seedhouse, A. E. _et al._ Pauli Blockade in Silicon Quantum Dots with Spin-Orbit Control.  _PRX Quantum_ 2, 010303 (2021).
  * (4) Uhrig, G. S.  Exact results on dynamical decoupling by $\pi$ pulses in quantum information processes.  _New Journal of Physics_ 10, 083024 (2008).
  * (5) Biercuk, M. J., Doherty, A. C. & Uys, H.  Dynamical decoupling sequence construction as a filter-design problem.  _Journal of Physics B: Atomic, Molecular and Optical Physics_ 44, 154002 (2011).
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  * (8) Reed, M. _et al._ Reduced Sensitivity to Charge Noise in Semiconductor Spin Qubits via Symmetric Operation.  _Physical Review Letters_ 116, 110402 (2016).
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## Data availability

All data underlying this study will be made available in a Zenodo repository.

## Acknowledgements

We acknowledge the staff of the Binnig and Rohrer Nanotechnology Center for
their contributions to the sample fabrication and thank all members of the IBM
Research Europe - Zurich spin qubit team for useful discussions. A.F.
acknowledges support from the Swiss National Science Foundation through Grant
No. 200021 188752.

## Author contributions

N.W.H. performed the experiments and data analysis, with contributions from
G.S.; N.W.H. fabricated the sample with contributions from L.M., M.M., and
F.S.; S.P. contributed to the development of the experimental setup. S.W.B.
provided the heterostructures. N.W.H. wrote the manuscript with contributions
from G.S. and A.F. and input from all authors. N.W.H. and A.F. planned the
project.

## Competing Interests

The authors declare no competing interests.

## Additional information

Supplementary information is available with this paper.

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

$\partial\tensor{g}/\partial V_{i}$ | P2 | B2 | B12
---|---|---|---
$\partial\phi/\partial V_{i}$ (mrad$\cdot$mV-1) | -0.11(2) | 1.3(2) | -3.1(1)
$\partial\theta/\partial V_{i}$ (mrad$\cdot$mV-1) | 0.0008(5) | 0.005(4) | 0.028(3)
$\partial\zeta/\partial V_{i}$ (mrad$\cdot$mV-1) | 0.04(2) | -2.3(1) | 2.2(1)
$\partial g_{x^{\prime}}/\partial V_{i}$ (mV-1) | 0.000181(9) | -0.00028(7) | -0.00073(6)
$\partial g_{y^{\prime}}/\partial V_{i}$ (mV-1) | 0.000507(3) | 0.00037(2) | -0.00146(2)
$\partial g_{z^{\prime}}/\partial V_{i}$ (mV-1) | 0.0045(1) | -0.0001(8) | -0.0071(7)
Supplementary Table 1: Description of $\bm{\partial\tensor{g}/\partial V_{i}}$ of Q2 for a potential applied to gates P2, B2, and B12. Overview of the parameters describing the voltage-induced deformation of the $g$-tensor. Three Euler angles $\partial\zeta/\partial V_{i}$, $\partial\theta/\partial V_{i}$, and $\partial\phi/\partial V_{i}$ describing the rotation, as well as changes to the principle $g$-factors $\partial g_{x^{\prime}}/\partial V_{i}$, $\partial g_{y^{\prime}}/\partial V_{i}$, and $\partial g_{z^{\prime}}/\partial V_{i}$, for Q2.  Data set |  $\phi_{B}$ (deg.) |  $\theta_{B}$ (deg.)
---|---|---
| 0.0 | 90.03
| 0.0 | 85.50
| 0.0 | 95.63
| -105.0 | 95.63
| -105.0 | 90.00
| -105.0 | 89.91
| 0.0 | 91.13
| 0.0 | 91.08
Supplementary Table 2: Magnetic field parameters for data in Fig. 3 of the
main text. Magnetic field elevation $\theta_{B}$ and azimuth $\phi_{B}$ for
the different data sets in Fig. 3 of the main text, as indicated by the
coloured markers.  Clifford gates
---
I
-Z, X/2, X/2
-Z, X/2
-Z, X/2, -Z
-Z, X/2, -Z/2
-Z, X/2, Z/2
-Z/2
-Z/2, X/2
-Z/2, X/2, -Z
-Z/2, X/2, -Z/2
-Z/2, X/2, Z/2
X/2, X/2
X/2, X/2, Z/2
X/2
X/2, -Z
X/2, -Z/2
X/2, Z/2
Z
Z/2
Z/2, X/2, X/2
Z/2, X/2
Z/2, X/2, -Z
Z/2, X/2, -Z/2
Z/2, X/2, Z/2
Supplementary Table 3: Clifford set used for randomized benchmarking. The
Clifford set used for the randomized benchmarking consists out of $X_{\pi/2}$
pulses, and virtual $Z$ gates. The average number of physical gates per
Clifford equals 0.875.  Supplementary Figure 1: Exemplary EDSR resonance
traces used to fit the qubit $g$-tensors. a, Normalized charge sensor current,
as a function of the microwave frequency $f$, showing spin resonance for Q1.
We apply a frequency chirp to assure full spin inversion. b, Normalized charge
sensor current, as a function of the microwave frequency $f$ and magnetic
field azimuth $\phi_{\mathbf{B}}$. c, Illustration of all magnetic field
orientations at which $g^{*}$ was measured to fit $\tensor{g}$, for Q1 (top)
and Q2 (bottom). Each of the 402 (935) markers corresponds to a single
resonance measurement for Q1 (Q2) as in panel a.  Supplementary Figure 2:
Projections of the fitted $\bm{g}$-tensors. From the experimentally extracted
the qubit $g$-tensors, we calculate the qubit quantisation axis
$h\mathbf{f_{\text{Q}}}=\mu_{\text{B}}\tensor{g}\mathbf{B}$ as a function of
$\mathbf{B}$. a, Diagram illustrating the relevant angles.
$\theta_{\mathbf{B}}$, and $\phi_{\mathbf{B}}$ are the elevation and azimuth
angle of the applied magnetic field respectively.
$\theta_{\mathbf{f_{\text{Q}i}}}$, and $\phi_{\mathbf{f_{\text{Q}i}}}$ are the
elevation and azimuth angle of the resulting Larmor vector of Q$i$.
$\alpha_{Qi}$ is the angle between the applied magnetic field and the
resulting Larmor vector of Q$i$ and $\beta$ is the angle between the two
quantisation axes of the two qubits. b,c, Elevation angle of the Larmor vector
of Q1 (b) and Q2 (c), as a function of the orientation of the magnetic field.
As a result of the large anisotropy of the $g$-tensor, the quantisation axis
of the qubit rapidly flips from $\mathbf{f_{\mathbf{\text{Q}}}}\parallel z$ to
$\mathbf{f_{\mathbf{\text{Q}}}}\parallel-z$ as the magnetic field crosses the
equator of the $g$-tensor. d,e, The absolute angle between the qubit
quantisation axis and applied magnetic field direction
$\absolutevalue{\alpha_{Qi}}$ for Q1 (d) and Q2 (e). f, Angle between the
qubit quantisation axes $\beta$ as a function of the magnetic field
orientation. Near the principle axis directions of the qubit $g$-tensors, a
large misalignment between the two quantisation axes can be observed. g,
Colour plot of
$\absolutevalue{\hat{\mathbf{f}}_{\mathbf{Q1}}\cdot\hat{\mathbf{f}}_{\mathbf{Q2}}}$,
illustrating the orthogonality of the two qubit quantisation axes. For the
ring-shaped regions where this quantity equals $0$, around the $x$ principle
axis of the $g$-tensors, the qubit quantisation axes of Q1 and Q2 are
perpendicular to each other. Supplementary Figure 3: Hahn echo decay of Q2. a,
Hahn echo decay of Q2, for the magnetic field aligned with the hyperfine sweet
plane. Solid line is a fit to the data used to extract $T_{2}^{\text{H}}$. b,
Hahn echo decay of Q2, for the magnetic field aligned away from the hyperfine
sweet plane. A collapse-and-revival structure can be observed due to the
interaction of the qubit with the precession of the 73Ge nuclear spins. Solid
line is a fit to the data used to extract $T_{2}^{\text{H}}$.  Supplementary
Figure 4: Q2 coherence as a function of the different gate voltage
sensitivities a-c, Observed Hahn echo coherence time (data from Fig. 3g in the
main text), as a function of the relative qubit frequency sensitivity
$(\partial f_{\text{Q2}}/\partial V_{i})/f_{\text{Q2}}$ to a voltage
fluctuation on gate P2 (a), B2 (a), or B12 (a). Voltage sensitivities are
obtained from the data for gate P2 (Fig. 3e) from the fits of
$\delta\tensor{g}$ presented in Fig. 4 of the main text, as well as Fig. 8 of
this Supplementary Information for gates B2 and B12. Black lines are fits to a
power law $T_{2}^{H}=a\exp(((\partial f_{\text{Q2}}/\partial
V_{i})/f_{\text{Q2}})^{\alpha})$, with $a$ a scaling factor and $\alpha$ the
exponent. We find the correct trend with $\alpha=-1$ (see Methods) for a
voltage noise effectively coming from the plunger gate of the qubit,
indicating the dominant charge traps are located directly above our qubit for
most magnetic field orientations. For very specific field orientations where
the qubit sensitivity to noise from other directions is particularly strong,
such as for the pink data point, differently located charge traps could limit
$T_{2}^{H}$.  Supplementary Figure 5: Linearity of the qubit frequency shift.
We apply a Hahn echo sequence to the qubit, where a positive (negative)
voltage pulse $\delta V_{\text{P2}}$ is applied during the first (second) free
evolution, as illustrated in the schematic. As we vary the depth of the
voltage pulse, we observe a linear shift of the qubit frequency, which allows
us to extract $\partial f_{\text{Q2}}/\partial V_{\text{P2}}$ by evaluating
the frequency shift for a single point of $\delta V_{\text{P2}}$.
Supplementary Figure 6: Attenuation of the microwave pulses applied to
different gates. a-c, Charge sensor Coulomb peak as a result of a continuous
wave (CW) microwave excitation with a frequency of $f_{\text{CW}}=250$ MHz and
a peak-to-peak amplitude of $V_{\text{pp}}$, applied to gate B2 (a), B12 (b),
and P2 (c). Solid markers indicate the regime where the splitting $\delta
V_{\text{pp}}$>$\sigma$ with $\sigma$ the peak width. d-f, Charge sensor
Coulomb peak as a result of a continuous wave microwave excitation with a
frequency of $f_{\text{CW}}=1$ MHz and a peak-to-peak amplitude of
$V_{\text{pp}}$, applied to gate B2 (a), B12 (b), and P2 (c). Solid markers
indicate the regime where the splitting $\delta V_{\text{pp}}$>$\sigma$ with
$\sigma$ the Coulomb peak width. g Splitting of the Coulomb peak $\delta
V_{\text{pp}}$, as extracted by fitting each line to the expected shape of a
sine wave broadened Lorentzian, obtained by analytically evaluating the
convolution between a Lorentzian and the probability distribution of a sine.
We extract the relative slope $\alpha$ by fitting the data indicated by the
solid markers in panels a-f to $\delta V_{\text{P2}}=\alpha V_{\text{pp}}$.
Colour of the markers correspond to the coloured markers as used in panels
a-f. h The ratio between the attenuation at $f_{\text{CW}}=250$ MHz and
$f_{\text{CW}}=1$ MHz for the three different gates. We find attenuation
ratios of $\mu_{\text{P2}}=2.1$, $\mu_{\text{B12}}=2.5$, and
$\mu_{\text{B2}}=2.5$.  Supplementary Figure 7: Lever arm extraction of the
plunger gates. We extract the lever arm of the plunger gates by tuning the
charge sensor into a transport double quantum dot underneath gates SB1 and
SB2, which are lithographically identical to qubit gates P1 and P2. We measure
the bias triangles at $V_{\text{SD}}=1$ mV and find an average plunger lever
arm of $\alpha_{\text{P}}=7.4(8)$ %, with the uncertainty as obtained from the
scatter of $\alpha$ for gates SB1 and SB2.  Supplementary Figure 8:
Longitudinal and transverse components of the fitted
$\bm{\partial\tensor{g}/\partial V_{i}}$. a-c. Using the fitted
$\partial\tensor{g}/\partial V_{i}$, as detailed in Fig. 4 of the main text
and Supp. Table 1 , 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 $V_{i}$ on gate P2 (d), B2 (e), and B12 (f) for different
magnetic field orientations. g-i. Expected ratio of the transverse and
longitudinal components $(\partial f_{\text{Rabi}}/\partial V_{i})/(\partial
f_{\text{Q2}}/\partial V_{\text{P2}})$ 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.  Supplementary Figure 9: CPMG dynamical decoupling as a function of
magnetic field strength. a, Schematic illustrating the power spectrum of noise
assumed for the fitting. The spectrum consists of a $1/f$ charge noise
component with a power $S_{0,\text{E}}$ at 1 Hz, plus a Gaussian spectral line
at $f_{\text{Ge-73}}$ with width $\sigma_{\text{Ge-73}}$ and power
$S_{0,\text{hf}}$. b-f, Normalized charge sensor signal for a CPMG sequence
with 1 (b), 2 (c), 4 (d), 8 (e), and 16 (f) decoupling pulses respectively, as
a function of the spacing between two subsequent decoupling pulses $\tau$ and
$B$. $N\tau$ is the total evolution time. $\phi_{\mathbf{B}}=0\degree$ and
$\theta_{\mathbf{B}}=90\degree$. The inset displays the fit to the data from
which we extract $\gamma_{\text{Ge-73}}=1.48$ MHz/T and
$\sigma_{\text{Ge-73}}=17$ kHz. The extracted charge-induced noise spectrum
$S_{0,\text{E}}(B)$ is plotted in Supp. Fig. 10f.  Supplementary Figure 10:
Hyperfine interaction between the qubits and the 73Ge nuclear spins. a,b,
Normalized charge sensor signal for a Hahn echo experiment as a function of
the total free evolution time $2\tau$ (colour map in left subpanel) and the
qubit frequency $f_{\text{Q}i}$ (plot in right subpanel) as a function of the
elevation angle of the magnetic field for Q1 (a) and Q2 (b). $B=89$ mT is kept
constant throughout the measurement. For any $\theta_{\mathbf{B}}$ away from
the $x^{\prime}y^{\prime}$ plane of the $g$-tensor, we observe the collapse-
and-revival characteristic for a well-defined spectral component acting on the
qubit. Due to the small tilt between the two qubit $g$-tensors, a common
hyperfine sweet spot is defined by the intersection of two differently tiled
ellipsoids in the lab frame, at $\phi_{\mathbf{B}}=97.5\degree$,
$\theta_{\mathbf{B}}=89.7\degree$. Selecting this magnetic field orientation
allows to operate both qubits in their respective hyperfine sweet planes
simultaneously. Here, we find $T_{\text{2,Q1}}^{H}=8.1(2)\leavevmode\nobreak\
\mu$s and $T_{\text{2,Q2}}^{H}=11.5(6)\leavevmode\nobreak\ \mu$s for
$\mathbf{B}$ as specified. c,d, Normalized charge sensor signal for a CPMG
sequence with respectively 2 (c), and 8 (d) decoupling pulses, as a function
of the spacing between two subsequent decoupling pulses $\tau$ and
$\theta_{\mathbf{B}}$. $N\tau$ is the total evolution time. The magnetic field
strength is $B=133$ mT and the azimuth angle is
$\phi_{\mathbf{B}}=97.5\degree$. The inset displays the fit to the data. These
data complement the dataset displayed in Fig. 5c,d of the main text. e, Width
of the hyperfine line $\sigma_{\text{Ge-73}}$ as extracted from the CPMG
coherence as a function of $B$ (Supp. Fig. 9, $\phi_{\mathbf{B}}=0\degree$,
blue markers), and as extracted from the CPMG coherence as a function of
$\theta_{\mathbf{B}}$ (Fig. 5c,d of the main text,
$\phi_{\mathbf{B}}=97.5\degree$, black markers). We find
$\sigma_{\text{Ge-73}}$ to be independent of $\theta_{\mathbf{B}}$ in the
experimental range, but observe a significant difference for the different
azimuth directions. Dashed lines correspond to the average
$\overline{\sigma}_{\text{Ge-73}}=9.4$ kHz for $\phi_{\mathbf{B}}=97.5\degree$
and ${\overline{\sigma}_{\text{Ge-73}}}=17$ kHz for
$\phi_{\mathbf{B}}=0\degree$. f, The intensity of the $1/f$ component of the
spectrum at $1$ Hz extracted from the fit to the data in Supp. Fig. 9. We find
$S_{0,\text{E}}\propto B^{2}\propto f_{\text{Q2}}^{2}$, as expected for charge
noise with a $1/f$ spectrum.  Supplementary Figure 11: Extended Ramsey
measurement at $\bm{B=35}$ mT. Normalized charge sensor signal as a function
of the waiting time $\tau$ for a Ramsey experiment. The data constitute of an
average of 750 traces, for a total integration time of over 12 hours and we
find a coherence time of $T_{2}^{*}=9.2\leavevmode\nobreak\ \mu$s.
Supplementary Figure 12: CPMG dynamical decoupling as a function of the number
of decoupling pulses. Charge sensor signal as a function of the total
evolution time and number of decoupling pulses of a CPMG sequence. The
magnetic field is aligned to the hyperfine sweet spot. For low
$N_{\text{CPMG}}$ we observe an expected Gaussian decay due to charge noise,
but for increasing $N$ the sensitivity to the nuclear noise is enhanced and we
observe a sharp coherence collapse caused by the hyperfine interaction.
Supplementary Figure 13: Expected qubit coherence from extracted noise
parameters. We simulate the expected CPMG-1 decay using the filter formalism
as detailed in the Methods section, using the noise power parameters as
extracted from the CPMG experiment displayed in Fig. 5 of the main text:
$\sqrt{S_{V}}=24.7\leavevmode\nobreak\ \mu V/\sqrt{\text{Hz}}$,
$\gamma_{\text{Ge-73}}=1.48$ MHz/T, $S_{0,\text{hf}}=2.5\cdot
10^{12}\cos^{2}(\theta_{f_{\text{Q2}}})$, $\sigma_{\text{Ge-73}}=17$ kHz for
$\phi_{\mathbf{B}}=0\degree$, and $\sigma_{\text{Ge-73}}=9$ kHz for
$\phi_{\mathbf{B}}=-105\degree$. We then fit the simulated decay using the
same procedure as used in Fig. 3f of the main text to extract the envelope
$T_{2}^{\text{H}}$. Markers indicate the experimental results and the solid
lines correspond to the envelope decay time as predicted by the model. The
excellent agreement between the simulation and data, without the need for any
fitting parameters, confirms our understanding of the system. The
$1/f_{\text{Q2}}$ decay of the envelope coherence can be explained by an
effective voltage noise on plunger gate P2, while the low-$B$ drop-off is
caused by the finite spread of the 73Ge precession frequencies. The shaded
area indicates the uncertainty in $T^{2}_{\text{H}}$, given an uncertainty of
$\pm 20\leavevmode\nobreak\ \mu$T in the $z$-component of the magnetic field.
For very small $B$, this yields a significant uncertainty in
$\theta_{\mathbf{B}}$, thus complicating an accurate prediction of
$T^{2}_{\text{H}}$.