# Acceptor-based qubit in silicon with tunable strain

Shihang Zhang  Yu He<EMAIL_ADDRESS>Peihao Huang<EMAIL_ADDRESS>Shenzhen Institute for Quantum Science and Engineering, Southern University of
Science and Technology, Shenzhen 518055, China  International Quantum Academy,
Shenzhen 518048, China  Guangdong Provincial Key Laboratory of Quantum Science
and Engineering, Southern University of Science and Technology, Shenzhen
518055, China

###### Abstract

Long coherence time and compatibility with semiconductor fabrication make spin
qubits in silicon an attractive platform for quantum computing. In recent
years, hole spin qubits are being developed as they have the advantages of
weak coupling to nuclear spin noise and strong spin-orbit coupling (SOC), in
constructing high-fidelity quantum gates. However, there are relatively few
studies on the hole spin qubits in a single acceptor, which requires only low
density of the metallic gates. In particular, the investigation of flexible
tunability using controllable strain for fault-tolerant quantum gates of
acceptor-based qubits is still lacking. Here, we study the tunability of
electric dipole spin resonance (EDSR) of acceptor-based hole spin qubits with
controllable strain. The flexible tunability of LH-HH splitting and spin-hole
coupling (SHC) with the two kinds of strain can avoid high electric field at
the "sweet spot", and the operation performance of the acceptor qubits could
be optimized. Longer relaxation time or stronger EDSR coupling at low electric
field can be obtained. Moreover, with asymmetric strain, two "sweet spots" are
induced and may merge together, and form a second-order "sweet spot". As a
result, the quality factor $Q$ can reach $10^{4}$ for single-qubit operation,
with high tolerance for the electric field variation. Furthermore, the two-
qubit operation of acceptor qubits based on dipole-dipole interaction is
discussed for high-fidelity two-qubit gates. The tunability of spin qubit
properties in acceptor via strain could provide promising routes for spin-
based quantum computing.

††preprint: APS

##  I Introduction

Spin-based qubit in silicon is an important candidate platform for quantum
computation. The original framework of the spin qubit in semiconductor is
based on the electron spin and nuclear spin [1, 2]. In the past decades,
electron spin qubits in gate-defined quantum dots (QDs) are well developed [3,
4, 5, 6, 7]. In particular, high-fidelity ($>99\%$) single-qubit and two-qubit
gates are realized [8, 9]. In contrast, qubit based on hole spin is
established and developed [10, 11, 12, 13, 14, 15, 16, 17, 18] rather late and
less attention has been paid. However, strong intrinsic spin-orbit coupling of
holes allows all-electrical manipulation of qubits without additional
design[10]. Moreover, the suppressed coupling between a hole spin and nuclear
spins in host material also reduces pure spin dephasing [19, 20]. For hole
spin qubits, four-qubit device and long coherence time has been achieved [16,
17]. Except for spin qubit in gate-defined QDs, dopant-based spin qubit is
another potential option by using scanning tunneling microscope (STM)
lithography [21, 22, 23] or ion implantation [24]. And compared to the gate-
defined QDs, single-atom devices provide more a steady environment, which
induces a long relaxation time of qubits [25, 26]. And particles are confined
deeply by the potential of the atom nucleus, reducing the density of gates
[27]. High-fidelity two-qubit gates based on donor atoms have also been
realized [28]. Although both hole spin qubits and single-donor qubits are
developed intensively in experiments, single-acceptor-based hole spin qubits
are still less studied. Due to strong spin-orbit coupling and low density of
gates, acceptor-based qubits could have advantages for fast quantum gates,
long coherence time, and high scalability. It is shown that the operation of
acceptor-based qubits may be realized via electric dipole spin resonance
(EDSR) [12]. And a long coherence time of 10 ms of acceptor spins is
demonstrated [29].

Energy levels of spin states in the acceptor can be influenced by the
interaction with the electric field, magnetic field, and strain [30, 31]. The
acceptor system is more complex than donors due to these interactions and the
spin-3/2 system [32, 33, 34]. The hole spin states split into two degenerate
spin states, called heavy hole (HH) spin states and light hole (LH) spin
states (in host material). Experimentally, the readout of acceptor qubits in
Si:B device shows the special energy levels of the spin-3/2 system and long
relaxation time [35, 36, 29]. For now, the experiment on operations of
acceptor-based qubits is lacking. Traditionally, the qubit operation is via
electron spin resonance based on the oscillating magnetic field, which is hard
to generate locally and enhance. To avoid these difficulties, all-electrical
manipulation of spin qubits is expected. For electron spin qubit or hole spin
qubit in QD, the all-electrical qubit operation is realized by electric dipole
spin resonance (EDSR) based on spin-orbit coupling (SOC), engineered by
magnetic field gradient or hyperfine interaction (flip-flop) [37, 38, 39].
High-speed qubit operations based on EDSR are the key to high-fidelity qubit
gates. Similarly, the manipulation of the acceptor-based qubit states via EDSR
is proposed based on coupling between LH states and HH states with opposite
spin polarization, called spin-hole coupling (SHC) [12, 40].

However, SOC or SHC also makes qubits sensitive to charge noise, which induces
decoherence of qubits [4, 41]. Fortunately, the sensitivity of qubits to the
charge noise can be reduced even during operation with controlling pulse or
energy level engineering [42, 43, 44, 45]. For the acceptor qubit, there is an
operation point immune to the first-order electrical noise, named "sweet
spot", which was predicted theoretically [12, 40]. In the acceptor system, the
appearance of the sweet spot is a combined effect of LH-HH splitting and SHC.
And manipulation of acceptor-based qubits also depends on these mechanisms.
Thus, the LH-HH splitting and SHC are critical underlying physics for the spin
qubit operations. In the previous study, both of them are tuned solely by the
vertical electric field [40]. Consequently, the operation performance of the
qubits is limited since the electric field can not control the two quantities
(i.e. LH-HH splitting and SHC) independently. For example, in previous work,
to access the sweet spot too high electric field may be required for the
system, and the operation performance can not be improved easily [12], where
only a constant strain was introduced to make light hole states ground states
[12, 29]. Moreover, the tolerance to the electric field noise may be small. In
this work, we show these problems can be solved by introducing tunable strain
into the system. Furthermore, the tunability of the strain and the new SHC
mechanism induced by strain are considered in our work.

In this work, we study the electric manipulation of an acceptor spin qubit in
the presence of tunable strain. The operation performance of acceptor-based
qubits can be optimized by strain engineering. Firstly, strain can adjust the
LH-HH splitting $\Delta_{LH}$. When the qubits operate at the sweet spot, the
main decoherence comes from the relaxation due to phonon. Larger LH-HH
splitting induces longer relaxation time, which improves the coherence of
acceptor qubits. And importing SHC from asymmetric strain, the EDSR coupling
may be enhanced. Consequently, both single-qubit and two-qubit operation rates
are higher. With proper strain, two sweet spots for the electric field appear
at low electric field. The two sweet spots can merge together where a second-
order sweet spot immune to the charge noise appears. We find regions where the
spin qubit has high quality factor and high tolerance to the electric field at
the same time. As a result, with tunable strain, high-fidelity single-qubit
and two-qubit gates beyond the fault-tolerant threshold could be constructed
based on all-electrical manipulations. The two-qubit gates can be realized
with long-range coupling between two qubits. In conclusion, we demonstrate a
feasible scheme based on acceptor spin qubits for a large-scale fault-tolerant
quantum computer.

This paper is developed as follows: In Sec. II, the model and Hamiltonian of
the system with strain are introduced. Based on that, the qubit definition and
EDSR of the acceptor qubit are detailed in Sec. III. Decoherence of the qubit
is also introduced. In Sec. IV, the results with effects of the strain are
discussed. And two-qubit operation via electric dipole-dipole interaction is
introduced in Sec. V. In the last section, the conclusion of this work is
given. An outlook on future research on this topic is mentioned.

Figure 1: The schematic diagram of the model. (a) The schematic diagram of the
device. Boron acceptors are implanted in silicon near the Si/SiO2 interface
with the depth of $d$. They are separated by the distance of R. Top gates (TG)
generate the vertical electric field. Side gates (SG) are used to apply in-
plane electric fields. The gray layer can be piezoelectric materials, which
stretch the silicon layer to produce strain. (b) The schematic diagram of the
silicon layer. The green edge arrows indicate the static strain induced by the
mismatch of lattice parameters. And the purple edge arrows are asymmetric
strain in $\hat{y}$ direction. Notice that the tensile strain on
$\hat{z}$-axis is not shown. The spin-3/2 states of the holes (yellow) bound
to the acceptor atoms (silver) are spin qubits. (c) Schematic of energy levels
of hole spin states. Due to the spin-orbit coupling, states on the six-fold
degenerate valence band maximum (VBM) are split into a two-fold degenerate
state named split-off band and a four-fold degenerate state. Then the four-
fold degenerate states can be divided into the heavy hole and light hole
states, by strain or electric field. The magnetic field further splits the
2-fold degeneracies of the spin states of the heavy hole and light hole. In
this work, the heavy hole spin state is always grounded. The qubit states are
spin-hole mixed states due to interaction with asymmetric strain and
$T_{d}$-interaction with electric field $E_{z}$. The LH-HH transition due to
the in-plane electric field plays an important role in EDSR.

##  II Model

As shown in Fig. 1, boron atoms are placed near the interface in the silicon
with a depth of $d$. In principle, the boron atom can be placed by using ion
implantation or STM lithography [46, 47, 35, 36, 23]. On the top of the
device, gate electrodes are used to manipulate the acceptor qubit. A magnetic
field is applied along the $\hat{z}$-axis perpendicular to the surface (see
Fig. 1 (a)). Strain ($\epsilon=\epsilon_{0}+\sum_{i=x,y}\epsilon_{ii1}$) is
introduced to the system in two ways: (i). Growing the silicon heterostructure
on the $\mathrm{Si}_{x}\mathrm{Ge}_{1-x}$ or using thermal expansion of
different materials [48, 29]. The strain created in this way is static and
’symmetric’ ($\epsilon_{xx}=\epsilon_{yy}=-(C_{11}/2C_{12})\epsilon_{zz}$)
[49]. $C_{11}$ and $C_{12}$ are the elastic stiffness constants for the
strain-stress tensor. In this work, the strain created in this way is labeled
as $\epsilon_{0}=\epsilon_{xx0}+\epsilon_{yy0}-\epsilon_{zz0}$, named
symmetric strain. (ii). Applying piezoelectric material to produce strain in a
certain direction [50, 51]. In this case, asymmetric in-plane strain can be
obtained ($\epsilon_{xx}\neq\epsilon_{yy}$). And it can be tuned by electric
field [51]. In this work, strain induced in this way is labeled as
$\epsilon_{ii1}$ $(i=x,y)$. An asymmetric strain
($\epsilon_{xx}\neq\epsilon_{yy}$) can induce new coupling mechanism like SHC
independent on the vertical electric field [52]. To tune the SHC without
changing LH-HH splitting, strain should have opposite deformation:
$\epsilon_{xx1}=-\epsilon_{yy1}$, called asymmetric strain. For simplicity,
asymmetric strain is denoted as:
$\epsilon_{1}=\epsilon_{xx1}-\epsilon_{yy1}=2\epsilon_{xx1}$. The system
Hamiltonian based on the device is:

$H=H_{\mathrm{Lut}}+H_{\epsilon}+H_{Z}+H_{E}.$ (1)

$H_{\mathrm{Lut}}$ is Luttinger Hamiltonian [53]. It’s a 4x4 matrix describing
the heavy hole and light hole in bulk. The split-off band is ignored here as
their energy levels are well separated from the states of interest.
$H_{\epsilon}$ is Bir-Pikus Hamiltonian describing the interaction with
strain, detailed in Supplementary [54]:

$\displaystyle
H_{\epsilon}=a^{\prime}Tr[\epsilon]+b^{\prime}((J_{x}^{2}-\frac{5}{4}I)\epsilon_{xx}+c.p.)$
(2)
$\displaystyle+(2d^{\prime}/\sqrt{3})(\\{J_{x},J_{y}\\}\epsilon_{xy}+c.p.),$

where $\epsilon_{ij}$ is strain, c.p. means cyclic permutations, the
$a^{\prime}$, $b^{\prime}$ and $d^{\prime}$ are deformation potential,
$\\{J_{x},J_{y}\\}=1/2(J_{x}J_{y}+J_{y}J_{x})$ is anti-commutator of spin-3/2
operator. $H_{Z}$ is the Zeeman Hamiltonian describing the interaction with
the magnetic field:

$H_{Z}=\mu_{B}[g_{1}(J_{x}B_{x}+c.p.)+g_{2}(J_{x}^{3}B_{x}+c.p.)].$ (3)

Only the $\hat{z}$ direction and its linear term is considered (the g factor
$g_{1}=1.07>>g_{2}$) [55]. Here, $+\hat{z}$-axis is pointing down towards
silicon. $H_{E}=H_{C}+H_{if}+H_{gate}+H_{T_{d}}$ includes Hamiltonian
describing interaction with electric field: $H_{C}=e^{2}/4\pi\epsilon_{s}r$ is
Coulomb potential, $r$ is position of hole relative to the nuclei of the
acceptor and $\epsilon_{s}$ is static dielectric constant of semiconductor.
$H_{if}=U_{0}\Theta(-z)$ is interaction with the interface potential. And
$H_{gate}=e\mathbf{E\cdot r}$ represents the interaction with the interface
gate field. These interface terms play an important role due to the large
transition between the light hole and heavy hole states, called LH-HH
coupling. $H_{T_{d}}=2pE_{x}\\{J_{y},J_{z}\\}/\sqrt{3}+c.p.$ is interaction
with the electric field due to the tetrahedral ($T_{d}$) symmetry of acceptor
in silicon [56]. The c.p. is cyclic permutation and $\\{A,B\\}=(AB+BA)/2$. In
the equation, $p=e\int_{0}^{a}f^{*}(r)rf(r)\textrm{d}r$ is effective dipole
moment, where $a$ is the lattice constant and $f(r)$ is radial bound hole
envelope function. The $J_{i}$ are matrices of the spin-3/2 for $i=x,y,z$.

To construct the matrix of the Hamiltonian, we define the spin states of the
heavy hole and light hole as basis states. As shown in Fig. 1. (c), the six-
fold degenerate state of the valence band in silicon is split into a two-fold
degenerate state named split-off band, and a topmost four-fold degenerate
state due to the SOC [57]. Near the interface between the silicon and the
dielectric layer, the topmost four-fold degenerate state of the valence band
with $J=3/2$ is separated into two doubly degenerate states. They can be named
heavy hole (HH) states ($|H\pm\rangle=|m_{J}=\pm 3/2\rangle$) and light hole
(LH) states ($|L\pm\rangle=|m_{J}=\pm 1/2\rangle$). Spin-orbit coupling mixes
states with $\Delta L=0,\pm 2$. Consequently, the wavefunctions (in bulk) of
the orbital ground states are linear combinations of states with $L=0$ and
$L=2$ [55]. And the orbital excited states are also linear combinations of
states with different $L$.

The Hamiltonian in the subspace of the orbital ground states
$\\{|H+\rangle,|H-\rangle,|L+\rangle,|L-\rangle\\}$ is:

$H=\left(\begin{array}[]{lccc}\varepsilon_{\mathrm{H}+}&0&-ipE_{+}+\alpha
E_{-}&t^{*}\\\ 0&\varepsilon_{\mathrm{H}-}&t&-ipE_{-}-\alpha E_{+}\\\
ipE_{-}+\alpha E_{+}&t^{*}&\varepsilon_{\mathrm{L}+}&0\\\ t&ipE_{+}-\alpha
E_{-}&0&\varepsilon_{\mathrm{L}-}\end{array}\right),$ (4)

where $\varepsilon_{i}$ represents the energy of state $i$
($i=|H\pm\rangle,|L\pm\rangle$):
$\varepsilon_{\mathrm{H}\pm}=\varepsilon_{H}\pm(3/2)\varepsilon_{Z}$,
$\varepsilon_{\mathrm{L}\pm}=\varepsilon_{L}\pm(1/2)\varepsilon_{Z}$
($\varepsilon_{Z}=g\mu_{B}B$). In this work, we let $g=1.07$ for holes in
silicon [58]. The off-diagonal terms (SHC)
$t=\mathrm{i}pE_{z}+\frac{\sqrt{3}}{2}b^{\prime}(\epsilon_{xx}-\epsilon_{yy})$
mix hole and spin states provided by not only $T_{d}$ interaction with
$E_{z}$, but also interaction with asymmetric strain. These terms mix
$|H+\rangle$ ($|H-\rangle$) and $|L-\rangle$ ($|H+\rangle$) defining the qubit
states, which will be discussed later. The $E_{\pm}=E_{x}\pm iE_{y}=Ee^{\pm
i\theta_{E}}$ are related to in-plane electric field. $\theta_{E}$ defines the
direction of the in-plane electric field. Terms related to $E_{\pm}$, include
two parts: One is the $T_{d}$ interaction with the in-plane electric field
with $p$. Another is interaction with interface potential and electric field
with coefficient $\alpha$, which is obtained by projecting the orbital first-
excited states onto the ground states via Schrieffer–Wolff transformation [59,
40]. Both of them couple the HH and LH states with spin states unchanged and
produce an LH-HH coupling, which will be a key mechanism to drive spin qubits
in this paper. Therefore, the time-dependent in-plane electric field can be
utilized to drive qubits. Combining the effect and the SHC terms, the
manipulation of acceptor qubit by a process similar to electric dipole spin
resonance (EDSR) can be achieved [37], which will be introduced in the
following section. The splitting between the heavy hole and light hole state
is [12]:

$\Delta_{LH}=\varepsilon_{L}-\varepsilon_{H}=\Delta_{if}+\Delta(E_{z})+\Delta_{\epsilon},$
(5)

where $\Delta_{if}$ is from the interface potential [60, 61], $\Delta(E_{z})$
depends on the gate electric field and
$\Delta_{\epsilon}=b^{\prime}(\epsilon_{xx}+\epsilon_{yy}-\epsilon_{zz})=b^{\prime}(\epsilon_{0}+\epsilon_{xx1}+\epsilon_{yy1})$.
In the model, tuning of strain can change the LH-HH splitting $\Delta_{LH}$
and SHC $t$ individually, which plays an important role in qubit operation. In
the following section, qubit definition and operation will be introduced.

##  III Qubit and operation

This section is the basis for the following section. IV. In this section, the
qubit, its operation, and decoherence with strain will be discussed. The
effect of LH-HH splitting and spin-hole coupling on operation performance will
be highlighted, for a better understanding of the results in the next section.

###  III.1 Qubit definition

Figure 2: Explanation of appearance of sweet spots without or with strain
($\epsilon_{xx}=-\epsilon_{yy}=-10^{-5}$) at $B=0.5$ T, $d=4.6$ nm. (a) LH-HH
splitting and SHC without strain ($\Delta_{0}$, $t_{0}$) and with strain
($\Delta_{\epsilon}$, $t_{\epsilon}$) are plotted as function of vertical
electric field $E_{z}$. In particular, SHC appears without the electric field
in the presence of asymmetric strain. (b) Proportion of $|H\pm\rangle$ in the
qubit states $|g\pm\rangle$ without strain ($N_{g\pm,0}$) and with strain
($N_{g\pm,\epsilon}$) are plotted as function of vertical electric field
$E_{z}$. (c) Qubit splitting without ($\hbar\omega_{0}$) and with strain
($\hbar\omega_{\epsilon}$) are plotted as a function of the vertical electric
field $E_{z}$ with strain. Corresponding with the variation of proportions of
states, qubit splitting increases at the low electric field. Then it turns to
decrease and is the same trend along the situation without strain.

From Eq. (4), we can define the acceptor qubit by diagonalization of static
parts of the Hamiltonian. Then, the manipulation of the qubit can be induced
by the time-dependent in-plane electric field, which will be included after
the diagonalization. To make sure the HH spin states are the ground states, we
assumed that $\Delta_{LH}>2\varepsilon_{Z}$. Then, after diagonalization, the
eigenvalues are:

$\displaystyle\lambda_{g+}$
$\displaystyle=\frac{1}{2}\left[\varepsilon_{\mathrm{H}+}+\varepsilon_{\mathrm{L}-}-\tilde{\Delta}_{-+}\right],$
(6) $\displaystyle\lambda_{g-}$
$\displaystyle=\frac{1}{2}\left[\varepsilon_{\mathrm{H}-}+\varepsilon_{\mathrm{L}+}-\tilde{\Delta}_{+-}\right],$
$\displaystyle\lambda_{\mathrm{e}+}$
$\displaystyle=\frac{1}{2}\left[\varepsilon_{\mathrm{H}-}+\varepsilon_{\mathrm{L}+}+\tilde{\Delta}_{+-}\right],$
$\displaystyle\lambda_{\mathrm{e}-}$
$\displaystyle=\frac{1}{2}\left[\varepsilon_{\mathrm{H}+}+\varepsilon_{\mathrm{L}-}+\tilde{\Delta}_{-+}\right],$

where $\tilde{\Delta}_{\mp\pm}=\sqrt{\Delta_{\mp\pm}^{2}+4t^{2}}$,
$\Delta_{\mp\pm}=\varepsilon_{\mathrm{L}\mp}-\varepsilon_{\mathrm{H}\pm}$ and
$t=\sqrt{(3/4)b^{{}^{\prime}2}\epsilon_{1}^{2}+p^{2}E^{2}}$ is SHC. And
corresponding eigenvectors:

$\begin{split}\left|g+\right\rangle&=\frac{1}{N_{1}}\left(\begin{array}[]{c}-\frac{\Delta_{-+}+\tilde{\Delta}_{-+}}{2}\\\
0\\\ 0\\\ t\end{array}\right)=\frac{1}{N_{1}}\left(\begin{array}[]{c}-a\\\
0\\\ 0\\\ t\end{array}\right),\\\
\left|g-\right\rangle&=\frac{1}{N_{2}}\left(\begin{array}[]{c}0\\\
\frac{\Delta_{+-}+\tilde{\Delta}_{+-}}{2}\\\ -t^{*}\\\
0\end{array}\right)=\frac{1}{N_{2}}\left(\begin{array}[]{c}0\\\ c\\\ -t^{*}\\\
0\end{array}\right),\\\
\left|\mathrm{e}+\right\rangle&=\frac{1}{N_{2}}\left(\begin{array}[]{c}0\\\
-t\\\ -\frac{\Delta_{+-}+\tilde{\Delta}_{+-}}{2}\\\
0\end{array}\right)=\frac{1}{N_{2}}\left(\begin{array}[]{c}0\\\ -t\\\ -c\\\
0\end{array}\right),\\\
\left|\mathrm{e}-\right\rangle&=\frac{1}{N_{1}}\left(\begin{array}[]{c}t^{*}\\\
0\\\ 0\\\
\frac{\Delta_{-+}+\tilde{\Delta}_{-+}}{2}\end{array}\right)=\frac{1}{N_{1}}\left(\begin{array}[]{c}t^{*}\\\
0\\\ 0\\\ a\end{array}\right),\end{split}$ (7)

where $N_{i}$ (i=1, 2) is normalization coefficients:

$\displaystyle N_{1}=\sqrt{|t|^{2}+a^{2}},N_{2}=\sqrt{|t|^{2}+c^{2}}.$ (8)

As mentioned in the last section, $T_{d}$ interaction with the vertical
electric field and asymmetric strain mix the hole and spin states. We define
our qubits on the lowest two states:
{$|0\rangle=|g-\rangle,|1\rangle=|g+\rangle$}. The qubit states are mixing of
the spin-hole states illustrated in Fig. 1. (c). They are mostly the HH spin
states. The qubit splitting can be obtained:

$\hbar\omega=\lambda_{g+}-\lambda_{g-}=2\varepsilon_{Z}-\tilde{\Delta}_{-+}+\tilde{\Delta}_{+-}.$
(9)

The qubit splitting depends directly on the LH-HH splitting $\Delta_{LH}$, SHC
$t$, and the magnetic field $B_{z}$. The derivative
$\partial\hbar\omega/\partial E_{z}=0$ defines the so-called sweet spot.
Operation of qubit at the sweet spot can be immune to electrical noise.
Moreover, when $\partial\hbar\omega/\partial E_{z}=0$ and
$\partial^{2}\hbar\omega/\partial E_{z}^{2}=0$, qubit is immune to the second
order electrical noise, defines the second-order sweet spot. To explain the
existence of the sweet spot, the compositions of qubit states should be
emphasized. Acceptor qubit state $|g+\rangle$ $(|g-\rangle)$ is a mixture of
$|H+\rangle$ $(|H-\rangle)$ and $|L-\rangle$ $(|L+\rangle)$. Therefore, the
energy splitting of eigenstates could be changed by tuning their mixing
proportion. The proportions of states in the qubit states are determined by
the relative strength of $\Delta_{LH}$ and $t=|t_{\pm}|$. With proper strength
of $\Delta_{LH}$ and $t$, the proportion of states will not be varied with the
electric field. Mathematically, the sweet spot appears at the extreme point of
qubit splitting. An example is given in Fig. 2 (c). For $d=4.6$ nm without
strain (blue line), there is a minimum around $E_{z}=12$ MV/m for qubit
splitting, which is the sweet spot. The mechanism behind this is the
competition between the SHC $t_{0}$ and LH-HH splitting $\Delta_{0}$. As shown
in Fig. 2 (a), the dominant factor is varying in different regions. When the
electric field $E_{z}<12$ MV/m, the mixing of states is enhanced due to the
increasing SHC $t_{0}$. However, when $E_{z}>12$ MV/m, the mixing of states is
reduced. That is because where the LH-HH splitting $\Delta_{0}$ increases
significantly and plays a dominant role. Thus, the sweet spot appears around
$E_{z}=12$ MV/m, which is the extreme point of qubit splitting.

The following section will show that another sweet spot could exist and can
merge with the first sweet spot in presence of asymmetric strain. We compare
the situation with and without strain ($\epsilon_{xx}=-\epsilon_{yy}=-10^{-3}$
% ), shown in Fig. 2. At $E_{z}=0$ point, despite the electric field
disappearing, the SHC $t$ still exists due to the asymmetric strain. Thus, at
$E_{z}=0$, the states are already mixed. In Fig. 2 (b), the components of
$|H+(-)\rangle$ in $|g+(-)\rangle$, defined as
$N_{H+(-)}=|a(c)/N_{1(2)}|^{2}$, is not equal to 1 in the absence of electric
field $E_{z}$. As mentioned above, the qubit splitting is determined by SHC
$t$ and LH-HH splitting $\Delta$. In Fig. 2. (c), with asymmetric strain
$\epsilon_{1}$ and within $0<E_{z}<2.5$ MV/m, the qubit splitting
$\hbar\omega_{\epsilon}$ increases as the electric field $E_{z}$ increases. In
the region, the qubit splitting $\hbar\omega_{\epsilon}$ is mainly affected by
the variation of LH-HH splitting $\Delta_{LH}$. That is because as $E_{z}$
increases, $\Delta_{LH}$ is increased while the variation of the SHC
$t_{\epsilon}$ is negligible, where the strain-induced SHC dominates over the
electric-field-induced SHC. However, when the electric field $2.5<E_{z}<10$
MV/m, the qubit splitting $\hbar\omega_{\epsilon}$ starts decreasing because
the SHC $t_{\epsilon}$ is increasing, where the electric-field-induced SHC
becomes dominant. In addition to the sweet spot discussed in the last
paragraph, a new sweet spot appears at around $E_{z}=2.5$ MV/m, in the
presence of the asymmetric strain. The two sweet spots are getting close to
each other for larger SHC induced by the strain. Later on, we show that the
two sweet spots can merge together and form a second-order sweet spot with
tunable strain.

###  III.2 Operation

Figure 3: Explanation of variation of the EDSR coupling when $d=4.6$ nm, B=0.5
T. To simplify the process, we assume that there is no strain in (a) and (b).
(a) The variation of the ratio factor of states as $E_{z}$ increases. (b) The
EDSR coupling $D$ and its factors $p_{eff}$ and $C_{1}$. (c). The enhancement
of EDSR coupling $D$ by asymmetric strain with the vertical electric field
$E_{z}=7$ MV/m. The asymmetric strain ($\epsilon_{xx}=-\epsilon_{yy}$) is
assumed to introduce SHC without changing LH-HH splitting.

The qubit operation of the acceptor qubit is induced by utilizing the LH-HH
transition, providing state mixing. As mentioned above, the qubit states are
the spin-hole mixed states. The $|H\mp\rangle$ in $|g\mp\rangle$ can be
coupled to $|L\mp\rangle$ in $|g\pm\rangle$ by LH-HH transition, which can be
used to realize single-qubit operation. Specifically, an applied AC electric
field can modulate the LH-HH transition. And thus, the LH-HH transition
provides coherent driving of the acceptor qubit. Single qubit operation is
obtained by writing these terms on the qubit basis:

$H_{E_{\parallel}}^{\prime}=\left(\begin{array}[]{lccc}0&C_{1}(\alpha
E_{-}-ipE_{+})&C_{2}(\alpha E_{-}-ipE_{+})&0\\\ C_{1}^{*}(\alpha
E_{+}+ipE_{-})&0&0&-C_{2}(\alpha E_{+}+ipE_{-})\\\ C_{2}(\alpha
E_{+}+ipE_{-})&0&0&C_{1}(\alpha E_{+}+ipE_{-})\\\ 0&-C_{2}(\alpha
E_{-}-ipE_{+})&C_{1}^{*}(\alpha E_{-}-ipE_{+})&0\end{array}\right),$ (10)

where $C_{1}=t^{*}(a-c)/N_{1}N_{2}$, $C_{2}=(|t|^{2}+ac)/N_{1}N_{2}$. The EDSR
dipole moment is

$D=p_{eff}|C_{1}|=|t|p_{eff}\frac{|a-c|}{N_{1}N_{2}},$ (11)

where $p_{eff}=\sqrt{\alpha^{2}+p^{2}+2\alpha p\sin(2\theta_{E})}$ is the
effective LH-HH transition. In this work, we assume that $E_{x}=E_{y}$, i.e.
$p_{eff}=|\alpha+p|$. Once the electric dipole moment $D$ is obtained, the
acceptor qubit operation can be modulated by the electric dipole transition
$DE_{ac}$, where $E_{ac}$ is an in-plane alternating electric field. The Rabi
frequency can be calculated: $\omega_{R}=DE_{ac}/h$. The strength of $D$ is
determined by $p_{eff}$ and $C_{1}$. The $p_{eff}$ includes $p$ and $\alpha$,
which depends on the electric field $E_{z}$. However, the interface-induced
$\alpha$ is much larger than $p$ induced by electric field $E_{z}$ [40].
Remarkably, $C_{1}$ comes from difference between $a$ and $c$. Here, the
$\frac{|t|a(c)}{N_{1}N_{2}}$ represents the transition between $|H+(-)\rangle$
and $|L+(-)\rangle$ in qubit states, which depends on the magnetic field $B$,
LH-HH splitting $\Delta_{LH}$ and SHC $t$. The influence of $\Delta_{LH}$ and
$t$ is plotted in Fig. 3. (a). The dependences of $a$ and $c$ on the LH-HH
splitting $\Delta_{LH}$ and spin-hole coupling $t$ are different. The $a$ is
more sensitive to the change due to the smaller splitting between $|H+\rangle$
and $|L+\rangle$. As shown in Fig. 3 (b), the trend of the $p_{eff}$ and
$|C_{1}|$ is opposite in two regions. Thus, EDSR coupling $D$, which is the
product of them, has two peaks in each region. In Fig. 3 (b), there is a dip
around the vertical electric field of 2.5 MV/m. This is mainly because
$p_{eff}$ approaches zero around 2.5 MV/m. Moreover, the larger magnetic field
can enhance the difference between a and c. That is because larger Zeeman
splitting reduces $a$ (less $|H+\rangle$ in $|g+\rangle$), and enhances $c$
(more $|H-\rangle$ in $|g-\rangle$). In conclusion, $D$ can be enhanced by the
magnetic field, the LH-HH transition $p_{eff}$ and SHC $t$, and be lowered by
LH-HH splitting $\Delta_{LH}$.

For example, SHC $t$ induced by asymmetric strain can enhance the EDSR
coupling, shown in Fig. 3. (c). However, the enhancement is maximized around
$|\epsilon_{xx}|=0.01$ %. That is because the coupling between $|H+\rangle$
and $|L-\rangle$ in $|g+\rangle$ stops increasing when the SHC is large
enough. Meanwhile, the coupling in between $|H-\rangle$ and $|L+\rangle$ in
$|g-\rangle$ is still increasing. Then, the difference $C_{1}$ is reduced.

###  III.3 Decoherence

Coherence time $T_{2}$ is as crucial as operation speed for quantum
computation. The performance of the qubits can be estimated by the quality
factor $Q=\omega_{R}T_{2}/2\pi$, which is the operation times of a full
rotation before the qubit states decohere. In silicon, the dephasing of hole
spin qubit due to the hyperfine interaction can be reduced by isotopic
purification [62, 63, 64]. Thus, the pure dephasing of the acceptor qubit is
mainly induced by the charge noise. The pure dephasing rate $1/T_{\phi}=\delta
E^{2}\tau/(2\hbar^{2})$ [41] is calculated in supplementary materials [54],
considering the energy fluctuation $\delta E$ caused by the extra electric
field to the second order. The electric field due to the defect is assumed as
3380 V/m [40]. The energy fluctuation depends on the derivative
$\partial\hbar\omega/\partial E_{z}$ and $\partial^{2}\hbar\omega/\partial
E_{z}^{2}$. As mentioned above, for the acceptor-based qubit, there exist
sweet spots where the qubit splitting is insensitive to the variation of the
electric field. Thus, operation at the sweet spot can reduce the dephasing
related to the electrical noise. From our model, the pure dephasing is greatly
suppressed at the sweet spot, where the relaxation becomes the main
decoherence source of the qubit. This is further illustrated in the following
section.

The main source of relaxation of hole spin qubits is spin-phonon interaction
via deformation potential [65, 66]. The spin relaxation of the qubits can be
calculated as

$1/T_{1}=\frac{(\hbar\omega)^{3}}{20\hbar^{4}\pi\rho}(C_{1})^{2}\left[2d^{{}^{\prime}2}\left(\frac{2}{3v_{l}^{5}}+\frac{1}{v_{t}^{5}}\right)\right],$
(12)

where $\rho=2330$ kg/m3 is the mass density, $d^{\prime}=-3.7$ eV is the
deformation potential [67], $v_{l}=899$ m/s and $v_{t}=1.7v_{l}$ are the
longitudinal and transverse sound velocities in silicon. The relaxation rate
has a quadratic dependence on the $C_{1}$. By contrast, EDSR coupling $D$
depends on $C_{1}$ linearly. As mentioned above, $C_{1}$ is determined by SHC
$t$ and LH-HH splitting $\Delta_{LH}$. That means relaxation of qubit will be
more sensitive to the variation of parameters of the system, than the EDSR
operation rate. For example, as shown in Sec. IV of supplementary material
[54], $T_{1}$ is enhanced four orders of magnitude when $E_{z}=10$ MV/m,
compared to the case in the absence of strain. While the EDSR coupling is
enhanced two orders of magnitude with the same condition in Fig. 4. (a). As a
result, the quality factor (the number of operations within coherence time) is
enhanced when the larger LH-HH splitting is induced by strain.

##  IV Effect of strain

Figure 4: Effect of symmetric and asymmetric strain. $B=0.5$ T,
$E_{ac}=10^{4}$ V/m, $d=4.6$ nm. (a), (b), (c) are all plotted as function of
symmetric strain $\epsilon_{0}$ and vertical electric field $E_{z}$. (a) The
EDSR coupling $D$. (b) Decoherence time $T_{2}$ depends on the relaxation time
$T_{1}$ and the pure dephasing time $T_{\phi}$. At the sweet spot, the
relaxation due to phonon leads the decoherence. (c) The quality factor $Q$. In
general, the quality factor $Q$ is enhanced from hundreds to $10^{4}$. The
performance of the qubit operation is better with the strain. However,
observing (b), the sweet spot requires a larger electric field with higher
static strain, which can be solved by SHC induced by asymmetric strain, seeing
(d), (e), (f). For these figures, the spin-hole coupling is enhanced by the
difference between the strain in $\hat{x}$ and $\hat{y}$ direction. And they
are all plotted as a function of asymmetric strain $\epsilon_{1}$ and vertical
electric field $E_{z}$. (d) The EDSR coupling $D$. (e) The decoherence time
$T_{2}$. There are two sweet spots for proper strain. (f) The quality factor
$Q$. The quality factor is almost unchanged and slightly smaller with
asymmetric strain.

This section shows the effect of strain and is divided into two parts. In the
first part, we discuss qubit operation performance by tuning LH-HH splitting
$\Delta_{LH}$ and SHC $t$ with strain separately. And then, the LH-HH
splitting and SHC will be tuned simultaneously. The optimal points for qubit
operation are found by plotting them as a function of strain and electric
field.

In this work, we assume: The depth of the acceptor $d=4.6$ nm, the magnetic
field $B=0.5$ T, and the strength of the in-plane electric field
$E_{ac}=10^{4}$ V/m. For silicon, the Bir-Pikus deformation potential
$b^{\prime}=-1.42$ eV and $d^{\prime}=-3.7$ eV[67]. The LH-HH splitting
$\Delta_{LH}$, effective dipole moment $p$, and interface-induced spin-hole
coupling $\alpha$ depend on the wavefunctions of hole spin states. They could
be calculated numerically [55, 61]. Given that they have been already
calculated, we take them from a previous work [40]. In the first part, the
symmetric strain is set as $\epsilon_{0}\in[-0.1,0.003]$ $\%$, to ensure heavy
hole states are the ground states. And the asymmetric strain is set as
$\epsilon_{1}\in[-0.003,0.003]$ $\%$.

###  IV.1 Effect of ’symmetric’ and ’asymmetric’ strain

To show the effect of tuning of LH-HH splitting on the qubit operation, a
’symmetric’ strain is considered, which means $\epsilon_{xx}=\epsilon_{yy}$.
For simplicity, the strain is denoted as
$\epsilon_{0}=\epsilon_{xx}+\epsilon_{yy}=2\epsilon_{xx}$. Thus the LH-HH
splitting induced by the strain (mostly compressed) is
$\Delta_{\epsilon}=b^{\prime}\epsilon_{0}$. Note that $b^{\prime}$ is
negative, $\Delta_{\epsilon}$ grows as strain decreases. The EDSR coupling
$D$, decoherence time $T_{2}$ and quality factor $Q$ are plotted in (a), (b)
and (c) of Fig. 4. EDSR coupling is reduced due to the larger LH-HH splitting
in the presence of strain. In Fig. 4. (b), the coherence time $T_{2}$ is
dominated by the dephasing. However, at sweet spots, the decoherence is mainly
due to relaxation. The relaxation time $T_{1}$ is reduced with more negative
symmetric strain. From Section. II, when the LH-HH splitting increases (strain
$\epsilon_{0}$ decreases), the proportion number $|C_{1}|$ increases.
Consequently, the EDSR coupling decreases and relaxation time increases,
inferred from their dependences $|C_{1}|$. Moreover, the variation of
relaxation time is square to that of EDSR coupling. In general, the quality
factor $Q$ raises with more negative strain, as recognized in Fig. 4. (c). And
from Fig. 4. (b), the sweet spot requires a higher vertical electric field
$E_{z}$ with stronger symmetric strain $\epsilon_{0}$. Therefore, there is a
trade-off between the quality factor and the required vertical electric field
and strain. This is because the splitting $\Delta_{LH}$ increases as
$\epsilon_{0}$ becomes more negative, then the strength of the SHC required to
obtain the sweet spot is larger. For now, the only source of SHC is
interaction with the vertical electric field $E_{z}$. In short, qubit
operation with high quality factor $Q$ can be obtained with large symmetric
strain, despite requiring a high vertical electric field $E_{z}$.

By applying the strain in x and y direction with opposite deformation
($\epsilon_{xx1}=-\epsilon_{yy1}$, that is asymmetric strain), the spin-hole
coupling, $t=\sqrt{3}b^{\prime}\epsilon_{1}/2+ipE_{z}$ is tuned independently
by asymmetric strain, while LH-HH splitting by strain is fixed. The EDSR
coupling $D$, decoherence time $T_{2}$ and quality factor $Q$ are plotted in
(d), (e) and (f) of Fig. 4. Certainly, the effect of SHC on the qubit
operation is symmetric for positive and negative $\epsilon_{1}$. In Fig. 4
(d), in general, EDSR coupling is enhanced. And the EDSR coupling is weak
around $E_{z}=2.5$ MV/m. The enhancement in EDSR coupling is due to the larger
$C_{1}$ with extra SHC from asymmetric strain. For weak electric fields, the
electric field-induced SHC is weak. Thus, the enhancement due to strain is
more significant. However, when the electric field is large, there is no
significant influence of SHC induced by strain on EDSR coupling. The dip
appears despite enhanced $C_{1}$ due to the weak $p_{eff}$. The other impact
of tuning of SHC is on the decoherence of the qubit. In Fig. 4 (f), with the
tuning of strain, high values of $Q$ can be found in a wide region of the
electric field $E_{z}\in[0,11]$ MV/m. The quality factor $Q$ is slightly
reduced, compared to the case without asymmetric strain $\epsilon_{1}$. That
is because $|C_{1}|$ is enhanced by SHC induced by asymmetric strain. And
$p_{eff}$ is weaker at low electric fields. However, the electric field at the
first-order sweet spot can be tuned by asymmetric strain. And the appearance
of the second-order sweet spot, induced by asymmetric strain, protects the
qubit in a wider electric field region. Moreover, the quality factor $Q$ is
not reduced too much at the second-order sweet spot.

Figure 5: The combined effect of the strain. $B=0.5$ T, $E_{ac}=10^{4}$ V/m,
$d=4.6$ nm. The window of the vertical electric field is set as $[1,20]$ MV/m.
The figures are plotted as a function of strain $\epsilon_{xx}$ and
$\epsilon_{yy}$. (a) The largest quality factor $Q_{max}$ is obtained by
varying with strain $\epsilon_{xx}$ and $\epsilon_{yy}$. For $Q_{max}<10$, let
$log_{10}(Q_{max})=1$. (b) The vertical electric field $E_{z}$ corresponds to
$Q_{max}$, is denoted as $E_{max}$. The electric field $E_{max}$ becomes
weaker away from the diagonal. For $Q_{max}<10$, let $E_{max}=10$ MV/m. (c)
The efficiency of $Q_{max}$ on electric field $E_{z}$. Figure 6: Qubit
performance around an optimal region for operation. $B=0.5$ T, $E_{ac}=10^{4}$
V/m, $d=4.6$ nm. The asymmetric strain is set as $\epsilon_{yy}=-0.1$ %. And
the figures are plotted as a function of the vertical electric field $E_{z}$
and strain $\epsilon_{xx}$. (a). The EDSR coupling $D$. (b). The decoherence
time $T_{2}$. Two sweet spots appear. (c). The quality factor $Q$.

###  IV.2 Combined effect of strain

According to the results shown in the last subsection, a ’symmetric’ strain
brings high quality factor requiring a large vertical electric field. An
’asymmetric’ strain tunes the location of sweet spots without changing $Q$ too
much. In this part, we show that qubit operation can be optimized by tuning
the LH-HH splitting and SHC with strain. In Fig. 5, quantities are plotted as
function of $\epsilon_{xx}$ and $\epsilon_{yy}$. In Fig. 5. (a), $Q_{max}$ is
the maximum of quality factor $Q$ relative to the vertical electric field
$E_{z}\in[1,20]$ MV/m, with a given strain. High $Q_{max}$ is concentrated
around the diagonal area, called high-Q area ($Q_{max}>10^{3}$) below. The
high $Q_{max}$ is obtained by operating qubits at the sweet spot. The high-Q
area is at the first sweet spot. That is because $Q$ is larger for larger LH-
HH splitting $\Delta_{LH}$ at the first sweet spot. Out of the area, there is
no sweet spot for the strain condition, or the electric field corresponds to
it out of the range [1,20] MV/m. In Fig. 5. (b), the electric field $E_{max}$
corresponding to the $Q_{max}$ is plotted in Fig. 5. (b). In the high-Q area
as shown in Fig. 5. (a), the $E_{max}$ becomes smaller with larger difference
between $\epsilon_{xx}$ and $\epsilon_{yy}$. And in the upper right corner,
$Q_{max}$ can exceed 1000 with weak strain and electric field.

Figure 7: Operation performance for $\sqrt{\text{SWAP}}$ gate. $B=0.3$ T,
$E_{ac}=10^{4}$ V/m, $d=4.6$ nm and $R_{12}=25$ nm. The window of the vertical
electric field is set as $[1,20]$ MV/m. (a) The fidelity
$F_{\sqrt{\text{SWAP}}}$, corresponding to the $Q_{max}$ for
$\sqrt{\text{SWAP}}$ gate. (b) The SWAP operation time
$\tau_{\sqrt{\text{SWAP}}}$ corresponds to $Q_{\sqrt{\text{SWAP}}}$. (c) The
dephasing time $log_{10}(T_{\phi,J})$ due to the variation of $J_{dd}$
corresponding to $Q_{\sqrt{\text{SWAP}}}$. The dephasing is weaker than
relaxation at the sweet spot. The $\sqrt{\text{SWAP}}$ operation time
$\tau_{\sqrt{\text{SWAP}}}$ corresponds to $Q_{\sqrt{\text{SWAP}}}$.

To clarify the trade-off between the quality factor and electric field,
$log_{10}(Q_{max})/E_{max}$ is plotted in Fig. 5. (c). There is a highly
efficient line preserving high quality factors with low electric fields.
Actually, it is reasonable. The electric field $E_{max}$ on the bright line
corresponds to the ’sweet region’, where the two sweet spots merge together.
And similarly, to find the optimal point for tolerance to the electric field.
The full width at half maximum (FWHM) of $E_{max}$, defined as $E_{tol}$, is
plotted in Sec. V of the supplementary material [54]. We find $E_{tol}>1000$
V/m in most of the high-Q area, which is important for a robust qubit
operation. In conclusion, an optimal region for qubit operation is found.

We choose a region in Fig. 5 that assuming $\epsilon_{yy}=-0.1$ $\%$ and
$\epsilon_{xx}\in[-0.09,-0.08]$ $\%$. The results are plotted in Fig. 6. In
Fig. 6. (a), the EDSR coupling is slightly reduced around the sweet spot,
compared with that without strain. In Fig. 6. (b), due to the enhancement of
LH-HH splitting, relaxation is weaker in general. And with the aid of
asymmetric strain, sweet spots can be obtained with a lower electric field.
Around $\epsilon_{0}=-0.085$ %, there is a second-order sweet spot with a high
tolerance for the vertical electric field $E_{z}$. The decoherence of the
qubit is determined by both relaxation and dephasing. Thus, around the sweet
spots, the quality factor $Q$ can exceed $10^{4}$ in a wide region. For a
small electric field, $Q$ still can exceed $10^{3}$.

##  V Two-qubit entanglement: dipole-dipole interaction

Long-range entanglement of spin qubits is crucial for the realization of
scalable quantum circuits. Compared with short-range entanglement via
Heisenberg exchange [68, 69], the long-range scheme mitigates the difficulties
of fabrication by reducing the density of gates [27]. For acceptor-based spin
qubits, long-range entanglement can be realized by electric dipole-dipole
interaction [12, 70]. The interaction is based on the Coulomb interaction of
qubits with each other. A $\sqrt{\text{SWAP}}$ gate can be constructed with
coupling $J_{dd}\approx\frac{D^{2}}{4\pi\epsilon R_{12}^{3}}$, see Sec. VI of
the supplementary material [54]. In the equation, $D$ is EDSR coupling,
$\epsilon$ is permittivity for silicon, and $R_{12}$ is the distance between
the acceptors. Operation time of $\sqrt{\text{SWAP}}$ gate is:
$\tau_{\sqrt{\text{SWAP}}}=h/4J_{dd}$. The fidelity $F_{\sqrt{\text{SWAP}}}$
corresponding to $Q_{max}$ of $\sqrt{\text{SWAP}}$ gate is plotted as function
of the strain $\epsilon_{xx}$ and $\epsilon_{yy}$. The distance between the
acceptor atoms $R_{12}$ is 25 nm. And the magnetic field $B$ is 0.3 T, which
is smaller than that used for single-qubit gates. A smaller magnetic field can
enhance the fidelity of the qubit operation, see Sec. II of the supplementary
material [54]. As a result, the fidelity $F_{\sqrt{\text{SWAP}}}$ can exceed
the fault-tolerance threshold [71]. In particular, the fidelity can even reach
up to 99.9 %, when the strain
$-0.07\%<\epsilon_{xx}\approx\epsilon_{yy}<-0.05\%$.

The region with high quality factor for $\sqrt{\text{SWAP}}$ is similar to the
high-Q area for single-qubit operation. The enhancement effect of symmetric
strain on $Q_{\sqrt{\text{SWAP}}}$ is less substantial for the
$\sqrt{\text{SWAP}}$ gate. That is because the strength of the coupling
$J_{dd}$ has a quadratic dependence on the EDSR coupling $D$. When the
relaxation time increases, the operation time $\tau_{\sqrt{\text{SWAP}}}$
increases by same magnitude, see Fig. 7. (b). When the magnitude of strain is
weak, the fidelity $F_{\sqrt{\text{SWAP}}}$ is enhanced by asymmetric strain,
which is different from single-qubit gates. Except for those decoherence
sources for single-qubit operation, the decoherence of two-qubit entanglement
is actually induced by the variation of $\sqrt{\text{SWAP}}$ coupling
$J_{dd}$, in this region of weak strain. Similar to qubit splitting
$\hbar\omega$, the coupling $J_{dd}$ is influenced by charge noise, see
Supplementary [54]. When the strain is small, the pure dephasing due to
$J_{dd}$ could dominate the decoherence. However, when strain is more negative
($|\epsilon_{xx}|,|\epsilon_{yy}|>0.05\%$), the main decoherence is still from
relaxation due to phonon. In conclusion, a technical route to construct long-
range, high-fidelity, and highly feasible two-qubit gates is proposed.

##  VI Discussion and Conclusion

The tunable strain provides a knob for controlling the electric manipulation
of the boron-based heavy hole spin qubit and better feasibility compared with
the previous work [40, 12]. One may further optimize the operation performance
through variation of depth of the acceptor atom, choice of materials, and
magnetic field orientation. Moreover, the flexible tunability of the strain is
not limited to heavy-hole bound to boron in silicon. It is expected that the
qubit operation based on light hole spin could also be optimized with tunable
strain when the light hole spin states are the ground states. Light hole-based
qubit is also promising since it can be manipulated faster [12]. However,
there are two drawbacks for light hole qubits: (i) the light hole-based qubit
is more sensitive to the charge noise, and (ii) the sweet spots of light hole-
based qubits require much higher electric fields to produce. According to the
results of this paper, by introducing strain, we expect these problems could
be solved.

In conclusion, we investigate the effect of tunable strain on the acceptor
hole spin qubit. Compared to the vertical electric field, strain provides a
way to tune the key quantities: LH-HH splitting and spin-hole coupling of the
system independently. With aid of strain, the required electric field for
sweet spots can be lowered. And a second-order sweet spot appears, where the
qubit coherence is improved. At sweet spots, the relaxation, which dominates
the qubit decoherence, can be suppressed by tuning strain, and quality factor
$Q$ can be enhanced accordingly. A concrete parameter regime is specified for
high-fidelity quantum gates of hole spin qubits. For the strain
$-0.07\%<\epsilon_{xx}\approx\epsilon_{yy}<-0.05\%$, and $d=25$ nm, the
fidelity above 99.99 % for single-qubit operation and 99.9 % for two-qubit
operation can be achieved. Thus, all-electric qubit operations can be
constructed with fidelity well beyond the fault-tolerant threshold of quantum
computing, at a low electric field, and with a large separation between the
qubits, which is crucial for scaling up quantum processors. The proposed
scheme of the boron-based spin qubit with tunable strain could pave a way for
building a large-scale fault-tolerant spin-based quantum computer.

###### Acknowledgements.

This work is supported by the National Natural Science Foundation of China
(Grant Nos. 11904157, 62174076, 92165210), Shenzhen Science and Technology
Program (No. KQTD20200820113010023), and Guangdong Provincial Key Laboratory
(No. 2019B121203002).

## Supplementary information

##  Appendix A Strain

Interaction of holes with strain plays an important role. The interaction is
described by the Hamiltonian [52]:

$H_{\epsilon}=a^{\prime}Tr[\epsilon]+b^{\prime}((J_{x}^{2}-\frac{5}{4}I)\epsilon_{xx}+c.p.)+(2d^{\prime}/\sqrt{3})(\\{J_{x},J_{y}\\}\epsilon_{xy}+c.p.),$
(1)

where c.p. means cyclic permutations, $a^{\prime}$, $b^{\prime}$ and
$d^{\prime}$ are deformation potential,
$\\{J_{x},J_{y}\\}=1/2(J_{x}J_{y}+J_{y}J_{x})$ is anti-commutator. The matrix
form of it in the basis of hole spin states $\\{\Psi_{+3/2}$, $\Psi_{-3/2}$,
$\Psi_{+1/2}$, $\Psi_{-1/2}\\}$:

$\displaystyle H_{\epsilon}$
$\displaystyle=a^{\prime}\left(\begin{array}[]{cccc}\epsilon&0&0&0\\\
0&\epsilon&0&0\\\ 0&0&\epsilon&0\\\
0&0&0&\epsilon\end{array}\right)+b^{\prime}\left(\begin{array}[]{cccc}-\dfrac{1}{2}(\epsilon_{xx}+\epsilon_{yy})+\epsilon_{zz}&0&0&\dfrac{\sqrt{3}}{2}(\epsilon_{xx}-\epsilon_{yy})\\\
0&-\dfrac{1}{2}(\epsilon_{xx}+\epsilon_{yy})+\epsilon_{zz}&\dfrac{\sqrt{3}}{2}(\epsilon_{xx}-\epsilon_{yy})&0\\\
0&\dfrac{\sqrt{3}}{2}(\epsilon_{xx}-\epsilon_{yy})&\dfrac{1}{2}(\epsilon_{xx}+\epsilon_{yy})-\epsilon_{zz}&0\\\
\dfrac{\sqrt{3}}{2}(\epsilon_{xx}-\epsilon_{yy})&0&0&\dfrac{1}{2}(\epsilon_{xx}+\epsilon_{yy})-\epsilon_{zz}\end{array}\right)$
(2)
$\displaystyle+\dfrac{d^{\prime}}{\sqrt{3}}\left(\begin{array}[]{cccc}0&0&\sqrt{3}(\epsilon_{xz}-i\epsilon_{yz})&-i\sqrt{3}\epsilon_{xy}\\\
0&0&i\sqrt{3}\epsilon_{xy}&-\sqrt{3}(\epsilon_{xz}+i\epsilon_{yz})\\\
\sqrt{3}(\epsilon_{xz}+i\epsilon_{yz})&-i\sqrt{3}\epsilon_{xy}&0&0\\\
i\sqrt{3}\epsilon_{xy}&-\sqrt{3}(\epsilon_{xz}-i\epsilon_{yz})&0&0\end{array}\right).$

The Hamiltonian can be divided into three parts respectively corresponding to
$a^{\prime}$, $b^{\prime}$, and $d^{\prime}$, the three Bir-Pikus deformation
potentials. The first part is trivial in the form of the identity matrix. And
in the second part, there are two kinds of terms. The diagonal terms split
heavy hole and light hole. And the off-diagonal terms rotate spin and hole
state simultaneously, which is similar to $pE_{z}$-term from $H_{T_{d}}$.
These terms appear only if $\epsilon_{xx}$ and $\epsilon_{yy}$ are different,
which means linear strain in the x-y plane is asymmetric. The form of the last
part is similar to that of $H_{T_{d}}$. Again, there are similar terms for
rotating spin and hole state simultaneously. And the rest of the terms change
hole state the with orientation of spin reserved.

Figure 1: An extended version of Fig. 1 (c) in the main text. Schematic of
energy levels of hole spin states. Due to the spin-orbit coupling, states on
the six-fold degenerate valence band maximum (VBM) are split into a two-fold
degenerate state named split-off band and a four-fold degenerate state. Then
the four-fold degenerate states can be divided into the heavy hole and light
hole states, by strain or electric field. The magnetic field further splits
the 2-fold degeneracies of the spin states of the heavy hole and light hole.
In this work, the heavy hole spin state is always grounded. The qubit states
are spin-hole mixed states due to interaction with asymmetric strain and
$T_{d}$-interaction with electric field $E_{z}$. The LH-HH transition due to
the in-plane electric field plays an important role in EDSR.

##  Appendix B EDSR coupling

From Sec. III.2 of the main text, the EDSR coupling D is:

$D=p_{eff}C_{1}=|t|p_{eff}\frac{|a-c|}{N_{1}N_{2}}.$ (3)

The form here can be transformed to a more practical form with realistic
parameters:

$D=|\frac{|t|p_{eff}(-4\varepsilon_{Z}+\sqrt{(\Delta-2\varepsilon_{Z})^{2}+4|t|^{2}}-\sqrt{(\Delta+2\varepsilon_{Z})^{2}+4|t|^{2}})}{2\sqrt{[|t|^{2}+(\frac{\Delta-2\varepsilon_{Z}+\sqrt{(\Delta-2\varepsilon_{Z})^{2}+4|t|^{2}}}{2})^{2}][t^{2}+(\frac{\Delta+2\varepsilon_{Z}+\sqrt{(\Delta+2\varepsilon_{Z})^{2}+4|t|^{2}}}{2})^{2}]}}|,$
(4)

where the $\Delta$ is the LH-HH splitting, $t$ is SHC, $\epsilon_{Z}$ is the
Zeeman energy, and $p_{eff}$ is the effective LH-HH transition. The dependence
of the EDSR coupling $D$ on the magnetic field $B$ is plotted in Fig. 3. (a).
The EDSR coupling is almost linearly dependent on the magnetic field. Later
on, we show that the relaxation rate is more sensitive to the magnetic field,
showed in Fig. 3. (b). Thus, the quality factor can be enhanced by a smaller
magnetic field. High fidelity of the qubit operations can be achieved.

##  Appendix C Dephasing due to the charge noise

All-electrical manipulation of spin qubits can be controlled easily and is
beneficial to the integration of qubits. However, it is unavoidable to bring
decoherence from charge noise by the same path. Defects in the crystal induce
unexpected electric field $\mathcal{F}_{z}$, which is the main source of
charge noise. The dephasing rate is $1/T_{\phi}=V^{2}\tau/(2\hbar^{2})$ [41].
To calculate the energy fluctuation $V$ of the unexpected electric field on
qubit splitting, the derivative of the qubit splitting relative to the
electric field is needed:

$V=\dfrac{\partial\hbar\omega}{\partial
E_{z}}E_{z}+\dfrac{\partial^{2}\hbar\omega}{\partial E_{z}^{2}}E_{z}^{2}.$ (5)

Figure 2: Dephasing of spin qubits from $J_{dd}$. $B=0.5$ T, $E_{ac}=10^{4}$
V/m, $d=4.6$ nm and $R_{12}=25$ nm. The window of the vertical electric field
is set as $[1,20]$ MV/m. (a) The max value of the dephasing time
$log_{10}(T_{\phi,J}^{max})$. (b) The minimum value of the dephasing time
$log_{10}(T_{\phi,J}^{min})$. Except for the sweet spot, Dephasing from
$J_{dd}$ is weaker than that from qubit splitting $\hbar\omega$. (c) The
dephasing time due to the charge noise $T_{2,J}$ corresponds to
$Q_{\sqrt{\text{SWAP}}}$.

Qubit splitting is dependent on the Zeeman energy, LH-HH splitting
$\Delta_{LH}$, and spin-hole coupling $t$. Except for Zeeman energy and terms
from interaction with strain, all of them are relative to the electric field.
Thus, it is hard to show a compacted form of
$\frac{\partial\hbar\omega}{\partial E_{z}}$ and
$\frac{\partial^{2}\hbar\omega}{\partial E_{z}^{2}}$ here. However, we found
that the dephasing due to the second-order electrical noise is much weaker.
The first-order electrical noise plays an important role in decoherence.
Moreover, for the $\sqrt{\text{SWAP}}$ gate realized by dipole-dipole
interaction, the coupling strength $J_{dd}$ is also fluctuated by charge
noise. And the dephasing rate can be obtained similarly. The dephasing due to
the charge noise on $J_{dd}$ is weaker compared to that on qubit splitting out
of the sweet spot.

##  Appendix D Relaxation due to the interaction with phonon

When the hole spin qubit is operated at the sweet spot, the relaxation may
lead to the decoherence of the qubit. In this work, relaxation is obtained by
considering the interaction of holes with phonons. The relaxation rate is
calculated by the Fermi’s golden rule:

$\dfrac{1}{T_{n\rightarrow n^{\prime}}}=\dfrac{2\pi}{\hbar}|\langle
n^{\prime}|H^{\prime}|n\rangle|^{2}\rho(E_{n^{\prime}}).$ (6)

The relaxation is due to the interaction with phonon. Thus, perturbation
$H^{\prime}$ is hole-phonon interaction. Fermi’s golden rule transforms to:

$\dfrac{1}{T_{n\rightarrow
n^{\prime}}}=\dfrac{2\pi}{\hbar}\sum_{i,j,s,\mathbf{q}_{s}}|\langle
n^{\prime},n_{\mathbf{q}}+1|H_{\epsilon
ijs}|n,n_{\mathbf{q}}\rangle|^{2}\times\delta(E_{n}-E_{n}^{\prime}-\hbar\omega_{qs}).$
(7)

The transition from $|n\rangle$ to $|n^{\prime}\rangle$ is complicated by
emission of phonon. The energy of phonon is $\hbar\omega_{qs}=\hbar
v_{s}q_{s}$. The sum in the formula is the density of states with proper
energy. $\mathbf{q}_{s}$ is the phonon wave vector. And $s=l,t_{1},t_{2}$ are
the phonon polarization.
$\sum_{i,j}H_{\epsilon}=\sum_{i,j}D_{ij}\epsilon_{ijs}$ is the electron-phonon
interaction, where $D_{ij}$ are deformation potential matrices from Bir-Pikus
Hamiltonian. After some simplification and calculation steps [66, 65, 40], the
relaxation rate is:

$\frac{1}{T_{1}}=\frac{(\hbar\omega)^{3}}{20\hbar^{4}\pi\rho}\left[\sum_{i}\left|\left\langle
n^{\prime}\left|D_{ii}\right|n\right\rangle\right|^{2}\left(\frac{2}{v_{l}^{5}}+\frac{4}{3v_{t}^{5}}\right)+\sum_{i\neq
j}\left|\left\langle
n^{\prime}\left|D_{ij}\right|n\right\rangle\right|^{2}\left(\frac{2}{3v_{l}^{5}}+\frac{1}{v_{t}^{5}}\right)\right].$
(8)

Figure 3: EDSR coupling $D$ and relaxation rate $1/T_{1}$ due to interaction
with phonon plotted as functions of magnetic field B. Here, we take the
logarithm to the base 10 of them to clarify the difference. (a) The EDSR
coupling $D$ is almost linearly dependent on the magnetic field $B$. (b) The
blue (orange) line is calculated in the exact (perturbed) way. They have
similar dependence on the magnetic field. As the magnetic field increases,
relaxation from exact calculation becomes higher. In conclusion, the quality
factor $Q$ can be enhanced with the weaker magnetic field $B$.

Then, the relaxation time of a hole spin qubit is calculated. For the system,
$\left\langle n^{\prime}\left|D_{ij}\right|n\right\rangle$ is $\left\langle
g-\left|D_{ij}\right|g+\right\rangle$, which is obtained by Schrieffer-Wolff
transformation of $\tilde{H}=\tilde{H}_{op}+\tilde{H}_{\epsilon ijs}$
($\tilde{H}=U^{\dagger}_{t0}HU_{t0}$). Thus, for heavy hole qubit:

$\langle
g-|D_{ij}|g+\rangle=\dfrac{1}{E_{g}-E_{e}}(H^{\prime}_{g-,e-}H^{\prime}_{e-,g+}+H^{\prime}_{g-,e+}H^{\prime}_{e+,g+}).$
(9)

And the deformation potential:

$\displaystyle D_{ii}$ $\displaystyle=b^{\prime}(J_{i}^{2}-\dfrac{5}{4}),$
(10) $\displaystyle D_{ij}$
$\displaystyle=\dfrac{2d^{\prime}}{\sqrt{3}}\\{J_{i},J_{j}\\},(i\neq j).$

For heavy hole qubit:

$\displaystyle|\langle g-|D_{xx}|g+\rangle|^{2}$ $\displaystyle=|\langle
g-|D_{yy}|g+\rangle|^{2}=|\langle g-|D_{zz}|g+\rangle|^{2}=|\langle
g-|D_{xy}|g+\rangle|^{2}=0,$ (11) $\displaystyle|\langle
g-|D_{yz}|g+\rangle|^{2}$ $\displaystyle=|\langle
g-|D_{zx}|g+\rangle|^{2}=16d^{{}^{\prime}2}(\dfrac{\tilde{\varepsilon}_{Z}}{\tilde{\Delta}})^{2}.$

where $\tilde{\varepsilon}_{Z}=\varepsilon_{Z}t/\tilde{\Delta}$,
$\tilde{\Delta}=\sqrt{\Delta^{2}+t^{2}}$. Phonon-induced relaxation of heavy
hole spin qubit:

$\frac{1}{T_{1}}=\frac{(\hbar\omega)^{3}}{20\hbar^{4}\pi\rho}(\dfrac{\tilde{\varepsilon}_{Z}}{\tilde{\Delta}})^{2}\left[32d^{{}^{\prime}2}\left(\frac{2}{3v_{l}^{5}}+\frac{1}{v_{t}^{5}}\right)\right].$
(12)

According to previous paragraphs, $H_{Z}$ should be treated as a perturbation
to induce relaxation. However, $H_{Z}$ can be exactly treated to obtain the
relaxation, which is more precise than that from the perturbation step. The
main difference between the exact and perturbed way is deformation potential
$\langle g-|D_{ij}|g+\rangle$. Now, Zeeman Hamiltonian $H_{Z}$ is considered
exactly, which means $H_{hp}$ is projected into the exact form of acceptor
qubit. The deformation potential can be obtained without using the Schrieffer-
Wolff transformation:

$\displaystyle|\langle g-|D_{xx}|g+\rangle|^{2}=|\langle
g-|D_{yy}|g+\rangle|^{2}=|\langle g-|D_{zz}|g+\rangle|^{2}=|\langle
g-|D_{xy}|g+\rangle|^{2}=0,$ (13) $\displaystyle|\langle
g-|D_{yz}|g+\rangle|^{2}=|\langle
g-|D_{zx}|g+\rangle|^{2}=d^{{}^{\prime}2}(\dfrac{|t|(a-c)}{N_{1}N_{2}})^{2}.$

Then, substituting them into Eq. (8), relaxation of heavy hole acceptor qubit:

$\frac{1}{T_{1}}=\frac{(\hbar\omega)^{3}}{20\hbar^{4}\pi\rho}[\dfrac{|t|(a-c)}{N_{1}N_{2}}]^{2}\left[2d^{{}^{\prime}2}\left(\frac{2}{3v_{l}^{5}}+\frac{1}{v_{t}^{5}}\right)\right].$
(14)

To verify the validity of the exact method, relaxation from the exact or
perturbed way is compared in Fig. 3. They are almost consistent when the
magnetic field is weak. That means the perturbed result matches up to the
first-order part of the exact result. It is reasonable to estimate the
relaxation due to phonon in the exact way introduced in this subsection.

Figure 4: The relaxation rate due to interaction with phonon. $B=0.5$ T,
$E_{ac}=10^{4}$ V/m, $d=4.6$ nm. Here, we take the logarithm to the base 10 of
them to clarify the difference. (a) $T_{1}$ is plotted as function of
symmetric strain $\epsilon_{0}$ and vertical electric field $E_{z}$. The
relaxation time is longer with stronger strain. (b) $T_{1}$ is plotted as
function of strain $\epsilon_{xx}$ and $\epsilon_{yy}$.

##  Appendix E Tolerance to the electric field at sweet spot

With aid of asymmetric strain, there can be two sweet spots for the single-
qubit operation of the acceptor-based qubit. And as the strain increases, the
two sweet spots can merge together and form a second-order sweet spot. Around
the region of the second-order sweet spot, the qubits are still insensitive to
the charge noise. The coherence performance of the qubits is preserved. In
Fig. 5, the $E_{tol}$ is defined as full width at half maximum of $E_{max}$.
Around most of the high-Q area, the $E_{tot}$ exceeds 1000 V/m.

Figure 5: Tolerance to the electric field $E_{tol}$. $B=0.5$ T,
$E_{ac}=10^{4}$ V/m, $d=4.6$ nm. The window of the vertical electric field is
set as $[1,20]$ MV/m. The figures are plotted as functions of strain
$\epsilon_{xx}$ and $\epsilon_{yy}$.

##  Appendix F Two qubit gate: dipole-dipole interaction

The two-qubit gate might be realized by the electric dipole of acceptors and
spin-orbit coupling. To import this effect, we assume two acceptors are
separated by the distance of R, which can make exchange interaction ignored.
And the interaction between the acceptors is Coulomb interaction $V_{12}$.
Hamiltonian of the two acceptor qubits is:

$H=H_{op}^{1}+H_{op}^{2}+V_{12},$ (15)

where $H_{op}^{1(2)}$ is the operation Hamiltonian of acceptor 1 (2). Two
qubits subspace is contained in a 16$\times$16 product subspace
$\\{|mn\rangle\\}=\\{|m^{1}\rangle\\}\otimes\\{|n^{2}\rangle\\}$.
$m,n\in\\{g+,g-,e+,e-\\}$. In the subspace of the single qubit, the
Hamiltonian is obvious:

$\langle
m(n)|H_{op}^{1(2)}|m^{\prime}(n^{\prime})\rangle=\begin{pmatrix}\varepsilon_{g+}&0&0&0\\\
0&\varepsilon_{g-}&0&0\\\ 0&0&\varepsilon_{e+}&0\\\
0&0&0&\varepsilon_{e-}\end{pmatrix}.$ (16)

Hamiltonian of the Coulomb interaction is:

$\langle mn|V_{12}|m^{\prime}n^{\prime}\rangle=\int
dr_{1}^{3}dr_{2}^{3}\dfrac{e^{2}\Psi_{m}^{\dagger}(\mathbf{r}_{1})\Psi_{n}^{\dagger}(\mathbf{r}_{2})\Psi_{m^{\prime}}(\mathbf{r}_{1})\Psi_{n^{\prime}}(\mathbf{r}_{2})}{4\pi\epsilon|\mathbf{r}_{1}\mathbf{r}_{2}|}.$
(17)

And we suppose the separation $\mathbf{R}_{12}=\mathbf{r}_{1}-\mathbf{r}_{1}$
between the acceptors is larger than the dipole moment of the acceptors. Then
the Coulomb interaction can be expanded by multi-pole approximation. The
lowest-order non-zero term is:

$\langle mn|V_{12}|m^{\prime}n^{\prime}\rangle=\dfrac{e^{2}}{4\pi\epsilon
R_{12}^{5}}[R^{2}_{12}\langle\delta\mathbf{r}_{1}\rangle_{nn^{\prime}}\langle\delta\mathbf{r}_{2}\rangle_{mm^{\prime}}-3(\langle\delta\mathbf{r}_{1}\rangle_{nn^{\prime}}\cdot\mathbf{R}_{12})(\langle\delta\mathbf{r}_{2}\rangle_{mm^{\prime}}\cdot\mathbf{R}_{12})],$
(18)

where

$\langle\delta\mathbf{r}_{1}\rangle_{nn^{\prime}}=\int
dr_{i}^{3}(\mathbf{r}_{i}-\mathbf{r}_{i})\Psi_{n}^{\dagger}(\mathbf{r}_{i})\Psi_{n^{\prime}}(\mathbf{r}_{i}).$
(19)

Defining $\mathbf{r}^{\prime}_{i}=\mathbf{r}_{i}-\mathbf{R}_{i}$ the hole
coordinate relative to the ion, and assuming the relative position in the x-y
plane: $\mathbf{R}=R\cos(\theta_{E})\hat{x}+R\sin(\theta_{E})\hat{y}$, the
Coulomb interaction:

$\langle
mn|V_{12}|m^{\prime}n^{\prime}\rangle=\dfrac{(1-3\cos^{2}(\theta_{E}))\langle
m|ex_{1}^{\prime}|m^{\prime}\rangle\cdot\langle
n|ex_{2}^{\prime}|n^{\prime}\rangle+(1-3\sin^{2}(\theta_{E}))\langle
m|ey_{1}^{\prime}|m^{\prime}\rangle\cdot\langle
n|ey_{2}^{\prime}|n^{\prime}\rangle+\langle
m|ez_{1}^{\prime}|m^{\prime}\rangle\cdot\langle
n|ez_{2}^{\prime}|n^{\prime}\rangle}{4\pi\epsilon R^{3}}.$ (20)

The dipole matrix required by the Coulomb interaction:

$\langle
m(n)|ex^{\prime}|m^{\prime}(n^{\prime})\rangle=\alpha\begin{pmatrix}0&C_{1}&C_{2}&0\\\
C_{1}^{*}&0&0&-C_{2}\\\ C_{2}&0&0&C_{1}\\\ 0&-C_{2}&C_{1}^{*}&0\end{pmatrix},$
(21)

and

$\langle
m(n)|ey^{\prime}|m^{\prime}(n^{\prime})\rangle=\alpha\begin{pmatrix}0&-iC_{1}&-iC_{2}&0\\\
iC_{1}^{*}&0&0&iC_{2}\\\ iC_{2}&0&0&-iC_{1}\\\
0&-iC_{2}&iC_{1}^{*}&0\end{pmatrix}.$ (22)

Substituting them into the Eq. (15), and projecting into two-qubit subspace
$\\{|g-g-\rangle,|g-g+\rangle,|g+g-\rangle,|g+g+\rangle\\}$:

$\langle
mn|H|m^{\prime}n^{\prime}\rangle=\begin{pmatrix}\varepsilon^{a}_{g-}+\varepsilon^{b}_{g-}&0_{1}&0&3d_{11}e^{-2i\theta_{E}}\\\
0&\varepsilon^{a}_{g-}+\varepsilon^{b}_{g+}&-d_{11}&0\\\
0&-d_{11}&\varepsilon^{a}_{g+}+\varepsilon^{b}_{g-}&0\\\
3d_{11}e^{2i\theta_{E}}&0&0&\varepsilon^{a}_{g+}+\varepsilon^{b}_{g+}\end{pmatrix},$
(23)

where a,b corresponding to acceptor 1,2, and $d_{11}$ is defined as
$d_{11}=\dfrac{\alpha^{2}C_{1}^{a}C_{1}^{b}}{4\pi\epsilon R_{12}^{3}}$.
Actually, $d_{11}$ can be approximated to:

$d_{11}\approx\dfrac{D^{2}}{4\pi\epsilon R_{12}^{3}}.$ (24)

From the above, we know that form of entanglement of acceptor qubits is
determined by relative separation between acceptors. However, entanglement
between $|g-g+\rangle$ and $|g+g-\rangle$ is independent on that.
$\sqrt{\text{SWAP}}$ gate can be constructed by assuming
$\varepsilon^{a}_{g-}+\varepsilon^{b}_{g+}=\varepsilon^{a}_{g+}+\varepsilon^{b}_{g-}$.

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