v_{\alpha}},\theta,\phi\right)\right)=\sum_{\alpha}\left(\frac{\Delta\mathbb{E}^{3}}{8\pi^{2}\hbar^{4}\rho
v_{\alpha}^{5}}\,\Omega_{\alpha}\\!\left(\frac{\Delta\mathbb{E}}{\hbar
v_{\alpha}}\right)\right)$

where $\Omega$ denotes the angular integration.

#### Dipole approximation.

Alternatively, one can expand $e^{i\mathbf{q}\cdot\mathbf{r}}\approx
1(+i\mathbf{q}\cdot\mathbf{r}-|\mathbf{q}\cdot\mathbf{r}|^{2}+..)$ which
simplifies the product
$\left|\left\langle\mathbb{0}\right|e^{i\mathbf{q}\cdot\mathbf{r}}H^{LKBP}_{strain,\alpha}\left|\mathbb{1}\right\rangle\right|^{2}\approx\left|\left\langle\mathbb{0}\right|H^{LKBP}_{strain,\alpha}\left|\mathbb{1}\right\rangle+i\left\langle\mathbb{0}\right|\mathbf{q}\cdot\mathbf{r}H^{LKBP}_{strain,\alpha}\left|\mathbb{1}\right\rangle-\left\langle\mathbb{0}\right||\mathbf{q}\cdot\mathbf{r}|^{2}H^{LKBP}_{strain,\alpha}\left|\mathbb{1}\right\rangle+..\right|^{2}$.
In sharp contrast to electron spin-$\frac{1}{2}$ qubits, where the local
strain is a diagonal tensor and the leading zeroth order term
$\left\langle\mathbb{0}\left|H_{\varepsilon}\right|\mathbb{1}\right\rangle$
vanishes; for the spin-$\frac{3}{2}$ holes the leading term is the zeroth
order, resulting in a $\alpha_{r}B^{3}+\beta_{r}B^{4}+\gamma_{r}B^{5}$
relaxation rate variation, compared to $B^{5}$ variation in electron
spin-$\frac{1}{2}$ qubits. While the $B^{3}$ and $B^{5}$ dependence are
explained by the first two terms in dipole approximation, the orbital B-terms
in the qubit admixture give rise to the $B^{4}$ dependence.

##  Appendix D Random Telegraph Noise(RTN) Dephasing: Screened Potential of a
charge defect

Taking into account the screening effect of the 2DHG being formed in Ge, the
Fourier $q$-space form of the potential of single defect with charge $e$ is
given by:

$U_{scr}(q)=\frac{e^{2}}{2\epsilon_{0}\epsilon_{r}}e^{-qd}\frac{\Theta\left(2k_{F}-q\right)}{q+q_{TF}}.$
(D64)

where $U_{scr}(q)$ is known as the Thomas-Fermi screened potential, $q$ is the
Fourier space variable, $q_{\text{TF}}{=}0.49\,\text{nm}^{-1}$ is the Thomas-
Fermi wave vector in germanium independent of the density of holes. $k_{F}$ is
the Fermi wave vector which is estimated to be $0.1\text{nm}^{-1}$ in our
calculations, and $\Theta$ is the Heaviside Theta step function. Considering
constant screening, which only holds for $q{<}2k_{F}$, the screened potential
in real space under the limit $d{\ll}r$ is approximated to be:

$U_{s}(\bm{r})=\frac{e^{2}}{4\pi\epsilon_{0}\epsilon_{r}}\frac{1}{q_{\text{TF}}^{2}}\left(\frac{1}{\left|\mathbf{r}-\mathbf{r}_{D}\right|^{3}}+d\frac{q_{\text{TF}}}{\left|\mathbf{r}-\mathbf{r}_{D}\right|^{3}}\right)$
(D65)

The relative electrical permeability of germanium is $\epsilon_{r}=15.8$;
$\epsilon_{0}$ being the vacuum electrical permeability.
$\mathbf{r}_{D}{=}\\{80\,\text{nm},80\,\text{nm},0\\}$ is the vector denoting
the distance of the in-plane charge defect from the center of the QD.

##  Appendix E $\bm{g}$-factor anisotropy and fitting parameters

In the main text, Fig.8, we mentioned that the fitting paramters are not
unique. Here we suggest some other possible configurations to fit the
parameters. The main fitting parameters are dot size ($L_{x},L_{y},L_{z}$),
the electric field can be tuned via gate to modulate the $g$-factors.

  1. 1.

$L_{x}$=40 nm, $L_{y}$=60 nm $L_{z}$ = 10 nm

  2. 2.

$L_{x}$=30 nm, $L_{y}$=50 nm $L_{z}$ = 10.5 nm

  3. 3.

$L_{x}$=44 nm, $L_{y}$=60 nm $L_{z}$ = 9.5 nm

Note that there are many other possible combinations of parameters if the
strains (both uniaxial strain and shear strain) are included.