# Tunable hole spin-photon interaction based on
$\operatorname{\mathsf{g}}$-matrix modulation

V. P. Michal  J. C. Abadillo-Uriel  Univ. Grenoble Alpes, CEA, IRIG-MEM-L_Sim,
Grenoble, France.  S. Zihlmann  R. Maurand  Univ. Grenoble Alpes, CEA,
Grenoble INP, IRIG-Pheliqs, Grenoble, France.  Y.-M. Niquet  M. Filippone
<EMAIL_ADDRESS><EMAIL_ADDRESS>Univ. Grenoble Alpes, CEA, IRIG-MEM-
L_Sim, Grenoble, France.

###### Abstract

We consider a spin circuit-QED device where a superconducting microwave
resonator is capacitively coupled to a single hole confined in a semiconductor
quantum dot. Thanks to the strong spin-orbit coupling intrinsic to valence-
band states, the gyromagnetic $\operatorname{\mathsf{g}}$-matrix of the hole
can be modulated electrically. This modulation couples the photons in the
resonator to the hole spin. We show that the applied gate voltages and the
magnetic-field orientation enable a versatile control of the spin-photon
interaction, whose character can be switched from fully transverse to fully
longitudinal. The longitudinal coupling is actually maximal when the
transverse one vanishes and vice-versa. This “reciprocal sweetness” results
from geometrical properties of the $\operatorname{\mathsf{g}}$-matrix and
protects the spin against dephasing or relaxation. We estimate coupling rates
reaching $\sim 10\,{\rm MHz}$ in realistic settings and discuss potential
circuit-QED applications harnessing either the transverse or the longitudinal
spin-photon interaction. Furthermore, we demonstrate that the
$\operatorname{\mathsf{g}}$-matrix curvature can be used to achieve parametric
longitudinal coupling with enhanced coherence.

Hybrid circuit quantum electrodynamics (cQED) investigates the interaction
between microwave photons in resonators and diverse quantum excitations in
solid state devices [1]. This field inherits from cavity QED [2] its
fundamental interest in light-matter interactions, and sustains the
development of quantum information [3] by extending the research initiated
with superconducting circuits [4]. Special efforts are currently devoted to
the coupling of microwave photons with the charge [5, 6, 7, 8] and spin [9,
10, 11, 12, 13] degrees of freedom of a localized electron or hole. The spin
of a single carrier is indeed of particular interest as it is one of the most
elemental realizations of a two-level system [14].

Light acts on two-level systems either transversally or longitudinally, that
is by changing or preserving the level occupations, respectively. These
actions trigger different dynamics; transverse coupling underpins high-ﬁdelity
readout [15] and control operations [16, 17, 16, 18, 19, 20, 21], while
longitudinal coupling has recently emerged as a valuable resource for fast
quantum non-demolition measurements [22, 23, 24, 25, 26, 27] and entangling
gates [28, 29, 30, 31, 32, 23, 33, 34, 35, 36, 37, 38, 39]. Yet, controlling
the relative strength of longitudinal and transverse couplings requires
specific circuit design [22, 23, 24, 25, 26, 27], driving protocols [40] or
additional elements such as micro-magnets [41]. Relying on fundamental and
simple physical mechanisms to switch from one coupling to the other would thus
open new opportunities to explore and harness various aspects of light-matter
interaction.

In this work, we show that a single hole confined in a compact semiconductor
nanostructure exhibits a controllable spin-photon interaction, without the
need for additional driving or circuit elements [12, 11]. In such a spin cQED
system, the key to tunability is the presence of an intrinsically strong spin-
orbit coupling (SOC) allowing for all-electrical modulation of the hole
gyromagnetic $\operatorname{\mathsf{g}}$-matrix [42, 43, 44, 45, 46]. This
strong SOC underpins hole-spin qubit control in silicon [47, 48, 49] and
germanium [50, 51], as well as multi-qubit logic [52, 53]. Thanks to SOC,
photons in a microwave resonator directly couple to the hole spin. With the
minimal device depicted in Fig. 1a, we demonstrate that the gate voltages and
the orientation of the applied magnetic field provide flexible control over
the spin-photon interaction. In particular, we find that the spin-photon
coupling can be switched from fully transverse to fully longitudinal owing to
geometrical properties of the hole spin $\operatorname{\mathsf{g}}$-matrix.
Eventually, we show that strong parametric longitudinal coupling with long
coherence can also be achieved. With realistic parameter estimates we discuss
potential applications such as quantum-state transfer and CZ gates. Our work
thus highlights the potential of hole quantum dots in semiconductors as
compact and versatile platforms for cQED experiments with spins.

We start by discussing the physics of the hole qubit device sketched in Fig.
1, and outline how it features all the ingredients needed to achieve tunable
spin-photon coupling. It is well established [42, 43, 54, 48, 44, 45, 46, 55,
56] that strong SOC in the valence band of semiconductors leads to an
anisotropic Zeeman interaction between the spin of a single hole and an
external magnetic field $\mathbf{B}$. This interaction is generally described
by a two-level Hamiltonian

$\mathcal{H}=\frac{\mu_{B}}{2}\,\boldsymbol{\sigma}\cdot\operatorname{\mathsf{g}}(V)\cdot\mathbf{B}\,,$
(1)

where $\boldsymbol{\sigma}$ is the vector collecting the Pauli matrices of the
hole spin, $\mu_{B}$ is the Bohr magneton, and $\operatorname{\mathsf{g}}(V)$
is the $\operatorname{\mathsf{g}}$-matrix [57, 48, 44]. Differently from
electrons, the $\operatorname{\mathsf{g}}$-matrix is not a number
($g_{e}\simeq 2$) and it strongly depends on the voltage $V$ applied to
confine the hole.

In the thin dot regime $L_{z}\ll L_{x,y}$, the spin wave-function has a
dominant heavy-hole character [58] along the strong confinement axis taken as
$z=[001]$. The behavior of the $\operatorname{\mathsf{g}}$-matrix of such
heavy-hole devices has been extensively analyzed with different methods [44,
45, 46, 59, 55] reproducing experimental observations [48, 56]. We provide
details about our theoretical derivation of the
$\operatorname{\mathsf{g}}$-matrix in the Supplemental Material (SM) [60].
Given the freedom of choice on the magnetic axes and two-level basis set, the
$\operatorname{\mathsf{g}}$-matrix can always be cast in diagonal form

$\operatorname{\mathsf{g}}(V)=\mbox{diag}\,[\,g_{x}\,,\,g_{y}\,,\,g_{z}\,]\,,$
(2)

where $g_{z}>g_{x,y}$ is the fingerprint of the dominant heavy-hole character
of the wave-function, while the smaller $g_{x,y}$ arise from heavy/light hole
mixing, mainly controlled by lateral confinement in the $(xy)$ plane.
Ultimately, this $\operatorname{\mathsf{g}}$-matrix leads to a strongly
anisotropic Larmor pseudovector
$\boldsymbol{\omega}_{L}=\tfrac{\mu_{B}}{\hbar}\operatorname{\mathsf{g}}\cdot\mathbf{B}$
[57]. Figure 2a illustrates this anisotropy for a particular realization of
the device of Fig. 1, by showing the effective
$\operatorname{\mathsf{g}}$-factor
$\operatorname{\mathsf{g}}^{\ast}=|\operatorname{\mathsf{g}}\cdot\mathbf{b}|$
relating the Larmor angular frequency
$|\boldsymbol{\omega}_{L}|=\tfrac{\mu_{B}}{\hbar}\operatorname{\mathsf{g}}^{\ast}B$
to the amplitude $B=|\mathbf{B}|$ of the magnetic field for a given
orientation $\mathbf{b}=\mathbf{B}/B$.

We focus now on the coupling of the hole spin to photons confined in a
microwave resonator with angular frequency $\omega_{0}$. These photons give
rise to quantized voltage fluctuations $\delta V=V_{\rm zpf}(a+a^{\dagger})$
on the confining gate, where $V_{\rm zpf}=\omega_{0}\sqrt{\hbar Z_{r}/\pi}$ is
the zero point voltage fluctuation at the resonator edge, $Z_{r}$ is its
characteristic impedance and the operator $a^{\dagger}$ creates a photon [61,
62]. Such voltage fluctuations modulate the
$\operatorname{\mathsf{g}}$-matrix, leading to spin-photon coupling. Without
loss of generality, we separate the spin precession vector
$\boldsymbol{\sigma}$ into components longitudinal ($\parallel$) and
transverse ($\perp$) to the static Larmor vector
$\boldsymbol{\omega}_{L}(\operatorname{\mathnormal{V}_{\mathnormal{G}}^{0}})$
[42, 63], where $\operatorname{\mathnormal{V}_{\mathnormal{G}}^{0}}$ is the DC
bias on the gate. Expanding Eq. (2) to the leading order in $V_{\rm zpf}$, the
Hamiltonian for the spin-resonator dynamics reads

$\mathcal{H}=\frac{\hbar}{2}\omega_{L}\sigma_{\parallel}+\hbar\operatorname{\omega_{0}}a^{\dagger}a+\hbar\left[g_{\parallel}\sigma_{\parallel}+g_{\perp}\sigma_{\perp}\right](a+a^{\dagger})\,,$
(3)

where
$\omega_{L}=|\boldsymbol{\omega}_{L}(\operatorname{\mathnormal{V}_{\mathnormal{G}}^{0}})|$.
The spin-photon coupling thus involves longitudinal and transverse components,
see Fig. 1b. Defining the unit vector $\mathbf{n}$ along
$\boldsymbol{\omega}_{L}(\operatorname{\mathnormal{V}_{\mathnormal{G}}^{0}})$,
these couplings take the form

$g_{\parallel/\perp}=\operatorname{\mathnormal{V}_{\rm
zpf}}\frac{\mu_{B}B}{2\hbar}\beta_{\parallel/\perp}(\mathbf{b})\,,$ (4)

where we have introduced the parallel and transverse
$\operatorname{\mathsf{g}}$-susceptibilities

$\displaystyle\beta_{\parallel}$
$\displaystyle=\left(\operatorname{\mathsf{g}}^{\prime}(\operatorname{\mathnormal{V}_{\mathnormal{G}}^{0}})\cdot\mathbf{b}\right)\cdot\mathbf{n}\,,$
$\displaystyle\beta_{\perp}$
$\displaystyle=\left|\left(\operatorname{\mathsf{g}}^{\prime}(\operatorname{\mathnormal{V}_{\mathnormal{G}}^{0}})\cdot\mathbf{b}\right)\times\mathbf{n}\right|\,,$
(5)

with the shorthand notation
$\operatorname{\mathsf{g}}^{\prime}(\operatorname{\mathnormal{V}_{\mathnormal{G}}^{0}})=\partial_{V}\operatorname{\mathsf{g}}(V)|_{V=\operatorname{\mathnormal{V}_{\mathnormal{G}}^{0}}}$.
These susceptibilities characterize the spin-photon couplings per unit of
voltage and magnetic field.

Figure 1: (a) Sketch of the setup considered here. A single hole quantum dot
(red shape) is confined in a rectangular nanowire with sides $L_{y}>L_{z}$ by
a partly overlapping gate (light blue) with length $L_{x}$ and DC bias
$V=\operatorname{\mathnormal{V}_{\mathnormal{G}}^{0}}$. The hole is coupled to
photons in a microwave coplanar resonator connected to the gate. In this
configuration, the shape of the quantum dot (dashed lines) is primarily
controlled by the in-plane component $\operatorname{\mathcal{E}}_{y}$ of the
electric field of the gate [44, 45, 46], which modulates the Larmor vector
$\boldsymbol{\omega}_{L}$ of the hole spin owing to spin-orbit coupling. (b)
As a consequence, quantized voltage fluctuations $\delta V$ originating from
the resonator photon field can change the norm of $\boldsymbol{\omega}_{L}$
(longitudinal coupling $g_{\parallel}$) and/or its orientation (spin flip
transverse coupling $g_{\perp}$).

Figure 2: Maps of (a) the effective $\operatorname{\mathsf{g}}$-factor and
(b), (c) the perpendicular $\beta_{\perp}$ and parallel $|\beta_{\parallel}|$
susceptibilities as a function of the magnetic field orientation
$\mathbf{b}=\mathbf{B}/|\mathbf{B}|$. We consider an idealized silicon device
as in Ref. 46 with hard-wall confinement along $y$ and $z$ ($L_{z}=15\,{\rm
nm}$, $L_{y}=40\,{\rm nm}$), harmonic confinement along $x$ (harmonic length
$\ell_{x}=11.5\,{\rm nm}$, as defined in [46]), and a homogeneous static
electric field $\operatorname{\mathcal{E}}_{y}=0.12\,{\rm mV/nm}$ (we assume
$\partial\operatorname{\mathcal{E}}_{y}/\partial V=-1/L_{y}$ [60]). The
$\operatorname{\mathsf{g}}$-matrices are numerically computed from the
solutions of the four bands Luttinger-Kohn Hamiltonian [45, 46, 60]. The
longitudinal susceptibility $|\beta_{\parallel}|$ is maximum for specific
orientations (dephasing hot spots) where the transverse susceptibility
$\beta_{\perp}$ is zero, and vanishes along dephasing sweet lines. The
transverse susceptibility $\beta_{\perp}$ is maximum (fastest Rabi
oscillations) very close to such a sweet line.

The transverse ($\beta_{\perp}$) and longitudinal ($\beta_{\parallel}$)
susceptibilities show strong dependence on the orientation of the magnetic
field $\mathbf{B}$, as illustrated in Figs. 2b-c. In particular,
$\beta_{\perp}$ vanishes where $\beta_{\parallel}$ is maximum (longitudinal
sweet spot, along the device axes), while $|\beta_{\parallel}|$ vanishes next
to the maxima of $\beta_{\perp}$ (transverse sweet spot). At such sweet spots,
the hole spin is better protected against, respectively, relaxation
($\beta_{\perp}=0$) or dephasing ($\beta_{\parallel}=0$) due to electrical
noise. This “reciprocal sweetness” between the transverse and longitudinal
couplings is a key argument of the following discussion and has a robust
geometrical origin. Indeed, $\beta_{\parallel}$ and $\beta_{\perp}$ are
proportional to the projections of
$\operatorname{\mathsf{g}}^{\prime}\cdot\mathbf{b}$ onto vectors that are
respectively parallel and perpendicular to $\boldsymbol{\omega}_{L}$.
Therefore, whenever one of the couplings is zero because
$\operatorname{\mathsf{g}}^{\prime}\cdot\mathbf{b}$ and
$\boldsymbol{\omega}_{L}$ are either parallel or perpendicular, the other is
expected to be maximum. Reciprocal sweetness shall hence be ubiquitous in a
large variety of hole spin devices. The role played by the variations of
$|\operatorname{\mathsf{g}}^{\prime}\cdot\mathbf{b}|$ in the reciprocal
sweetness are discussed in the SM [60].

A non-zero $g_{\parallel}$ implies that voltage fluctuations modulate the
Zeeman splitting $|\boldsymbol{\omega}_{L}|$. Such modulations are forbidden
when the magnetic field is uniform and the electric field displaces the hole
wave-function as a whole without deforming it. In this usual “iso-Zeeman”
driving mode, only transverse coupling to the resonator is possible [64, 65,
48, 66]. Thus, $g_{\parallel}\neq 0$ must result from voltage-induced
deformations of the wave-function, which give rise to the so-called $g$-TMR
($g$-Tensor Modulation Resonance) introduced in the context of transverse
coupling [42, 43, 67, 48, 46].

In the device of Fig. 1, the deformation of the wave-function is controlled by
the joint action of the transverse electric field
$\operatorname{\mathcal{E}}_{y}$ and the anharmonic hard-wall confinement
across the channel width. The interplay between anharmonicity and symmetries
is illustrated in Fig. 3a, where we track the maximum
$\beta_{\parallel/\perp}^{\max}$ as a function of the transverse electric
field $\operatorname{\mathcal{E}}_{y}$. In the absence of electric field
($\operatorname{\mathcal{E}}_{y}=0$), both $\beta_{\parallel}$ and
$\beta_{\perp}$ vanish due to the existence of an inversion symmetry in the
confinement potential [44]. Once the electric field is turned on, the hole
gets progressively squeezed onto the side facet of the channel, and the spin-
photon couplings increase. Furthermore, $\beta_{\parallel}^{\max}$ and
$\beta_{\perp}^{\max}$ are of the same order of magnitude whatever
$\operatorname{\mathcal{E}}_{y}$. Ultimately, $\beta_{\parallel/\perp}$ tend
to zero at large $\operatorname{\mathcal{E}}_{y}$ as the heavily squeezed hole
hardly responds any more to fluctuations of the electric field. Both spin-
photon couplings are optimal at a comparable electric field
$\operatorname{\mathcal{E}}_{y}^{\ast}=0.12\,{\rm mV/nm}$, with $\mathbf{B}$
lying in the $(xy)$ plane for $\beta_{\parallel}$, and at an angle
$\delta\theta=\pm\arctan(\sqrt{|g_{y}/g_{z}|})$ from that plane for
$\beta_{\perp}$ [45, 46]. We emphasize that $\beta_{\perp}$ is typically
maximal at out-of-plane values of $B_{z}$ that remain compatible with the
critical fields of resilient superconducting resonators [68, 69].

Following Ref. 46, we confirm analytically that
$\beta_{\parallel}^{\max}=\beta_{\perp}^{\max}$ in the thin dot regime
$L_{z}\ll L_{x,y}$ [60]. The optimal susceptibilities
$\beta_{\parallel/\perp}^{\ast}=Ce/\Delta$ are achieved at the electric field
$e\operatorname{\mathcal{E}}_{y}^{\ast}\sim
1.5\pi^{3}\hbar^{2}/(m_{\parallel}^{h}L_{y}^{3})$, where $\Delta$ is the
heavy-hole/light-hole band gap, $m_{\parallel}^{h}$ is the in-plane heavy-hole
effective mass, and $C$ is a material-dependent constant of order unity
($C\approx 0.33$ for silicon and $C\approx 0.84$ for germanium).

Figure 3: (a) Maximum parallel and perpendicular susceptibilities
$\beta_{\parallel}^{\max}$ (red) and $\beta_{\perp}^{\max}$ (blue) as a
function of the static electric field $\operatorname{\mathcal{E}}_{y}$, for
the same parameters as in Fig. 2. They are computed at the magnetic field
orientations that maximize respectively $\beta_{\parallel}$ and
$\beta_{\perp}$ for each $\operatorname{\mathcal{E}}_{y}$. The dashed line
pinpoints the electric field $\operatorname{\mathcal{E}}_{y}^{\ast}$ that
optimizes $\beta_{\parallel}^{\max}$. (b) Second-order longitudinal
susceptibility $\gamma_{\parallel}$ at the optimal
$\operatorname{\mathcal{E}}_{y}=0$ and $\mathbf{B}\parallel\mathbf{y}$, as a
function of $L_{y}$. The dots are the numerical calculations with the
parameters of Fig. 2, and the line is an analytical fit
$\gamma_{\parallel}^{\max}=0.14m_{\parallel}^{h}L_{y}^{2}e^{2}/(\pi^{3}\hbar^{2}\Delta)$.

The thin dot regime is, however, usually suboptimal as the gap $\Delta\propto
L_{z}^{-2}$ may be large. In the more favorable conditions of Fig. 2, and
considering a typical $\operatorname{\mathnormal{V}_{\rm zpf}}=20\,{\rm\mu
V}$, the numerically computed couplings at
$\operatorname{\mathcal{E}}_{y}^{\ast}\simeq 0.12\,{\rm mV/nm}$ reach up to
$g_{\parallel/\perp}/2\pi\sim 10\,{\rm MHz}$ at $B=0.5\,{\rm T}$
($\omega_{L}/2\pi\approx 5\,{\rm GHz}$ for parallel and
$\omega_{L}/2\pi\approx 12\,{\rm GHz}$ for perpendicular coupling). We stress
that these values for single holes confined in a single quantum dot are
comparable to the transverse couplings reported in spin-cQED experiments with
electrons in silicon double quantum dots with micromagnets [11, 12]. We show
in the SM that these estimates are reproduced by numerical simulations of more
realistic devices with inhomogeneous lateral and vertical electric fields
[60].

The strong dependence of $g_{\parallel/\perp}$ on the electric field
$\operatorname{\mathcal{E}}_{y}$ can be used to turn on and off the coupling
between single hole spins and resonators. Moreover, owing to the reciprocal
sweetness between $\beta_{\parallel}$ and $\beta_{\perp}$, the nature of the
spin-photon coupling can be switched from purely transverse to purely
longitudinal by a rotation of the magnetic field. This provides a playground
for the exploration of the physics and applications of spin-photon
interactions in arbitrary regimes.

As an example, the sweet lines for transverse coupling ($\beta_{\parallel}=0$)
enable spin-microwave photon quantum state transfer [70] with fidelities
reaching $\sim 97\%$ for realistic noise, typically limited by the photon
losses in the resonator (see SM [60]). Single holes have already demonstrated
coherence times of the order of tens of $\mu$s on such sweet lines [56]. In
principle, it is also possible to leverage the longitudinal coupling at finite
$\operatorname{\mathcal{E}}_{y}$ to mediate CZ gates or ZZ interactions
between distant qubits [29, 30, 36, 34, 35, 60]. Nevertheless, regimes where
$\beta_{\parallel}$ is large are hot spots for two-level dephasing induced by
charge noise [56], which may compromise the experimental demonstration of
coherent spin manipulation.

This difficulty can be overcome by exploiting the second-order longitudinal
coupling proportional to the curvature
$\operatorname{\mathsf{g}}^{\prime\prime}(\operatorname{\mathnormal{V}_{\mathnormal{G}}^{0}})$
of the $\operatorname{\mathsf{g}}$-matrix [22, 29, 31, 32, 23, 33, 34, 35, 36,
37, 38, 39]. Curvature coupling indeed prevails at
$\operatorname{\mathcal{E}}_{y}=0$, where $\beta_{\parallel}=\beta_{\perp}=0$,
so that the qubit is insensitive (to first-order) to both charge noise
dephasing and relaxation.

In this regime, if the resonator is driven at its resonant frequency
$V\rightarrow V(t)=\operatorname{\mathnormal{V}_{\mathnormal{G}}^{0}}+V_{\rm
d}\cos(\omega_{0}t)$, the Hamiltonian (3) reduces to
$\mathcal{H}=\frac{\hbar}{2}\tilde{g}_{\parallel}\sigma_{\parallel}(a+a^{\dagger})$
in the frame rotating at angular frequency $\omega_{0}$ [22]. There the
parametrically strong longitudinal coupling $\tilde{g}_{\parallel}$ is
proportional to the curvature of the $\operatorname{\mathsf{g}}$-matrix 111We
stress that Eq. (6) assumes $\beta_{\parallel/\perp}=0$. A more general
expression valid for regimes where $\beta_{\parallel/\perp}\neq 0$ is given in
the SM [60].:

$\displaystyle\tilde{g}_{\parallel}$
$\displaystyle=\operatorname{\mathnormal{V}_{\rm zpf}}V_{\rm
d}\frac{\mu_{B}B}{2\hbar}\gamma_{\parallel}\,,$
$\displaystyle\gamma_{\parallel}$
$\displaystyle=\left(\operatorname{\mathsf{g}}^{\prime\prime}(\operatorname{\mathnormal{V}_{\mathnormal{G}}^{0}})\cdot\mathbf{b}\right)\cdot\mathbf{n}\,,$
(6)

where we use the shorthand notation
$\operatorname{\mathsf{g}}^{\prime\prime}(\operatorname{\mathnormal{V}_{\mathnormal{G}}^{0}})=\partial^{2}_{V}\operatorname{\mathsf{g}}(V)|_{V=\operatorname{\mathnormal{V}_{\mathnormal{G}}^{0}}}$.
Notice that there is a concomitant transverse curvature parameter
$\gamma_{\perp}$ in total analogy with Eq. (5). Nevertheless, reciprocal
sweetness applies to $\gamma_{\parallel/\perp}$ for the same geometrical
reasons as for $\beta_{\parallel/\perp}$. Namely, the orientation of the
magnetic field $\mathbf{B}$ can be always chosen such that $\gamma_{\perp}=0$
while $\gamma_{\parallel}$ is maximum, or vice-versa. The dependence of
$\gamma_{\parallel/\perp}$ on the magnetic field orientation is actually
similar to that shown in Fig. 2 for the linear susceptibilities
$\beta_{\parallel/\perp}$ [60].

The curvature coupling can be readily tuned (and switched on/off) with the
drive amplitude $V_{d}$. We also highlight that $\gamma_{\parallel}$ strongly
depends on the lateral confinement length $L_{y}$, and tends to increase when
widening the channel as shown in Fig. 3b. This trend can indeed be understood
from Fig. 3a, which suggests that
$\gamma_{\parallel}^{\max}\sim\beta_{\parallel}^{\rm
max}/(L_{y}\operatorname{\mathcal{E}}_{y}^{*})\sim
Cm_{\parallel}^{h}L_{y}^{2}e^{2}/(\pi^{3}\hbar^{2}\Delta)$ in the thin dot
regime. This estimate is however expected to be less accurate when $L_{z}$
gets comparable to $L_{y}$, as is the case in Fig. 3b, where we actually fit
the numerical data with $C=0.14$. Practically, the quantum dot size will
ultimately be limited by disorder-induced localization [59], and/or by the
larger magnetic fields needed to compensate for the decrease of
$\operatorname{\mathsf{g}}_{y}$.

For the device parameters of Figs. 2 and 3, assuming $V_{\rm zpf}=20\,{\rm\mu
V}$, a magnetic field $B=2\,{\rm T}$ along $y$ ($\omega_{L}/2\pi\sim 6\,{\rm
GHz}$), and a voltage drive $V_{\rm d}=2\,{\rm mV}$, the longitudinal coupling
is of the order of $\tilde{g}_{\parallel}/2\pi\sim 10\,{\rm MHz}$. Such
couplings would enable different operations, such as genuine quantum non-
demolition measurements [22], CZ gates and, more generally, non-resonant ZZ
interactions between distant spins connected to the same resonator [29, 30,
33, 36, 35]. We address the fidelity of a CZ gate leveraging the curvature
coupling in the SM [60], and find that the error may be brought down to the
percent range thanks to the high coherence at
$\operatorname{\mathcal{E}}_{y}=0$.

In conclusion, we have shown that various aspects of spin circuit-QED can be
efficiently addressed and exploited in minimal single hole semiconductor
devices. The all-electrical modulation of the hole
$\operatorname{\mathsf{g}}$-matrix allows to tune both transverse and
longitudinal spin-photon couplings, whose relative strengths can be controlled
by the orientation of the magnetic field thanks to their reciprocal sweetness.
In an experiment the couplings can be electrically switched on and off, either
by using the gate voltage dependence of the $\beta_{\parallel/\perp}$
susceptibilities (see Fig. 3), or by switching-off the AC modulation of the
gate voltage for parametric longitudinal coupling. The transverse coupling can
be harnessed to transfer the spin state to the microwave photon state, with
high fidelity when operating on the sweet lines where dephasing due to
electric noise is reduced. On the other hand the longitudinal coupling can be
used for fast quantum non-demolition readout and robust multi-spin
interactions [32, 22].

An interesting perspective would be to tune the relative strength of the
transverse and longitudinal couplings with a fully electrical knob instead of
the magnetic field orientation. This would enable rapid switching between both
couplings within single experiments, but calls for the ability to rotate
electrically the spheres in Fig. 2b-c around the $x$ or $y$ axis. This
operation is challenging in simple, few-gate geometries and we leave this
question open for further investigations.

While we have mainly considered single holes in silicon nanowire quantum dots,
the principles discussed in this work also apply to other hole spin qubit
systems, such as germanium nanowires and heterostructures [72, 73], or dopant-
based hole qubits [74, 75]. Further improvements in the quality of the
microwave resonators and the use of low-loss substrates will lead to
enhancements of the fidelities of multi-spin operations and allow progress
towards a highly connected spin-qubit architecture for quantum computation and
quantum simulation.

Acknowledgements – We thank Olivier Buisson, Silvano de Franceschi, Étienne
Dumur and Benoît Vermersch for useful comments and discussions. This work was
supported by the French National Research Agency (ANR) through the MAQSi
project, and by the European Union’s Horizon 2020 research and innovation
program under grant agreements No. 810504 (ERC project QuCube) and No. 759388
(ERC project LONGSPIN).

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## Supplemental Material

Here we give details on the hole $\operatorname{\mathsf{g}}$-factors in the
quasi-two-dimensional regime (Section .1), on the fidelity of the
spin/microwave-photon quantum state transfer protocol in the presence of
spin/photon decay and quasistatic charge noise (Section .2), and on the
fidelity of the two-qubit controlled-Z gate based on longitudinal couplings
(Section .3). Finally, we further support the proposal with numerical
simulations on a more realistic setup (Section .4).

###  .1 Holes in a thin channel

The hole $\operatorname{\mathsf{g}}$-factors in the quasi-two-dimensional
(thin dot) regime were analytically calculated in [46] with the four bands
Luttinger-Kohn Hamiltonian. There, the analytical expressions were benchmarked
against numerical methods developed in [44, 45]. For holes with mostly heavy
character, only the diagonal elements of the
$\operatorname{\mathsf{g}}$-matrix are non-zero and read:

$\displaystyle\operatorname{\mathsf{g}}_{x}$
$\displaystyle=\frac{6\gamma_{3}\hbar^{2}}{m_{0}\Delta}\left(\operatorname{\kappa_{3}}\langle
k_{y}^{2}\rangle-\operatorname{\kappa_{2}}\langle k_{x}^{2}\rangle\right)\,,$
(7a) $\displaystyle\operatorname{\mathsf{g}}_{y}$
$\displaystyle=\frac{6\gamma_{3}\hbar^{2}}{m_{0}\Delta}\left(\operatorname{\kappa_{2}}\langle
k_{y}^{2}\rangle-\operatorname{\kappa_{3}}\langle k_{x}^{2}\rangle\right)\,,$
(7b) $\displaystyle\operatorname{\mathsf{g}}_{z}$
$\displaystyle=-6\kappa+2\gamma_{h}+\delta\mathsf{g}_{z}\,.$ (7c)

Here $m_{0}$ is the bare electron mass, $\gamma_{1}$, $\gamma_{2}$,
$\gamma_{3}$ are the Luttinger parameters characterizing the hole masses, and
$\kappa$ is the valence band Zeeman parameter.
$\Delta=2\pi^{2}\gamma_{2}\hbar^{2}/m_{0}L_{z}^{2}$ is the energy splitting
between the heavy-hole and light-hole 2D subbands, and
$\operatorname{\kappa_{2}}=\kappa-2\gamma_{2}\eta_{h}$,
$\operatorname{\kappa_{3}}=\kappa-2\gamma_{3}\eta_{h}$, where $\gamma_{h}$ and
$\eta_{h}$ are dimensionless parameters defined in Refs. [76, 46]
($\gamma_{h}\approx 1.16$ and $\eta_{h}\approx 0.08$ in thin silicon films
with hard wall boundary conditions). The expectations values of
$k_{x}=-i\partial_{x}$ and $k_{y}=-i\partial_{y}$ are calculated for the
ground-state heavy-hole envelope. The term $\delta\mathsf{g}_{z}$ collects
corrections of order $\hbar^{2}\langle k_{x}^{2}\rangle/m_{0}\Delta$ and
$\hbar^{2}\langle k_{y}^{2}\rangle/m_{0}\Delta$ [58]. In this regime
$\operatorname{\mathsf{g}}_{x},\operatorname{\mathsf{g}}_{y}\ll\operatorname{\mathsf{g}}_{z}$
(see Figure 2 of the Main Text).

In this work, we assume (as in Ref. [46]) hard-wall confinement in a
rectangular nanowire with sides $L_{y}$ and $L_{z}$, and harmonic confinement
along $x$ with characteristic length $\ell_{x}\propto L_{x}$ ($\ell_{x}\approx
11.5$ nm for gate length $L_{x}\approx 35$ nm [44, 46]). Following Ref. [46],
we can then optimize the transverse and parallel susceptibilities with respect
to the magnetic field orientation, and find
$\beta_{\perp/\parallel}^{\max}=\max(\operatorname{\mathsf{g}}^{\prime}_{x}(\operatorname{\mathnormal{V}_{\mathnormal{G}}^{0}}),\operatorname{\mathsf{g}}^{\prime}_{y}(\operatorname{\mathnormal{V}_{\mathnormal{G}}^{0}}))$.
$\beta_{\parallel}$ is actually maximum for a magnetic field $\mathbf{B}$ in
the $(xy)$ plane, and $\beta_{\perp}$ for a magnetic field $\mathbf{B}$ at an
angle $\delta\theta=\pm\arctan(\sqrt{|g_{y}/g_{z}|})$ from that plane. Further
optimization with respect to the electric field
$\operatorname{\mathcal{E}}_{y}$ yields:

$\beta_{\parallel/\perp}^{\ast}\approx\frac{1.2\gamma_{3}\max(|\operatorname{\kappa_{2}}|,|\operatorname{\kappa_{3}}|)e}{(\gamma_{1}+\gamma_{2}-\gamma_{h})\Delta}\equiv\frac{Ce}{\Delta}\,,$
(8)

at $e\operatorname{\mathcal{E}}_{y}\sim
3\pi^{3}\hbar^{2}/(2m_{\parallel}^{h}L_{y}^{3})$, where
$m_{\parallel}^{h}=m_{0}/(\gamma_{1}+\gamma_{2}-\gamma_{h})$ is the in-plane
heavy-hole mass and $e>0$ is the elementary charge (see the corresponding
equation in the Main Text). We have assumed here that zero point gate voltage
fluctuations $\delta V=V_{\rm zpf}$ translate into zero point electric field
fluctuations $\delta\operatorname{\mathcal{E}}_{y}=-V_{\rm zpf}/L_{y}$. The
relevance of this assumption is confirmed by the simulations of section .4.
Eq. (8) is valid in the thin dot regime
$\Delta\gg\hbar^{2}\gamma_{3}/(m_{0}\ell_{\parallel}^{2})$, where
$\ell_{\parallel}$ is the minimal in-plane confinement length, which may
ultimately be limited by disorder [59]. Beyond the thin dot regime, the
Luttinger-Kohn equations can be solved numerically with Fourier-series
expansions, as done in [45, 46] and in Figs. 2 and 3 of the Main Text.

As discussed in the Main Text, one of the susceptibilities $\beta_{\parallel}$
or $\beta_{\perp}$ is expected to be maximum if the other vanishes because
$\operatorname{\mathsf{g}}^{\prime}\cdot\mathbf{b}$ and
$\boldsymbol{\omega}_{L}$ are either parallel ($\beta_{\perp}=0$) or
perpendicular ($\beta_{\parallel}=0$). In practice however, the actual maxima
of $|\beta_{\parallel}|$ or $\beta_{\perp}$ may be shifted from the zeros of
the other because $|\operatorname{\mathsf{g}}^{\prime}\cdot\mathbf{b}|$ also
depends on $\mathbf{b}$ [see Eq. (5)]. In the present case,
$|\operatorname{\mathsf{g}}^{\prime}\cdot\mathbf{b}|$ is only weakly dependent
on the orientation of the magnetic field, as shown in Fig. 4. The maxima of
$|\beta_{\parallel}|$ therefore coincide with the zeros of $\beta_{\perp}$,
because they lie on the high symmetry device axes, while the maxima of
$\beta_{\perp}$ are very slightly moved away from the sweet lines of
$\beta_{\parallel}$. In general, one may still expect a maximum of one
coupling in the close vicinity of a zero of the other, unless the variations
of $|\operatorname{\mathsf{g}}^{\prime}\cdot\mathbf{b}|$ are really strong
around that point.

Figure 4: Map of the norm of the Larmor vector derivative as a function of
magnetic field orientation, for the same parameters as in Fig. 2 of the Main
Text.

###  .2 Fidelity for the resonant spin/microwave-photon state transfer in the
presence spin/photon decay and quasistatic charge noise

Figure 5: Fidelity of the spin-microwave photon quantum state transfer
protocol as a function of the orientation of the magnetic field, using the
spin-photon couplings of Fig. 2 of the Main Text. We use the same convention
for the magnetic field angles $\theta$ and $\varphi$ as in Refs. 44, 45, 46:
$\theta$ is the angle between the $z$ axis and the magnetic field, and
$\varphi$ is the angle between its projection in the $(xy)$ plane and the $y$
axis. (a) Fidelity $\mathcal{\overline{F}}_{M}$ limited by the decay of the
microwave photon as given by Eq. (9) with parameters
$\operatorname{\mathnormal{V}_{\rm zpf}}=20\,{\rm\mu V}$ and
$\operatorname{\mathnormal{\kappa_{r}}}/2\pi=0.5\,{\rm
MHz}\gg\gamma_{\downarrow}/2\pi$, at constant Larmor frequency
$\omega_{L}/2\pi=\omega_{0}/2\pi=5\,{\rm GHz}$. (b) Fidelity
$\mathcal{\overline{F}}_{c}$ limited by quasistatic charge noise as given by
Eq. (11) with $\sigma_{V}=20\,{\rm\mu V}$ [77, 78]. The dashed blue lines are
the sweet lines where the first-order longitudinal susceptibility
$\beta_{\parallel}$ cancels and $\mathcal{\overline{F}}_{c}\approx 1$. The
blue dots are the points on these lines that maximize
$\mathcal{\overline{F}}_{M}\approx 97\%$.

As an example of a protocol realizable with the setup presented here, we
discuss the spin-microwave photon quantum state transfer [70] and compute the
fidelities achievable for realistic noise (Markovian and quasistatic). This
protocol makes use of the transverse coupling to map the quantum state
$|\psi_{\rm
in}\rangle=\cos(\zeta/2)|g,0\rangle+e^{i\xi}\sin(\zeta/2)|e,0\rangle$ to
$|\psi_{\rm tar}\rangle=\cos(\zeta/2)|g,0\rangle-
ie^{i\xi}\sin(\zeta/2)|g,1\rangle$ by performing a resonant $\pi$ rotation
between the states $|e,0\rangle$ and $|g,1\rangle$ over time
$t_{g}=\pi/2g_{\perp}$ [here $|g,n\rangle$ and $|e,n\rangle$ are respectively
the ground ($g$) and excited ($e$) qubit states with $n$ photons in the
resonator]. For that purpose, the resonant transverse coupling $g_{\perp}$
between the hole spin and the microwave photon can be electrically switched on
and off thanks to the voltage dependence of the hole
$\operatorname{\mathsf{g}}$-factors (see Fig. 3 of the Main Text) [44, 46].
Let us evaluate the fidelity of such an operation in the presence of noise. We
first consider the effect of damping due to photon decay and spin relaxation.
The fidelity is then defined as
$\overline{\operatorname{\mathcal{F}}}_{M}=\overline{\langle\psi_{\rm
tar}|\rho(t_{g})|\psi_{\rm tar}\rangle}$, where $\rho(t_{g})$ is the density
matrix at time $t_{g}$, and the overline denotes an uniform average over the
Bloch sphere parametrized by the angles $\zeta$ and $\xi$. The master equation
for the density matrix of the coupled spin qubit/microwave photon reads in the
presence of photon decay and qubit relaxation:

$\dot{\rho}(t)=-\frac{i}{\hbar}[\mathcal{H}^{\prime},\rho(t)]+\operatorname{\mathnormal{\kappa_{r}}}\mathcal{D}[a]\rho(t)+\gamma_{\downarrow}\mathcal{D}[\sigma_{-}]\rho(t)\,,$
(9)

where $\operatorname{\mathnormal{\kappa_{r}}}$ is the photon decay rate,
$\gamma_{\downarrow}$ is the relaxation rate of the spin qubit,
$\mathcal{D}[O]\rho=O\rho O^{\dagger}-\frac{1}{2}(O^{\dagger}O\rho+\rho
O^{\dagger}O)$ is the damping superoperator, and $\mathcal{H}^{\prime}$ is the
Hamiltonian that includes perturbative corrections such as the qubit Lamb
shift. Equation (9) is valid when $k_{B}T\ll\hbar\operatorname{\omega_{0}}$,
$k_{B}$ being the Boltzmann constant and $T$ the device temperature. This
equation can be solved exactly, which yields the average fidelity
$\overline{\operatorname{\mathcal{F}}}_{M}=\tfrac{1}{2}+\tfrac{1}{3}e^{-a/2}+\tfrac{1}{6}e^{-a}$,
with
$a=\pi(\operatorname{\mathnormal{\kappa_{r}}}+\gamma_{\downarrow})/4g_{\perp}$.
Hence, for $2g_{\perp}\ll\operatorname{\omega_{0}}$ and to first order in
$(\operatorname{\mathnormal{\kappa_{r}}}+\gamma_{\downarrow})t_{g}$, the
average error evaluates to
$1-\overline{\operatorname{\mathcal{F}}}_{M}=\pi(\operatorname{\mathnormal{\kappa_{r}}}+\gamma_{\downarrow})/12g_{\perp}$.
Here we assume that the fidelity is realistically limited by the microwave
photon decay [51], which for
$\operatorname{\mathnormal{\kappa_{r}}}/2\pi=0.5\,{\rm MHz}$ (resonator
quality factor $Q_{r}=\operatorname{\omega_{0}}/\kappa_{r}=10^{4}$ with
$\omega_{0}/2\pi=5\,{\rm GHz}$), results in a minimal process error of a few
percents (see Fig. 5a).

We furthermore analyze the error in the spin-photon transfer process due to
quasistatic charge noise on the gate. The process fidelity due to such
electric noise is defined as

$\overline{\operatorname{\mathcal{F}}}_{c}=\overline{\langle\psi_{\rm
tar}|{\rm E}[U_{g}(V_{G})|\psi_{\rm in}\rangle\langle\psi_{\rm
in}|U_{g}^{\dagger}(V_{G})]|\psi_{\rm tar}\rangle}\,,$ (10)

where $U_{g}(V_{G})$ is the evolution operator at gate voltage $V_{G}$ and
${\rm E}[.]$ is the ensemble average over the random gate voltage fluctuations
$\delta V_{G}=V_{G}-\operatorname{\mathnormal{V}_{\mathnormal{G}}^{0}}$ that
we assume Gaussian. The unitary evolution of the state $|\psi_{\rm
in}\rangle=\cos(\zeta/2)|g,0\rangle+e^{i\xi}\sin(\zeta/2)|e,0\rangle$ under
Hamiltonian (3) of the Main Text close to the resonant condition
$|\delta\omega_{L}|=|\omega_{L}(V_{G})-\operatorname{\omega_{0}}|\ll\operatorname{\omega_{0}}$
yields $U_{g}(V_{G})|\psi_{\rm
in}\rangle=\cos(\zeta/2)|g,0\rangle+\sin(\zeta/2)e^{i\xi}\left([\cos(\omega_{R}t_{g}/2)-i\cos\alpha\sin(\omega_{R}t_{g}/2)]|e,0\rangle-i\sin\alpha\sin(\omega_{R}t_{g}/2)|g,1\rangle\right)$,
with $\cos\alpha=\delta\omega_{L}/\omega_{R}$,
$\sin\alpha=2g_{\perp}/\omega_{R}$, and
$\omega_{R}=\sqrt{4g_{\perp}(V_{G})^{2}+\delta\omega_{L}(V_{G})^{2}}$. This
leads to the fidelity of the quantum state transfer averaged over the initial
state:

$\overline{\operatorname{\mathcal{F}}}_{c}=\frac{1}{3}\left(1+{\rm
E}[\sin\alpha\sin(\omega_{R}t_{g}/2)]+{\rm
E}[\sin^{2}\alpha\sin^{2}(\omega_{R}t_{g}/2)]\right)\,.$ (11)

For small deviation from resonance $|\delta\omega_{L}|\ll
2g_{\perp}(\operatorname{\mathnormal{V}_{\mathnormal{G}}^{0}})$ and $|\delta
g_{\perp}|\ll g_{\perp}(\operatorname{\mathnormal{V}_{\mathnormal{G}}^{0}})$,
the angular frequency of state rotation approximates as $\omega_{R}\approx
2g_{\perp}(\operatorname{\mathnormal{V}_{\mathnormal{G}}^{0}})+2\delta
g_{\perp}+\delta\omega_{L}^{2}/4g_{\perp}(\operatorname{\mathnormal{V}_{\mathnormal{G}}^{0}}),$
and at leading order in the noise
$1-\overline{\operatorname{\mathcal{F}}}_{c}=({\rm
E}[\delta\omega_{L}^{2}]+\pi^{2}{\rm E}[\delta
g_{\perp}^{2}])/8g_{\perp}^{2}$. Close to the optimal $g_{\perp}$
($\operatorname{\mathcal{E}}_{y}\simeq\operatorname{\mathcal{E}}_{y}^{\ast}$),
the second term of the latter expression is much smaller than the first one,
so that

$1-\overline{\operatorname{\mathcal{F}}}_{c}=\frac{1}{2}\left(\frac{\sigma_{V}\beta_{\parallel}}{\operatorname{\mathnormal{V}_{\rm
zpf}}\beta_{\perp}}\right)^{2}\,,$ (12)

with $\sigma_{V}^{2}={\rm E}[\delta V_{G}^{2}]$ the variance of $\delta
V_{G}$. A sweet condition occurs when the longitudinal coupling vanishes
($\beta_{\parallel}=0$) and the error cancels at this leading order in
$\sigma_{V}$. Figure 2 of the Main Text shows the presence of sweet lines in
the vicinity of the magnetic field orientations where the transverse coupling
is maximal. As discussed previously, such sweet regions are expected to occur
in a variety of systems and geometries [73, 56]. In order to assess the
robustness of the sweet lines, we compute the fidelity, Eq. (12), up to second
order in gate voltage fluctuations. Then
$\delta\omega_{L}=\omega_{L}(V_{G})-\operatorname{\omega_{0}}=\delta\omega_{\parallel}+\delta\omega_{\perp}^{2}/2\operatorname{\omega_{0}}$,
where the transverse noise is first-order with respect to the fluctuations,
$\operatorname{\delta\omega}_{\perp}=\frac{\mu_{B}B}{\hbar}\beta_{\perp}\delta
V_{G}$, while the longitudinal noise (as well as $\delta g_{\perp}$) is
expanded up to the second order,
$\operatorname{\delta\omega}_{\parallel}=\operatorname{\delta\omega}_{\parallel}^{(1)}+\operatorname{\delta\omega}_{\parallel}^{(2)}=\frac{\mu_{B}B}{\hbar}\left[\beta_{\parallel}\delta
V_{G}+\frac{1}{2}\gamma_{\parallel}\delta V_{G}^{2}\right]$. Figure 5b
displays the spin-photon quantum state transfer fidelities numerically
calculated this way. It reveals that the sweet lines are remarkably robust to
higher order charge noise so that the process fidelity is overall limited by
the resonator microwave photon decay.

###  .3 Fidelity of the CZ gates based on longitudinal first-order and second-
order parametric couplings

Figure 6: (a) Effective $\operatorname{\mathsf{g}}$-factor and (b) curvature
parameter $\tfrac{\mu_{B}}{2\hbar}\gamma_{\parallel}$ as functions of the
orientation of the magnetic field at $\operatorname{\mathcal{E}}_{y}=0$. The
parameters are the same as in Fig. 2 of the Main Text. (c) Curvature parameter
$\tfrac{\mu_{B}}{2\hbar}\gamma_{\parallel}$ as a function of the electric
field $\operatorname{\mathcal{E}}_{y}$ for different orientations of the
magnetic field. $\gamma_{\parallel}$ is optimal for $\mathbf{B}$ along $y$ and
for $\operatorname{\mathcal{E}}_{y}=0$ (where the first-order spin-photon
couplings $g_{\parallel/\perp}$ are zero).

So far, we have dealt with the coherent interaction between a microwave photon
and a single-hole qubit using the most common transverse coupling. However, at
particular magnetic field orientations, the single-hole qubit also exhibits
pure longitudinal spin-photon coupling [29, 30, 31, 32, 23, 33, 34, 35, 36,
37, 38, 39], either at first order (Fig. 2 of the Main Text) or at second
order (curvature or dispersive coupling, see Fig. 6) with respect to the gate
voltage. We comment on the formal correspondence between the two cases after
Eq. (16) below.

The longitudinal coupling provides a mechanism for efficient readout [22] and
for two-qubit gates based on geometric phases [29, 36, 35]. The CZ gate
mediated by longitudinal coupling for example has the important advantage to
allow for efficient operation even when the resonator state is thermal [34,
35], so that the latter does not need to be cooled down to its ground-state.
Moreover, the static Larmor frequencies of the qubits can be different, since
the longitudinal interaction can be harnessed non-resonantly. We focus here on
the curvature coupling because of its tunability (the interaction is
controlled by a drive) [36, 35], of the enhanced coherence (the curvature is
optimal where the first-order susceptibilities $\beta_{\parallel/\perp}$ are
zero), and of its relation to readout [22].

We address below the fidelity of the CZ gate. For that purpose, we consider an
ensemble of spins located at the anti-nodes of some mode $n$ of the resonator
with angular frequency $\omega_{n}=(n+1)\omega_{0}$ (not necessarily the
fundamental mode $n=0$). The longitudinal coupling between these spins can be
tuned and possibly enhanced by applying a time-dependent voltage
$V_{d}(t)=V_{d}\cos(\omega_{d}t)$ on the gates, with angular frequency
$\omega_{d}$ close to $\omega_{n}$ [36, 35]. The modulated longitudinal
coupling of spin $i$, $g_{\parallel
n}^{(i)}(t)=V_{n}\tfrac{\mu_{B}B}{2\hbar}\beta_{\parallel}^{(i)}+\tilde{g}_{\parallel
n}^{(i)}\cos(\omega_{d}t)$, acquires a time-dependent component proportional
to the curvature of the Larmor frequency:

$\tilde{g}_{\parallel
n}^{(i)}=\frac{1}{2}\frac{\partial^{2}\omega_{L}^{(i)}}{\partial\operatorname{\mathnormal{V}_{\mathnormal{G}}^{0}}^{2}}V_{n}V_{\rm
d}\,,$ (13)

where

$\frac{\partial^{2}\omega_{L}^{(i)}}{\partial\operatorname{\mathnormal{V}_{\mathnormal{G}}^{0}}^{2}}=\frac{\mu_{B}B}{\hbar}\gamma_{\parallel}$
(14)

and

$\gamma_{\parallel}=\left(\operatorname{\mathsf{g}}^{(i)^{\prime\prime}}(\operatorname{\mathnormal{V}_{\mathnormal{G}}^{0}})\cdot\mathbf{b}\right)\cdot\mathbf{n}+\frac{\left(\beta_{\perp}^{(i)}\right)^{2}}{\left|\operatorname{\mathsf{g}}^{(i)}\cdot\mathbf{b}\right|}\,,$
(15)

with $V_{n}=\sqrt{n+1}\operatorname{\mathnormal{V}_{\rm zpf}}$ the voltage
amplitude of mode $n$ at a resonator anti-node [62].

It has been shown that for two spins coupled to a common resonator a polaron
transformation [29, 30, 33, 36, 35] yields the effective, non-resonant two-
spin interaction $H_{\rm 12}=\hbar
J_{12}\sigma_{\parallel}^{(1)}\sigma_{\parallel}^{(2)}$, where
$J_{12}=-\tilde{g}_{\parallel n}^{(1)}\tilde{g}_{\parallel
n}^{(2)}/2\Delta_{n}$ and $\Delta_{n}=\omega_{n}-\omega_{d}$. This interaction
is realized at the discrete times $t_{p}=2\pi p/\Delta_{n}$ ($p=1,2,\dots$) at
which the resonator decouples from the spins dynamics (assuming here
$\Delta_{n}\,,\tilde{g}_{\parallel n}^{(i)}\ll\omega_{d}$). On the other hand,
the two spins are maximally entangled at the two-qubit CZ gate time $t_{\rm
CZ}=\pi/4|J_{12}|$. The two times match when
$\Delta_{n}=2\sqrt{p\tilde{g}_{\parallel n}^{(1)}\tilde{g}_{\parallel
n}^{(2)}}$.

Sources of error for the CZ gate include the decay of the resonator photon
that gives rise to a global dephasing channel for the two-qubit system
(measurement induced dephasing), and the dephasing rate $\gamma_{\phi}$ due to
residual noise on the individual qubits [33, 36, 34, 35, 37]. The fidelity of
the two-qubit CZ gate was calculated in Refs. [33, 36, 35], taking these
processes into account. When $\tilde{g}_{\parallel
n}^{(1)}=\tilde{g}_{\parallel n}^{(2)}=\tilde{g}_{\parallel n}$ and the
damping is small, the gate error is
$1-\overline{\operatorname{\mathcal{F}}}_{\rm
CZ}=\tfrac{4}{5}[\gamma_{\phi}+\operatorname{\mathnormal{\kappa_{\rm
eff}}}(\tilde{g}_{\parallel n}/\Delta_{n})^{2}]t_{\rm
CZ}\sim(2\sqrt{p}\gamma_{\phi}+\operatorname{\mathnormal{\kappa_{\rm
eff}}}/(2\sqrt{p}))/\tilde{g}_{\parallel n}$ [36], where
$\operatorname{\mathnormal{\kappa_{\rm
eff}}}=\operatorname{\mathnormal{\kappa_{r}}}[1+2n_{th}(\omega_{n})]$ is the
effective resonator damping rate that accounts for the finite thermal
occupation of the resonator mode [35]. Thus a larger resonator cycle count $p$
can improve the fidelity, optimal at
$p^{\ast}=\max(1,\,[\operatorname{\mathnormal{\kappa_{\rm
eff}}}/4\gamma_{\phi}])$, where $[\cdot]$ is the integer closest to a number.
This yields the minimal error:

$1-\overline{\operatorname{\mathcal{F}}}_{\rm CZ}^{\ast}\sim
2\max\left(\gamma_{\phi}\,,\sqrt{\operatorname{\mathnormal{\kappa_{\rm
eff}}}\gamma_{\phi}}\right)/\tilde{g}_{\parallel n}\,.$ (16)

We emphasize that close to zero electric field
$\operatorname{\mathcal{E}}_{y}$, the dipolar couplings vanish, so that the
dephasing rate due to quasistatic charge noise $\gamma_{\phi}^{\rm
st}=\tfrac{\mu_{B}B}{\hbar}\gamma_{\parallel}\sigma_{V}^{2}$ is second-order
in $\sigma_{V}$. Moreover, a sufficiently large $p$ allows for the integration
of the entangling gate into a noise decoupling pulse sequence [36]. Indeed,
since the longitudinal coupling term commutes with the qubit Hamiltonian, a
spin echo protocol can make the CZ gate even more robust to quasistatic noise
and decrease the effective dephasing rate $\gamma_{\phi}$ [36, 35]. With the
numerically computed curvature parameter shown in Fig. 6, we estimate
$\tilde{g}_{\parallel n}/2\pi\sim 10\,{\rm MHz}$ with driving $V_{d}\sim
2\,{\rm mV}$ in a magnetic field $B=2\,{\rm T}$ along $y$
($\omega_{L}/2\pi\approx 6\,{\rm GHz}$). Along with rates $\gamma_{\phi}\sim
50\,{\rm kHz}$ [56] and $\operatorname{\mathnormal{\kappa_{\rm eff}}}/2\pi\sim
0.5\,{\rm MHz}$ ($p^{\ast}=16$) the CZ gate error is
$1-\overline{\operatorname{\mathcal{F}}}_{\rm CZ}^{\ast}\sim 1\%$, which could
be improved in the near term by reducing the loss rate of the resonator
microwave photons and by even further mitigating the dephasing noise.

Going back to the formal equivalence between the first and the second-order
longitudinal couplings, we point out that Eq. (16) can also be applied to the
first-order longitudinal interaction by replacing $\tilde{g}_{\parallel
n}\rightarrow 2g_{\parallel}$, $\Delta_{n}\rightarrow\omega_{0}$ [35], and by
properly adapting the dephasing rate parameter. In the quasistatic case, the
latter reads $\gamma_{\phi}^{\rm
st}=\tfrac{\mu_{B}B}{\hbar}\beta_{\parallel}\sigma_{V}$, which is now first-
order in $\sigma_{V}$.

Another interesting application of the longitudinal coupling is readout.
Compared to dispersive readout, performed by detuning a transversely coupled
qubit-cavity system, longitudinal readout yields a much larger signal to noise
ratio for similar parameters [22, 3]. To perform longitudinal readout, the
longitudinal coupling is driven in resonance with the resonator
($\omega_{d}=\omega_{n}$), which for a single-hole qubit, can be performed
with time-dependent gate voltages as above. The measurement is then performed
through homodyne detection and is non-demolition. Furthermore, unlike with
dispersive measurement, Purcell relaxation is not relevant since the
longitudinal interaction commutes with the qubit Hamiltonian.

We finally point out that the second-order longitudinal interaction described
in this section also has a transverse second-order counterpart that gives rise
to a two-photon term in the Hamiltonian,
$\mathcal{H}=\hbar\tilde{g}_{\perp}\sigma_{\perp}[a^{2}+(a^{\dagger})^{2}]$,
where

$\tilde{g}_{\perp}=V_{\rm
zpf}^{2}\frac{\mu_{B}B}{2\hbar}\gamma_{\perp}\,,~{}~{}\gamma_{\perp}=\left|\left(\operatorname{\mathsf{g}}^{\prime\prime}(\operatorname{\mathnormal{V}_{\mathnormal{G}}^{0}})\cdot\mathbf{b}\right)\times\mathbf{n}\right|\,.$
(17)

This allows for resonant two-photon processes relevant for the generation of
non-classical states of radiation [4]. The analysis of these processes is
beyond the scope of the article. The second-order longitudinal interaction
$\gamma_{\parallel}$ also gives rise to a dispersive shift of the spin Larmor
frequency of order $\mathcal{O}(V_{\rm zpf}^{2})$ [37], which is therefore
negligible when $V_{\rm zpf}\ll V_{d}$.

###  .4 Numerical simulation of a realistic device

In order to strengthen the conclusions of this work, we show that the trends
highlighted in Figs. 2 and 3 of the Main Text can be reproduced in a more
realistic setup with non-homogeneous lateral and vertical electric fields
(hence no bias with strict inversion symmetry).

Figure 7: The simulated device. The silicon channel is in red, the silicon
oxide in green and the gates in gray. The whole device is embedded in Si3N4
(not shown). The dot is controlled by the central front gate at potential
$\operatorname{\mathnormal{V}_{\mathnormal{G}}^{0}}=-50\,{\rm mV}$, while the
side gates are grounded. The substrate below is used as a back gate at
potential $V_{BG}$. The iso-probability surface of the ground-state wave-
function that encloses 90% of the hole charge is shown at (a)
$V_{BG}=-50\,{\rm mV}$ and (b) $V_{BG}=-85\,{\rm mV}$.

Figure 8: Maps of (a) the effective $\operatorname{\mathsf{g}}$-factor and
(b), (c) the perpendicular $\beta_{\perp}$ and parallel $|\beta_{\parallel}|$
susceptibilities as a function of the magnetic field orientation
$\mathbf{b}=\mathbf{B}/|\mathbf{B}|$, calculated in the device of Fig. 7 at
$V_{BG}=-77\,{\rm mV}$. (d) Maximum parallel and perpendicular
susceptibilities $\beta_{\parallel}^{\max}$ (red) and $\beta_{\perp}^{\max}$
(blue) as a function of the back gate voltage $V_{BG}$. (e) Curvature
parameter $\gamma_{\parallel}^{\max}$ (red) as a function of the back gate
voltage $V_{BG}$ ($\mathbf{B}\parallel\mathbf{y}$). The contribution from the
first term of Eq. (15) is also plotted in blue.

The device we consider is similar to the one studied in Ref. [44] (see Fig.
7). It consists in a rectangular, $[110]$ oriented silicon nanowire with
$(001)\times(\bar{1}10)$ facets and sides $L_{z}=15\,{\rm nm}$ and
$L_{y}=40\,{\rm nm}$. This nanowire lies on a $25\,{\rm nm}$ thick buried
oxide and a substrate used as a back gate at potential $V_{BG}$. A central
$L_{x}=40\,{\rm nm}$ long front gate overlaps the channel by $S_{y}=10\,{\rm
nm}$. It is insulated from the channel by a $4\,{\rm nm}$ thick SiO2 layer,
and biased at $\operatorname{\mathnormal{V}_{\mathnormal{G}}^{0}}=-50\,{\rm
mV}$.

Two other front gates laid $40\,{\rm nm}$ on the left and right mimic other
qubits or “access” gates to reservoirs. They are grounded in the present
study. The whole device is embedded in Si3N4. The central conductor of the
microwave resonator is connected to the central front gate.

Poisson’s equation for the electrostatic potential landscape in the device is
solved with a finite-volumes method as a function of the back gate voltage
$V_{BG}$. The wave-function of the single hole trapped under the central front
gate is then computed with a 6 bands $\mathbf{k}\cdot\mathbf{p}$ model
(including the nearby split-off bands) discretized using a finite-differences
scheme. The $\operatorname{\mathsf{g}}$-matrices and their derivatives are
finally calculated along the lines of Ref. [44].

The $g$-factor, perpendicular susceptibility $\beta_{\perp}$, and parallel
susceptibility $\beta_{\parallel}$ are plotted as a function of the
orientation of the magnetic field $\mathbf{b}$ in Figs. 8a-c, at back gate
voltage $V_{BG}=-77\,{\rm mV}$. They show the same trends as Fig. 2 of the
Main Text. The maximal susceptibilities $\beta_{\parallel}^{\max}$ and
$\beta_{\perp}^{\max}$ (with respect to the magnetic field orientation) are
plotted as a function of $V_{BG}$ in Fig. 8d. When sweeping the back gate
voltage from positive to negative values, the hole moves from the right facet
of the channel (under the front gate) to the left facet (opposite to the front
gate) [44]. At $V_{BG}\approx-68\,{\rm mV}$, the hole wave-function spans the
whole width of the channel. This is the situation closest to
$\operatorname{\mathcal{E}}_{y}=0$ in the model device of the Main Text,
although there is no strict inversion symmetry in the confinement potential at
this bias. Yet $\beta_{\parallel}^{\max}$ and $\beta_{\perp}^{\max}$ do vanish
as the orientation of $\operatorname{\mathsf{g}}^{\prime}\cdot\mathbf{b}$
reverses when the hole moves from one facet to the other. The longitudinal
curvature parameter $\gamma_{\parallel}^{\max}$ (for
$\mathbf{B}\parallel\mathbf{y}$) is also optimal at that back gate voltage
(see Fig. 8e). Therefore, the device of Fig. 7 behaves qualitatively as the
model device of the Main Text.

From a quantitative point of view, the optimal susceptibilities
$\beta_{\parallel}^{\ast}$, $\beta_{\perp}^{\ast}$ and
$\gamma_{\parallel}^{\ast}$ are of the same order of magnitude as in the model
device. Voltage drops in the oxides and other materials around the channel
however reduce the $\beta$’s by up to 40%. This is (over-)compensated for the
$\gamma$’s by enhanced non-linearities. The smaller the overlap between the
gate and the channel, the larger the ratio between the $y$ and $z$ components
of the electric field (the spin-photon couplings are significantly stronger
for a $S_{y}=10\,{\rm nm}$ than for a $S_{y}=20\,{\rm nm}$ overlap, by up to a
factor two for $\gamma_{\parallel}$). The present device might be further
optimized to limit voltage drops and enhance the electric field along $y$,
with, e.g., the introduction of “face-to-face” gate layouts [79], at the
expense of a more complex manufacturing.