Recent advances in hole-spin qubits

The description of confined hole systems can be based directly on the general four-band formalism just described, eventually supplemented by the split-off band. However, many of the hole-spin qubits of interest are fabricated from low-dimensional systems such as quantum wells or nanowires, where additional gates realize the zero-dimensional confinement. It is therefore useful to describe the effective Hamiltonians governing such 2D or 1D systems. Because such effective models are relatively simple, they are often taken as starting points to characterize various important effects, such as spin relaxation or electrical manipulation of spin qubits.

Quantum wells. For quasi-2D holes the potential in Eq. (2) may be taken as $V(z)$ if the growth direction is along a main crystal axis. Then, for ${\bf B}=0$ and zero in-plane wavevector, ${\bf{k}}_{\parallel}=(k_{x},k_{y})=(0,0)$, Eq. (2) simplifies considerably, giving two decoupled Schrödinger equations in 1D. The two sectors correspond to the $J_{z}=\pm 3/2$ and $J_{z}=\pm 1/2$ subspaces, characterized by different effective masses $m_{0}/(\gamma_{1}\pm 2\gamma_{2})$. The lowest-energy subband is formed by heavy-holes ($J_{z}=\pm 3/2$) and the energy gap to the lowest light-hole subband can be estimated as $\Delta_{\rm LH}\simeq 2\gamma_{2}(\pi\hbar/L)^{2}/m_{0}$, assuming an infinite potential well of width $L$. For more realistic scenarios, the value of $\Delta_{\rm LH}$ depends of the detailed shape of $V(z)$ and, as discussed already, is modified by uniaxial strain. In general, the four-dimensional degeneracy of the Luttinger Hamiltonian has been broken by the confining potential, generating naturally a two-level pseudospin. For the low-energy states, we can identify the effective spin up/down eigenstates with the $J_{z}=\pm 3/2$ eigenvalues, i.e., $\ket{\Uparrow}=\ket{+3/2}$ and $\ket{\Downarrow}=\ket{-3/2}$.

Despite the simple decoupling described above (which, we stress, is exact only at zero in-plane momentum), the light-hole/heavy-hole mixing terms still play a crucial role in the effective 2D Hamiltonian. As seen from Eq. (1), finite values of $k_{x,y}$ will induce matrix elements between the two subspaces, through the action of the $J_{x,y}$ operators. It is this coupling between light- and heavy-hole subbands what generates the desired effective spin-orbit interactions, to be harnessed for quantum computing. Typically, the light-hole/heavy-hole splitting $\Delta_{\rm LH}$ is much larger than energy scales associated with the in-plane motion (such as the 2DHG Fermi energy or, for spin qubits, the quantization energy of lateral quantum dots). Therefore, relevant values of $k_{x,y}$ are sufficiently small to only induce a weak mixing between heavy-holes and light-holes. In this limit, the pseudospin degree of freedom of the lowest-energy subband can still be associated with the $J_{z}=\pm 3/2$ quantum number, and the effective spin-orbit interactions can be conveniently derived by performing a perturbative elimination of higher-energy bands, e.g., with a Schrieffer-Wolff transformation. We list below some of the most relevant terms obtained through this procedure:

	$\displaystyle H^{\rm 2D}_{SOI}=$	$\displaystyle\beta_{D}(\pi_{+}\sigma_{-}+\pi_{-}\sigma_{+})+\beta_{3}(\pi_{+}\pi_{-}\pi_{+}\sigma_{-}+\pi_{-}\pi_{+}\pi_{-}\sigma_{+})$
		$\displaystyle+i\alpha_{R}(\pi_{+}^{3}\sigma_{-}-\pi_{-}^{3}\sigma_{+})+\gamma(B_{+}\pi_{+}^{2}\sigma_{-}+B_{-}\pi_{-}^{2}\sigma_{+}),$		(6)

where $v_{\pm}=v_{x}\pm iv_{y},{\boldsymbol{v}}\in\{{\boldsymbol{\sigma}},{\bf{B}},{\boldsymbol{\pi}}\}$. The first line of Eq. (1.1) lists the linear and cubic Dresselhaus terms, which originate from the lack of inversion symmetry of the crystalline potential, i.e., they derive from $H_{1}$, $H_{3}$, and $H_{E}$ [39]. The spin-orbit interactions listed in the second line appear also in crystals with a center of inversion (such as Si and Ge), and the form given here is obtained from the spherical approximation of the Luttinger Hamiltonian (i.e., assuming $\gamma_{2}=\gamma_{3}$). Specifically, the $\alpha_{R}$ term is the Rashba spin-orbit coupling of heavy-holes, caused by an asymmetric confinement potential [$V(z)\neq V(-z)$]. In contrast to electrons, this type of Rashba spin-orbit coupling is cubic in momentum, rather than linear [45, 46]. The last term of Eq. (1.1), which is proportional to the in-plane components of the magnetic field ${\bf B}$, was found to be relevant when modeling transport of holes in quantum point-contacts [47, 48]. Here, a $z\to-z$ asymmetry is induced by the vector potential which, for an in-plane magnetic field and using the Landau gauge, has the form ${\bf A}=(B_{y}z,-B_{x}z,0)$. A detailed analysis of these effective spin-orbit interactions, including estimates of various coupling coefficients and the leading corrections beyond the spherical approximation, can be found in [36, 39, 40] (and references therein).

Regardless of their origin, all coupling coefficients in Eq. (1.1), and not only $\alpha_{R}$, are strongly affected by the confinement potential $V(z)$. This becomes clear from their perturbative expressions, which depend on matrix elements and energy differences between 2D subband states. In general, the energy denominators of the perturbative expansion are on the order of the heavy-hole/light-hole splitting $\Delta_{\rm LH}$. Since $\Delta_{\rm LH}$ ranges from a few meV to several tens of meV, much less than the bulk band gap, spin-orbit interactions of holes are generally stronger than for $s$-type conduction electrons. Furthermore, it may be expected that a large value of $\Delta_{\rm LH}$ (due to, e.g., a thin well or large strain) will suppress spin-orbit interactions. This general tendency is true, but must be carefully weighted against the strength of the inversion-symmetry breaking mechanism. An interesting example is the non-monotonic dependence of $\alpha_{R}$ on the gate electric field, which was predicted theoretically [45] and observed experimentally [49]. While a moderate gate field enhances the spin-orbit coupling $\alpha_{R}$, due to the stronger asymmetry of $V(z)$ and larger heavy-hole to light-hole matrix elements, the limit of large fields is dominated by the growth of $\Delta_{\rm LH}$, which eventually leads to a decrease of $\alpha_{R}$. A different conclusion is found for the pseudospin-electric coupling contribution to $\beta_{D}$ which, despite being a Desselhaus term, is proportional to the applied gate field as well. Here, the growth of off-diagonal matrix elements was found to dominate over the energy denominators, leading to a monotonic dependence with gate field [39].

Nanowires. The case of one-dimensional systems is less straightforward to discuss since, assuming a confining potential $V(y,z)$ (with holes free to move along $x$), a factorization analogous to the 2D case does not happen at $k_{x}=0$. It is therefore necessary to consider in detail how all values of $J_{z}$ enter the low-energy eigenstates. Nevertheless, several general points discussed above are still valid. A finite spin-orbit coupling in the 1D subbands can be generated either by lack of bulk inversion symmetry or through an asymmetric external gate field, e.g., described by a term $eV(z)=eE_{z}z$ in Eq. (2). In the important case of Si and Ge nanowires, the former type (Dresselhaus spin-orbit interaction) is absent in the bulk, and the main spin-orbit mechanism is expected to be of the Rashba type. However, in contrast to quantum wells, where the Rashba spin-orbit coupling is cubic in momentum [see Eq. (1.1)], for nanowires it depends linearly on $k_{x}$.

The different (cubic or linear) dependence can be related to the exact decoupling between heavy-holes and light-holes at zero subband momentum, which only happens in 2D. For a quantum well, we see in Eq. (1) that the terms with a linear dependence on the in-plane compnents $k_{i}$ (where $i=x,y$) are proportional to $\{J_{i},J_{z}\}$, thus vanish in the $J_{z}=\pm 3/2$ subspace (i.e., at first-order in perturbation theory). These terms can only contribute to the effective spin-orbit interaction at higher-order, introducing a non-linear dependence on $k_{i}$¹¹1For 2D holes, a Rashba interaction linear in momentum is not forbidden by symmetry [36]. However, it requires the contribution of distant conduction bands (see the coupling coefficient $r_{51}^{7h7h}$, given in Section 6.3.3 of Ref. [36]), thus is a small effect which goes beyond the Luttinger Hamiltonian.. In 1D the same argument does not hold: The $V(y,z)$ confinement potential will induce a finite mixing of heavy- and light-holes in the lowest subband, even if $k_{x}=0$. The terms of the Luttinger Hamiltonian proportional to $\{J_{x},J_{j}\}$ (where $j=y,z$) can generate a Rashba spin-orbit interaction linear in $k_{x}$.

Another simple calculation which illustrates the appearance of a linear term in 1D considers a potential $V(y,z)=V_{s}(z)+V_{w}(y)$, where the vertical confinement along $z$ is much stronger than the confinement in the lateral direction, thus justifying the use of an effective 2D Hamiltonian. To first approximation, neglecting spin-orbit interactions and taking ${\bf B}=0$, the lowest 1D subbands are spin degenerate and have orbital wavefuction $\propto\psi_{0}(y)e^{ik_{x}x}$, where $\psi_{0}(y)$ is determined by $V_{w}(y)$ and by the in-plane effective mass. Using the Rashba spin-orbit coupling as described in Eq. (1.1), we obtain an effective 1D Hamiltonian by computing expectation values with respect to the wavefunction along $y$ (note that $\bra{\psi_{0}}p_{y}\ket{\psi_{0}}=\bra{\psi_{0}}p_{y}^{3}\ket{\psi_{0}}=0$, since $\ket{\psi_{0}}$ is a bound state):

$\displaystyle H^{\rm 1D}_{R}\simeq i\alpha_{R}\bra{\psi_{0}}(p_{+}^{3}\sigma_{-}-p_{-}^{3}\sigma_{+})\ket{\psi_{0}}\simeq-6\alpha_{R}\bra{\psi_{0}}p_{y}^{2}\ket{\psi_{0}}p_{x}\sigma_{y}+\ldots,$

(7)

where we only show the linear term, which dominates at small values of $k_{x}$. Including the cubic terms predicts a crossing point at larger $k_{x}$, which has experimental consequences in transport [50]. The above simple argument shows explicitly how the lateral confinement along $y$ changes the dependence on momentum from cubic (in 2D) to linear (in 1D). Equation (7) also indicates that the spin-orbit splitting grows for a tighter confinement along $y$, due to the increase of $\bra{\psi_{0}}p_{y}^{2}\ket{\psi_{0}}$. For actual nanowires, where the confinement energies along $z$ and $y$ are roughly equivalent, this simple approach is not applicable. Explicit calculations based on realistic geometries were carried on in detail, finding a ‘direct Rashba’ spin-orbit interaction linear in momentum which can be extremely large [42, 43]. For nanowires based on Ge and Si, the theoretical spin-orbit coupling energies can reach the meV scale. As we will discuss in Sec. 2.4, results from experiments in Ge/Si core/shell and hut-wire nanowires corroborate these theoretical predictions of a large spin-orbit coupling. In the literature, a spin-orbit coupling linear in momentum, e.g., with a dependence $\alpha_{\rm so}p_{x}\sigma_{y}$ as in Eq. (7), is often charachterized through the spin-orbit length $\lambda_{\rm so}=\hbar/m^{*}\alpha_{\rm so}$, where $m^{*}$ is the effective mass.

We now turn to quantum dots where, for a single hole confined in an external potential, the ground state is a degenerate Kramers doublet at ${\bf B}=0$. These doublets, which are split in energy by an external magnetic field, form a natural basis for a qubit. The qubit states can be labeled using the pseudospin-1/2 states, $\ket{\Uparrow}$ and $\ket{\Downarrow}$. It should be noted that, due to the presence of significant spin-orbit interactions, the states $\ket{\Uparrow}$ and $\ket{\Downarrow}$ cannot be decomposed into a product of spin and orbital degrees of freedom. This fact has important consequences on the properties of the effective $g$-tensor, electric manipulation of spin states, as well as spin relaxation and decoherence.

The $g$-factor of single holes We start by considering one of the most basic properties relevant for spin qubits: the Zeeman splitting induced by an external magnetic field. For holes, this splitting is found to depend on the direction of the magnetic field and on the details of the confinement potential. In large lateral quantum dots formed starting from a 2DHG, where the extent of the wavefunction is much larger in the $xy$ plane than along the $z$ direction, the lowest-energy states can be well approximated by pure heavy holes. Applying time inversion to the Zeeman term of Eq. (2), and projecting it to the heavy-hole subspace gives:

$$H_{Z}\simeq-\left(3\kappa+\frac{27}{4}q\right)\mu_{B}B_{z}\sigma_{z}-\frac{3}{2}q\mu_{B}(B_{x}\sigma_{x}-B_{y}\sigma_{y}),$$

(8)

where holes with $J_{z}=\pm 3/2$ are mapped to the $\sigma_{z}=\pm 1$ pseudospin states. The form of the transverse-coupling term ($\propto q$) reflects the fact that the zincblende and diamond lattices do not have 90-degree rotational symmetry (this term changes sign under 90-degree rotations). Since $\kappa\gg q$ (for example, $\kappa\simeq 1.2$ and $q\simeq 0.01$ in GaAs), the Zeeman splitting predicted by Eq. (8) is strongly anisotropic. It is important to note, however, that Eq. (8) neglects contributions from the admixture of light holes and heavy holes. These contributions can be important and must therefore be included in general.

Interestingly, for the $z$-direction, a finite correction to $g_{z}\simeq 6\kappa$ survives even in the theoretical limit of an infinitely thin well ($L\rightarrow 0$), when the admixture of light-hole states becomes negligible [51, 52, 53]. The finite correction arises in second-order perturbation theory, due to a cancellation between the large gap ($\Delta_{\rm LH}\propto 1/L^{2}$) and the squared transition matrix elements between heavy and light holes (which formally diverge as $1/L$). As the gaps between subbands and all transition matrix elements depend on the detailed confinement potential $V(z)$, the value of $g_{z}$ can be modulated by a top gate and a large variation between $g_{z}\simeq 1$ and $2.5$ was reported for the SiGe self-assembled quantum dots of Ref. [53]. The modulation of $g_{z}$ with gate voltage is a well-documented effect for spin qubits (e.g., in InGaAs [54] and GaAs [55] quantum dots), and we will further discuss it in Sec. 2. Similarly, the in-plane $g$-factors can be strongly affected by the confinement potential, as shown by the following simple example. We consider again a quasi-2D system and assume an elliptic confinement with principal axes along $x$ and $y$. Treating the $\gamma$ term of Eq. (1.1) as a perturbation, and taking the average over the orbital wavefunction, leads to:

$$\delta H_{Z}\simeq 2\gamma(\langle p_{x}^{2}\rangle-\langle p_{y}^{2}\rangle)(B_{x}\sigma_{x}+B_{y}\sigma_{y}).$$

(9)

In elliptical dots, where $\langle p_{x}^{2}\rangle\neq\langle p_{y}^{2}\rangle$, Eq. (9) modifies the in-plane $g$-factors. We see that the change of $g_{x,y}$ can be controlled through the $z$ confinement, entering the $\gamma$ coupling, as well as the $xy$ potential, affecting $\langle p_{x,y}^{2}\rangle\propto\omega_{x,y}$. Since Eq. (8) gives small values of $g_{x,y}$, corrections like Eq. (9) can be particularly relevant. A good example is a recent measurement in a Ge double quantum dot [56]. While $3q\simeq 0.18$ in Ge, the measured in-plane $g$-factors were $\simeq 0.26$ and $-0.16$ for the left and right dot, respectively. A detailed theoretical analysis of the $g$-factor, starting from the Luttinger Hamiltonian [Eq. (1)], finds a contribution $\propto\omega_{x}-\omega_{y}$, which is in agreement with Eq. (9). The interpretation given in Ref. [56] is that, since the two quantum dots are elliptical and have major axes which are approximately perpendicular to each other, the corrections to the in-plane $g$-factors can be comparable or larger than $3q$. Moreover, these corrections will have opposite sign for the two dots. We note that while the effect caputred by Eq. (9) is only relevant for elliptical quantum dots, other spin-orbit couplings [beyond the $\gamma$ term of Eq. (1.1)] can lead to a gate-dependent $g_{x}$ (as well as $g_{z}$) even for circular quantum dots [41].

Similar features are found for the $g$-factor of nanowires. In Ge/Si core/shell nanowires, the $g$-factor can be strongly quenched along the direction of the nanowire [57, 58]. A recent detailed analysis of $g$-factors in a silicon nanowire device, together with the dependence on various gate potentials, can be found in Ref. [59]. A thorough characterization of the gate dependence and anisotropy is important to determine suitable sweet spots, at which the hole-spin qubit is less sensitive to charge noise [60, 61, 41, 59].

Electric-dipole spin resonance. One of the main advantages of hole-spin qubits is the potential for electrical manipulation which, as we will see in Sec. 2, has been demonstrated with impressive performance. Electric-dipole spin resonance (EDSR) relies on a purely electrical drive coupling the hole-spin states. We consider a uniform alternating electric field, ${\bf{E}}(t)=E(t)\hat{x}$ and the associated potential $eE(t)x$. The basic mechanism for EDSR is a non-vanishing transition element $\bra{\Uparrow}eE(t)x\ket{\Downarrow}\neq 0$, which is generally allowed when the Zeeman-split levels are not pure spin states. The combination of spin-orbit coupling and electric field behaves as a net magnetic field, and purely electrical spin rotations can be accomplished when the frequency of the electric field matches the Zeeman splitting between the qubit states.

More detailed considerations are necessary to establish the effectiveness of the electrical drive, as this depends on the geometry of the dot and the type of spin-orbit interaction. Some intuition may be gained from discussing spin-orbit coupling as a perturbation. In this language we see that, unlike electron-spin resonance (ESR, its purely magnetic counterpart), EDSR involves orbital excited states. This is because in a quantum dot the expectation value of the spin-orbit interaction within any one level vanishes, but it causes spin-flip transitions between levels. In other words, its matrix elements are purely off-diagonal. It is simplest to assume a parabolic confinement potential for a circular quantum dot, with eigenstates given by the well-known Fock-Darwin states and where the qubit lies in the orbital ground state, labeled by the orbital quantum number $n=0$. The Dresselhaus terms of Eq. (1.1) connect the ground state $n=0$ to the first orbital excited state $n=1$, and the virtual transition is accompanied by a spin-flip. Since the electric potential $eE(t)x$ also connects the states with $n=0$ and $n=1$, but without an associated spin flip, virtual transitions from $n=0$ to $n=1$ and back appear in second order perturbation theory and couple opposite-spin states. For a circular dot, because of the presence of the electrical potential $eE(t)x$, the virtual transitions must involve the first orbital excited state, thus not every form of spin-orbit coupling leads to EDSR. Importantly, in Eq. (1.1) the Rashba term $\propto\alpha_{R}$ does not yield EDSR for a circular dot, since it connects $n=0$ to $n=3$. Thus, it does not contribute to second-order perturbation theory in conjunction with $eE(t)x$. In the seminal paper that introduced EDSR for holes [62] it is the comparatively weaker interaction $\propto\beta_{3}$ that rotates the spin (linear-$k$ terms were not included in Ref. [62]). If the linear Dresselhaus spin-orbit coupling dominates, the detailed analysis of EDSR becomes analogous to the case for conduction electrons [63].

As the $\alpha_{R}$ term turns out to be ineffective for circular dots, an interesting question concerns the mechanism of EDSR for materials where Dresselhaus terms are absent in the bulk, such as Si and Ge. In this case, a sizable $\beta_{D}$ interaction may be induced through interface inversion asymmetry [64] or through a finite dipolar coupling [39]. Even in the absence of Dresselhaus interactions, the Rashba spin-orbit coupling leads to EDSR if the 2D dot deviates from circular symmetry (and, by extension, in nanowires). Furthermore, the leading cubic-symmetry correction to the Rashba interaction, obtained by going beyond the spherical approximation of $H_{L}$, has the same formal dependence of the term $\propto\beta_{3}$ and has been shown to allow for efficient EDSR spin manipulation in Ge quantum dots [40].

Decoherence. In the earliest days of quantum-dot spin qubits the idea was prevalent that quantum-dot spins ought to exhibit strong coherence properties because the diagonal matrix elements of the spin-orbit interaction vanish (leading to a vanishing pure-dephasing rate [65, 66]). Over time it has been recognized that off-diagonal matrix elements of the spin-orbit coupling have a very strong effect on spin coherence times. Moreover, in contrast to EDSR where only some spin-orbit couplings have a non-vanishing contribution, all spin-orbit couplings can contribute to decoherence. In current hole-spin devices, spin-orbit coupling is usually regarded as the main decoherence engine since, as explained in detail in the next section, the $p$-character of their orbital wave functions ensures the contact hyperfine interaction vanishes, leaving only the weaker orbital and dipolar hyperfine couplings.

It is generally believed that relaxation is associated primarily with phonons, while the main dephasing mechanisms are phonons and background charge fluctuations. The electrical potentials describing phonons and charge noise connect the ground state to all higher orbital excited states. Consequently, phonons and charge noise combined with spin-orbit coupling affect the qubit subspace (in second order perturbation theory). The spin-flip matrix elements induced by phonons are analogous to those enabling EDSR, but become now responsible for qubit relaxation. For holes, the relaxation time $T_{1}$ due to the phonon mechanism has been shown to behave as $T_{1}\propto 1/B^{5}$ for cubic Dresselhaus, and $T_{1}\propto 1/B^{9}$ for Rashba dominated spin-orbit interaction [66]. Recent measurements of $T_{1}$ in a GaAs quantum dot find a dependence close to $\propto 1/B^{5}$, favoring the first scenario [67]. Note, however, that linear spin-orbit coupling gives a $\propto 1/B^{5}$ dependence as well [65]. Thus, the data might be compatible with a dominant $\beta_{D}$ term [39], see Eq. (1.1). For quantum dots based on Si and Ge, theoretical studies of the spin lifetime are also available [57, 60].

As discussed above in some detail, the $g$-factor depends on the electrostatic environment of the hole spin, and this dependence provides a natural decoherence mechanism. Like gate potentials, electric fluctuations induced by phonons and charge noise can modify the $g$-factor and lead to dephasing. For this reason, hole-spin dephasing is often related to the spectral properties of classical charge noise (see for example Refs. [68, 20, 59]). We also note that the EDSR strength is linear in the spin-orbit coupling, while the relaxation and dephasing rates are quadratic. Consequently, the existence of a favorable range of spin-orbit coupling strengths, where fast EDSR is possible with relatively slow relaxation and dephasing rates, is generally expected.

Understanding the hyperfine coupling of electron and hole spins to an uncontrolled environment of nuclear spins is essential for a broad class of qubits based on quantum dots [69, 70, 71, 72]. This is especially true for electron spins in quantum dots based on III-V semiconductors since every naturally occurring isotope of every group III and group V element carries a finite nuclear spin [69], and the conduction band of III-V materials is predominantly $s$-like, resulting in a strong contact interaction. The significant dephasing induced by nuclear spins in early III-V spin qubits was a main motivation to develop alternative platforms, and hole spins in III-V or group IV quantum dots show great promise in this respect. Since the contact hyperfine coupling vanishes for the predominantly $p$-like states of the valence band, the influence of nuclear spins is expected to be generally weaker for holes [73, 33, 74, 75], and the anisotropy of the hole-spin hyperfine interaction admits motional averaging that is not possible for an isotropic contact interaction [73, 76, 77, 78].

A direct approach to suppress decoherence from nuclear spins is through isotopic purification, which is possible for devices based on Si and Ge. However, even a trace amount (800 p.p.m.) of ²⁹Si in some isotopically enriched ²⁸Si devices has been shown to significantly influence electron-spin dynamics [79, 80]. As we will discuss, the hyperfine interaction in the conduction band of Si has a strength comparable to values predicted for the valence band of Si and Ge (see Table 1), suggesting that hyperfine interaction can affect the hole-spin dynamics in many practically relevant scenarios. As effective strategies to cope with charge noise are being developed, the role of hyperfine interaction will presumably grow in importance for Si and Ge hole-spin devices. In fact, coherence times compatible with hyperfine interaction timescales were already reported in Si FinFET devices [81]. Establishing a solid understanding of the hyperfine coupling in each material, and for each type of device, could also allow for alternative strategies, where the hyperfine coupling is exploited as a resource (e.g., by storing quantum information in the nuclear degrees of freedom [82, 80]), rather than suppressed as an unwanted source of decoherence.

Microscopic theory. For an electron moving in the electric and magnetic fields generated by a collection of spinful nuclei (spins $\left\{\mathbf{I}_{l}\right\}$ at sites $\left\{\mathbf{r}_{l}\right\}$), a non-relativistic expansion of the Dirac equation leads to a sum of three terms that couple the electron orbital ($\mathbf{r},\mathbf{p}$) and spin ($\mathbf{S}$) degrees of freedom to the nuclear spin ${\bf I}_{l}$:

$$\mathcal{H}_{\mathrm{hf}}=\sum_{l}\left(\mathcal{H}^{l}_{\mathrm{c}}+\mathcal{H}^{l}_{\mathrm{dip}}+\mathcal{H}^{l}_{\mathrm{orb}}\right).$$

(10)

The first contribution is the Fermi contact interaction, which determines the hyperfine interaction in the conduction band. The explicit expression is (with $\hbar=1$):

$$\mathcal{H}_{\mathrm{c}}^{l}=\frac{\mu_{0}}{4\pi}\frac{8\pi}{3}|\gamma_{S}|\gamma_{l}\delta_{\mathrm{T}}(\mathbf{r}-\mathbf{r}_{l})\mathbf{S}\cdot\mathbf{I}_{l},$$

(11)

where $\gamma_{S}=-2\mu_{\mathrm{B}}$ is the electron gyromagnetic ratio, $\gamma_{l}$ is the nuclear gyromagnetic ratio, and $\delta_{\mathrm{T}}(\mathbf{r})=\frac{1}{4\pi r^{2}}\frac{df_{\mathrm{T}}(r)}{dr}$ is a localized function. This function is written in terms of $f_{\mathrm{T}}(r)=r/(r+r_{\mathrm{T}}/2)$, where $r_{\mathrm{T}}=Z\alpha^{2}a_{B}$ is the Thomson radius for a nucleus of core charge $Z$, with $\alpha\simeq 1/137$ being the fine structure constant and $a_{B}$ the Bohr radius. Since the Fermi contact interaction vanishes in the valence band, the hyperfine interaction of hole spins is entirely due to the dipolar and orbital couplings. The corresponding expressions read:

$$\mathcal{H}_{\mathrm{dip}}^{l}=\frac{\mu_{0}}{4\pi}|\gamma_{S}|\gamma_{l}\mathbf{S}\cdot\overleftrightarrow{D}(\mathbf{r}-\mathbf{r}_{l})\cdot\mathbf{I}_{l},$$

(12)

where the dipolar tensor has elements $D_{\alpha\beta}({\bf r})=\frac{3r_{\alpha}r_{\beta}-r^{2}\delta_{\alpha\beta}}{r^{5}}f_{T}(r)$, and:

$$\mathcal{H}_{\mathrm{orb}}^{l}=\frac{\mu_{0}}{4\pi}|\gamma_{S}|\gamma_{l}\frac{1}{|\mathbf{r}-\mathbf{r}_{l}|^{3}}f_{\mathrm{T}}(|\mathbf{r}-\mathbf{r}_{l}|)\mathbf{L}_{l}\cdot\mathbf{I}_{l},$$

(13)

where $\mathbf{L}_{l}=\left(\mathbf{r}-\mathbf{r}_{l}\right)\times\mathbf{p}$ gives the electron orbital angular momentum about site $l$. Note that in the extreme non-relativistic limit, $r_{\mathrm{T}}\to 0$, $f_{\mathrm{T}}(r)\to 1$ and $\delta_{\mathrm{T}}(\mathbf{r})\to\delta(\mathbf{r})$ reduces to a 3D Dirac delta function. Provided the nonrelativistic Schrödinger equation is used consistently to solve for the electronic states, this limit can still lead to accurate results for the hyperfine parameters of small-$Z$ nuclei (with the small parameter $Z\alpha\ll 1$). However, the results become progressively quantitatively less accurate for large $Z$ [83]. Although it may be appropriate to use solutions to the non-relativistic Schrödinger equation to derive hyperfine couplings from the extreme non-relativistic form of the hyperfine interactions ($r_{\mathrm{T}}\to 0$), it is absolutely necessary to consistently include $r_{\mathrm{T}}\neq 0$ if relativistic solutions for the electronic states are found, since these relativistic solutions may vary rapidly on the scale $r\sim r_{\mathrm{T}}$ [34].

To make use of the microscopic Hamiltonian $\mathcal{H}_{\mathrm{hf}}$ for a qubit described by two energetically well-isolated states $\ket{\Uparrow},\ket{\Downarrow}$, we derive an effective Hamiltonian via the projector $P=\sum_{s=\Uparrow,\Downarrow}\outerproduct{s}{s}$:

$$H_{\mathrm{hf}}=P\mathcal{H}_{\mathrm{hf}}P=\sum_{l}\left(\mathbf{s}\cdot\overleftrightarrow{A}_{l}\cdot\mathbf{I}_{l}+\mathbf{B}_{l}\cdot\mathbf{I}_{l}\right).$$

(14)

Here, we have introduced a pseudospin $\mathbf{s}$ that acts in the space of qubit states [e.g., $s_{z}=(\ket{\Uparrow}\bra{\Uparrow}-\ket{\Downarrow}\bra{\Downarrow})/2$]. For holes, the hyperfine tensor elements $A_{l}^{\alpha\beta}$ and the field components $B_{l}^{\alpha}$ are found from matrix elements of Eqs. (12) and (13) with respect to the qubit states. When $\ket{\Uparrow},\ket{\Downarrow}$ describes a Kramers doublet (two states related by time-reversal), $\mathbf{B}_{l}=0$ identically, which follows from the fact that all terms in $\mathcal{H}_{\mathrm{hf}}$ are odd under time reversal [34]. Furthermore, as discussed, the $\ket{\Uparrow},\ket{\Downarrow}$ states of a hole qubit established in a quantum dot are commonly described through the envelope function approximation. Explicitly, we have ($s=\Uparrow,\Downarrow$):

$$\innerproduct{\sigma\mathbf{r}}{s}\simeq\sqrt{v_{0}}\sum_{j_{z}=-3/2}^{3/2}F^{(s)}_{j_{z}}(\mathbf{r})u_{j_{z}}(\mathbf{r},\sigma),$$

(15)

where $u_{j_{z}}(\mathbf{r},\sigma)$ are the lattice-periodic Bloch amplitudes at crystal momentum ${\bf k}=0$. The Bloch amplitudes are normalized to $\sum_{\sigma}\int_{\Omega}d^{3}r|u_{j_{z}}(\mathbf{r},\sigma)|^{2}=n_{a}$ over the unit cell volume $\Omega$, where $n_{a}$ is the number of atoms in the unit cell and $v_{0}=\Omega/n_{a}$ is the volume per atom. The envelope functions satisfy the normalization condition $\sum_{j_{z}}\int d^{3}r|F^{(s)}_{j_{z}}({\bf r})|^{2}=1$. When the qubit states can be expressed as in Eq. (15), the hyperfine tensor elements can be calculated in a straightforward two-step process. First, a microscopic description of the electronic structure for the translationally invariant crystal gives an estimate for the Bloch amplitudes, $u_{j_{z}}(\mathbf{r},\sigma)$, allowing for a calculation of the material-dependent (but device-independent) part of the hyperfine tensor. Second, modeling of the device or nanostructure may give a reasonable description of the envelope functions, allowing for the final device-dependent hyperfine tensor to be found in terms of the device-independent material parameters.

Hyperfine parameters of holes. Carrying out the above procedure, the device-independent hyperfine interaction with nucleus of isotope $i_{l}$ at site $l$ leads to the following contribution that can be added to the effective Hamiltonian [Eq. (2)] for the envelope functions [33, 34]:

$$\mathrm{H}^{l}=\left[\left(\frac{1}{3}A_{\parallel}^{i_{l}}-\frac{3}{2}A_{\perp}^{i_{l}}\right)\mathbf{J}\cdot\mathbf{I}_{l}+\frac{2}{3}A_{\perp}^{i_{l}}\boldsymbol{\mathcal{J}}\cdot\mathbf{I}_{l}\right]v_{0}\delta({\bf r}-{\bf r}_{l}).$$

(16)

Here, the Dirac delta function bears no relation to the Fermi contact interaction, since Eq. (16) is only determined by the dipolar and orbital terms. The presence of the delta function reflects the fact that the envelope functions are approximately constant over the scale of a unit cell, thus the hyperfine interaction acts on the 4-component spinor $\left(F^{(s)}_{3/2}(\mathbf{r}_{l}),F^{(s)}_{1/2}(\mathbf{r}_{l}),F^{(s)}_{-1/2}(\mathbf{r}_{l}),F^{(s)}_{-3/2}(\mathbf{r}_{l})\right)^{T}$ at the position of the nuclear spin. Notably, Eq. (16) shows that the hyperfine tensor for a general hole-spin qubit (any linear combination of heavy holes and light holes) can be expressed in terms of just two material parameters per isotope: $A_{\parallel}^{i},\,A_{\perp}^{i}$. The dependence on the angular momentum matrices $J_{i}$ takes the same form as in the Zeeman term, see Eq. (2), as it is determined by the tetrahedral point-group symmetry of the zincblende lattice. In particular, a nonzero value of $A_{\perp}^{i}$ is due to the hybridization of $p$ and $d$ orbitals induced by the crystal field [33, 34]. Specializing to an ideal heavy-hole qubit and, assuming for simplicity a spin-independent envelope function $F(\mathbf{r})$, one obtains [33, 34]:

$$H_{\mathrm{hf}}^{\mathrm{HH}}=\sum_{l}v_{0}\left|F(\mathbf{r}_{l})\right|^{2}\left[A^{i_{l}}_{\parallel}s_{z}I^{z}_{l}+A^{i_{l}}_{\perp}\left(s_{x}I^{x}_{l}-s_{y}I^{y}_{l}\right)\right].$$

(17)

Similarly, projecting ${\rm H}^{l}$ to the light-hole subspace gives [33, 34]:

$$H^{\mathrm{LH}}_{\mathrm{hf}}=\frac{1}{3}\sum_{l}v_{0}\left|F(\mathbf{r}_{l})\right|^{2}\left[\left(A^{i_{l}}_{\parallel}-4A^{i_{l}}_{\perp}\right)s_{z}I^{z}_{l}+\left(2A^{i_{l}}_{\parallel}+A_{\perp}^{i_{l}}\right)\left(s_{x}I^{x}_{l}+s_{y}I^{y}_{l}\right)\right],$$

(18)

which corrects the expression given in Appendix D of Ref. [34]. Since typically $A_{\perp}\ll A_{\parallel}$, the above limiting cases lead to hyperfine interactions of very different character. For heavy holes, the Ising term ($\propto s_{z}I^{z}_{l}$) dominates for all nuclear species listed in Table 1. For light holes instead, the in-plane coupling ($\propto s_{x}I^{x}_{l},s_{y}I^{y}_{l}$) is always larger. Interestingly, for the Ga site of GaAs an approximate cancellation of the Ising term occurs in Eq. (18), based on the density-functional theory results of  Ref. [34] (collected in Table 1). Due to these different behaviors, a significant heavy-hole/light-hole mixing will result in a nontrivial form of the hyperfine interaction. For spin qubits based on Si and Ge (where the effect of $d$-orbital hybridization is small, see Table 1), a carefully chosen confinement geometry and magnetic field setting can lead to a qubit that is comparatively insensitive to nuclear-field fluctuations [84]. On the other hand, for self-assembled quantum dots the effect of heavy-hole/light-hole mixing was found to be smaller than $d$-orbital hybridization [33, 85].

Table 1 shows a summary of all hyperfine parameters found within density functional theory in Ref. [34], as well as the valence-band hyperfine parameters obtained in Ref. [86] for germanium. For reference, we also list the hyperfine parameters obtained for the conduction band, which compare well with early estimates in GaAs [87] and Knight shift measurements (see Ref. [34] for details). Remarkably, the strength of the hyperfine coupling for holes in silicon is found to be comparable to the strength of the hyperfine coupling for electrons in a silicon conduction band. This is a consequence of $s$-$p$ hybridization in the conduction band of silicon, resulting in an overall reduction in the contact interaction, rather than any enhancement of the hole hyperfine coupling. The hyperfine parameters in the valence band of Ge are comparable in magnitude to those for Si, and show a similar high degree of anisotropy ($A^{i}_{\parallel}\gg A^{i}_{\perp}$). An important next step will be to experimentally measure the hole hyperfine parameters listed in Table 1. Until now, the transverse “flip-flop” components of the hyperfine tensor have been largely inaccessible, with most measurements focusing on the longitudinal Overhauser field [88, 89, 74, 75]. Hole spin echo envelope modulations (HSEEM) [90] should be highly sensitive to these transverse terms and may provide a path to extracting these parameters in the near future.

Coherent spin manipulation of single holes was first demonstrated in 2011, based on self-assembled InGaAs/GaAs quantum dots [91, 92]. Figure 1(a) shows a typical self-assembled quantum dot, which is obtained upon epitaxial deposition of a thin ‘wetting layer’ of InAs on a GaAs substrate [72]. Due to the lattice mismatch, small InAs islands form spontaneously. The quantum dots are subsequently capped by GaAs which, thanks to the smaller bandgap of InAs, provides three-dimensional confinement for both electrons and holes. The small size of the dot results in large orbital energies $\sim 10$ meV, and allows for operation at liquid-helium temperatures instead of the mK regime.

Single-spin manipulation. A central feature of these spin qubits is that they are optically addressable [72]. Single-spin manipulation can be realized with optical techniques on an ultrafast (ps) timescale and, to achieve this, the system is usually operated in the Voigt geometry, i.e., an in-plane magnetic field is applied perpendicular to the optical axis, see Fig. 1(a). As explained in Sec. 1, see Eq. (17) and Table 1, the Overhauser field acting on heavy-holes is predominantly along $z$. As a consequence, the choice of a Voigt geometry also greatly improves coherence, by suppressing the effect of hyperfine interaction [73, 75]. The relevant states and optical transitions are illustrated in Fig. 1(b) and involve single-hole spin states $\ket{\Uparrow},\ket{\Downarrow}$ and trion states $\ket{\Uparrow\Downarrow\uparrow},\ket{\Uparrow\Downarrow\downarrow}$, where an additional electron-hole pair is present in the quantum dot. By driving resonantly the $\ket{\Downarrow}\leftrightarrow\ket{\Uparrow\Downarrow\downarrow}$ transition, the state $\ket{\Uparrow}$ can be initialized with high fidelity through optical pumping [93]. Readout can be performed in a similar manner, by monitoring the population of $\ket{\Downarrow}$ through the fluorescent light emitted by $\ket{\Uparrow\Downarrow\downarrow}$. As the transition $\ket{\Uparrow}\leftrightarrow\ket{\Uparrow\Downarrow\uparrow}$ remains undriven, $\ket{\Uparrow}$ does not give a readout signal. Finally, spin manipulation can be achieved through short (a few ps) off-resonant optical pulses [91, 92, 94]. As specified in Fig. 1(b), in the Voigt geometry all four transitions between $\ket{\Uparrow},\ket{\Downarrow}$ and $\ket{\Uparrow\Downarrow\uparrow},\ket{\Uparrow\Downarrow\downarrow}$ are dipole-allowed for light with linearly polarization along $x$ or $y$. The selection rules are such that a circularly polarized broadband pulse will drive the stimulated Raman transition $\ket{\Uparrow}\leftrightarrow\ket{\Downarrow}$, thanks to the two available $\Lambda$-systems (for example, $\ket{\Uparrow}\leftrightarrow\ket{\Uparrow\Downarrow\downarrow}\leftrightarrow\ket{\Downarrow}$). Such pulses implement effective Rabi oscillations between $\ket{\Uparrow}$ and $\ket{\Downarrow}$, with a rotation angle which depends on the applied power. Combining the optically induced rotations (along $z$) with the free precession induced by the external magnetic field (along $x$), it is possible to achieve full control on the Bloch sphere. Recently, as an alternative to broadband pulses, a Raman laser technique was introduced for electron spins in InGaAs/GaAs quantum dots [95] and successfully applied to single holes [96, 97].

Coherence. These hole spin qubits display good coherence properties compared to their electron spin counterparts, where strong hyperfine interaction leads to dephasing times $T_{2}^{*}$ of only a few ns. Spectroscopic studies of single holes, based on coherent population trapping, are consistent with dephasing times of at least $100-500~{}$ns [98, 99, 75]. However, shorter values are observed in pulsed experiment. For example, a typical $T^{*}_{2}\sim 20$ ns was extracted from the decay of Ramsey fringes [92], and similar values were obtained from spin-flip Raman scattering [100]. A non-monotonic dependence on the applied magnetic field was explicitly demonstrated in [68], with the longest coherence time $T_{2}^{*}\simeq 70$ ns occurring at $B_{x}=4$ T [68]. Consistently with early reports [91, 92], the high-field dephasing is attributed to to voltage fluctuations affecting the $g$-factor, while nuclear spins likely dominate noise at low fields [68]. Coherence can be effectively prolonged through dynamical decoupling, giving $T^{\rm HE}_{2}\sim 1-2~{}\mu$s for the Hahn-echo decay time [91, 68]. A coherence time of $T_{2}^{\rm CP}\simeq 4~{}\mu$s was achieved through a Carr and Purcell (CP) decoupling sequence of $N_{\pi}=8$ pulses [68]. Further increasing $N_{\pi}$ is not beneficial, due to limitations in the $\pi$-pulse fidelity. Several recent experiments have reported a $\pi$-pulse fidelity around $F_{\pi}\simeq 90\%$ [68, 96, 97].

Entangling operations. For self-assembled quantum dots, scaling-up of single qubits through fabrication is challenging. While quantum dot molecules of holes (formed in two vertically stacked dots) are actively studied [101, 102, 103] and, in fact, the coherent exchange interaction of two holes was demonstrated early on [92], an attractive alternative is to establish entanglement through the emitted photons. In 2016, an entangled state of two distant hole spins (5 m apart) was generated through an heralded protocol, based on the emission of Raman photons [104]. Applying weak pulses resonant with the $\ket{\Downarrow}\leftrightarrow\ket{\Uparrow\Downarrow\downarrow}$ transition results in a small probability of spin flip, $\ket{\Downarrow}\rightarrow\ket{\Uparrow}$, accompanied by the emission of a single Raman photon. If two distant quantum dots initialized in the $\ket{\Downarrow}$ state are simultaneously driven in this manner, a Raman photon can be emitted by either of the dots (emission of two photons is highly unlikely). The experimental apparatus is carefully designed to erase the ‘which-path’ information, such that detection of a single Raman photon heralds the desired entangled state, ideally of the form $(\ket{\Uparrow}\ket{\Downarrow}+e^{-i\theta}\ket{\Downarrow}\ket{\Uparrow})/\sqrt{2}$ [104]. In the actual experiment a fidelity $\sim 55~{}\%$ could be achieved [104].

Hole spins as quantum emitters. Self-assembled quantum dots are excellent quantum emitters of single photons [105, 106, 107]. A number of interesting protocols is based on this property, allowing to generate spin-photon entanglement as well as various classes of entangled photon states. A basic process is the same type of Raman scattering useful for remote entanglement. Upon application of a driving pulse on the ‘diagonal’ transition $\ket{\Downarrow}\leftrightarrow\ket{\Uparrow\Downarrow\uparrow}$, see Fig. 1(b), Raman scattering with probability $p_{1}$ leads to a final state of the form $\sqrt{1-p_{1}}\ket{\Downarrow}\ket{0}+\sqrt{p_{1}}\ket{\Uparrow}\ket{1}$, where the states $\ket{0},\ket{1}$ refer to the emitted photon [108]. Crucially, the process where a photon is emitted but the hole spin returns to the initial state $\ket{\Downarrow}$ should be suppressed, which is achieved by making the ratio $C=\gamma_{v}/\gamma_{d}$ between the vertical/diagonal decay rates of the $\ket{\Uparrow\Downarrow\uparrow}$ trion as large as possible. While in free space $C=1$, a Purcell enhancement of $\gamma_{v}$ by a factor of about 4 was achieved by embedding the quantum dot in a micropillar cavity [109, 108]. Furthermore, by coupling the quantum dot to a photonic crystal waveguide, a large cyclicity $C=14.7$ could be demonstrated [96].

The above scheme can be extended to multiple pulses. A second pulse can further deplete $\ket{\Downarrow}\ket{0}$, leading to a state of the form $\sqrt{1-p_{1}-p_{2}}\ket{\Downarrow}\ket{00}+\ket{\Uparrow}(\sqrt{p_{2}}e^{i\phi}\ket{10}+\sqrt{p_{1}}\ket{01})$. Here we indicate with $\ket{n_{2}n_{1}}$ the photon occupations in the temporal modes after the first ($n_{1}$) and second ($n_{2}$) pulse. The states $\ket{01}\equiv\ket{e}$ and $\ket{10}\equiv\ket{l}$ (‘early’ and ‘late’ phonon, respectively) serve as a basis for a photonic time-bin qubit. An arbitrary time-bin state can be prepared by a suitable choice of $p_{1,2}$ and $\phi$, controlled by the intensities and relative phase of the two pulses [108]. Continuing the process allows to generate time-bin encoded W-states, and sequences up to 4 pulses were implemented [109].

Such protocols can also take advantage of active spin-manipulation. A recent demonstration of spin-photon entanglement [97] starts from $(\ket{\Downarrow}+\ket{\Uparrow})/\sqrt{2}$, obtained after applying a $\pi/2$ spin rotation to $\ket{\Downarrow}$. By driving the ‘vertical’ transition $\ket{\Uparrow}\leftrightarrow\ket{\Uparrow\Downarrow\uparrow}$, a state of the form $(\ket{\Downarrow}\ket{0}+\ket{\Uparrow}\ket{1})/\sqrt{2}$ is obtained. Afterwards, a $\pi$ rotation of the hole spin swaps $\ket{\Uparrow},\ket{\Downarrow}$ and a second excitation pulse is applied, with a controlled phase difference $\phi$. Ideally, the final result is a maximally entangled state $(e^{i\phi}\ket{\Uparrow}\ket{l}-\ket{\Downarrow}\ket{e})/\sqrt{2}$ between the hole-spin and the time-bin qubit. In practice, the state was obtained with a $F_{\rm Bell}\simeq 68\%$ fidelity, mainly limited by incoherent spin flips degrading the gate fidelity of the hole spin [97]. In principle, with additional rotations and excitation pulses, the protocol can be extended to generate GHZ and linear cluster states of multiple time-bin qubits [97, 110].

Photonic cluster states can also be generated following an early ‘photonic machine gun’ proposal [111], which was recently implemented using a hole-spin [112]. At variance with previous discussions, this protocol is based on the optical selection rules of spin states $\ket{\Uparrow}_{z},\ket{\Downarrow}_{z}$, i.e., with quantization axis along $z$. Furthermore, the in-plane magnetic field should be sufficiently small, allowing to neglect its effect during a fast process of optical excitation to the trion and subsequent decay. Under such conditions, linearly polarized resonant pulses induce the entanglement process $\ket{\Uparrow}=(\ket{\Uparrow}_{z}+\ket{\Downarrow}_{z})/\sqrt{2}\to(\ket{\Uparrow}_{z}\ket{\sigma^{-}}+\ket{\Downarrow}_{z}\ket{\sigma^{+}})/\sqrt{2}$, where the photonic qubit is defined by the circular polarization ($\sigma_{\pm}$) of emitted photons. Each pulse adds a photon to the output state and synchronizing the pulses with $\pi/2$ spin rotations along $x$ (induced by the weak magnetic field) generates a one-dimensional photonic cluster state [111]. The experimental protocol applies three excitation pulses and demonstrates significant improvements from a previous realization based on a dark exciton [113]. An extrapolated entanglement length of $\xi_{\rm LE}\simeq 10$ photons is obtained at $B=0.1$ T, substantially extending the previous $\xi_{\rm LE}\simeq 3.2$ [113]. Furthermore, relying on the hole spin allows to reach GHz photon generation rate with a much higher photon indistinguishability [112].

Hole-spin qubits in III-V semiconductors can also be formed in gated structures, which offer better prospects for scalability through nanofabrication. Despite the early success of GaAs quantum dots hosting electron-spin qubits [2], the corresponding hole-spin devices only witnessed significant progress in recent years (see [23] for a detailed review). After developing more stable devices based on undoped heterostrucures, where the 2DHG is formed at a GaAs/AlGaAs interface using a top accumulation gate [114, 115], the charge stability diagram of a GaAs double quantum dot could be mapped in 2014 down to the single-hole regime [116].

At the moment of writing, hole-based double quantum dots in GaAs have been well characterized but spin manipulation is only at a preliminary stage, especially when compared to planar quantum dots in strained Ge quantum wells (see the next section). Besides fabrication issues, a disadvantage of GaAs is that every isotope of Ga and As has non-vanishing spin. Therefore, in contrast to silicon and germanium nanostructures, the nuclear-spin environment cannot be eliminated via isotopic purification. Evidence of dynamic nuclear polarization was reported in Ref. [23] but, as the strength of the hyperfine interaction is much weaker than for electron spins, nuclear spins should not prevent substantial progress in the near future. In this section we will also introduce several basic concepts relevant for all gated quantum dots.

General device operation. An example of a planar double quantum dot in GaAs is shown in Fig. 2(a). The holes are strongly confined along the $z$ direction (perpendicular to the 2DHG) thanks to the barrier induced by band offset at the GaAs/AlGaAs heterointerface, together with the voltage applied to a global top gate [not shown in Fig. 2(a)]. In addition, there is a smooth in-plane potential generated by voltages applied to the top gates labelled in Fig. 2(a). The 2DHG is locally depleted, to confine holes around minima of the potential in the $xy$-plane. The top gates enable the control of on-site energies of individual quantum dots, as well as tunneling barriers between them. It is usual to specify the charge state with the occupation number of each dot, e.g., $(n_{\rm L},n_{\rm R})$ for a double quantum dot (where L/R refers to the left/right dot). Figure 2(b) shows the stability diagram obtained by controlling the voltages applied to two of the gates. In this case, changes in the charge state are inferred from the current $I_{\rm CS}$ through a nearby quantum point contact, serving as a charge sensor. Besides charge sensing, the system can be probed by measuring the current $I_{\rm DOT}$ through the device [see Fig. 2(a)], under various operating conditions. This tunneling spectroscopy approach is very versatile. Application of a finite source-drain bias, or even oscillating voltages at the gates (e.g., inducing EDSR or photon-assisted tunneling [118]), makes it possible for $I_{\rm DOT}$ to probe the excited states. The knowledge of spectral properties of the double quantum dot allows to extract various important parameters, such as tunneling amplitudes and the $g$-factors.

Spin-flip tunneling. The (1,0) and (0,1) charge configurations, where only one hole is present, are already of considerable interest, due to a non-trivial interplay of Zeeman and tunneling interactions [118, 55, 67, 119]. The (1,0) configuration is realized at sufficiently large negative detuning $\epsilon=\epsilon_{\rm L}-\epsilon_{\rm R}$ (where $\epsilon_{\rm L/R}$ are on-site energies). In this case, the ground state at zero magnetic field is a Kramers doublet commonly denoted $\ket{L\Uparrow},\ket{L\Downarrow}$. Instead, a positive detuning gives the (0,1) single-hole states $\ket{R\Uparrow},\ket{R\Downarrow}$. Tunneling becomes important around $\epsilon\approx 0$, where the hole is in a coherent superposition of L/R states. In the device of Fig. 2(a), it was demonstrated that the tunnel amplitudes can be modulated through gate C in a large interval between $0.1-100~{}\mu$eV [23]. Furthermore, in terms of the above basis states, tunneling is characterized by a ‘spin-conserving’ ($t_{\rm N}$) and ‘spin-flip’ ($t_{\rm F}$) amplitude. This should be expected in general since, as discussed at length, $\ket{L\Uparrow},\ket{L\Downarrow}$, $\ket{R\Uparrow},\ket{R\Downarrow}$ are not pure spin states in the presence of strong spin-orbit interactions. Remarkably, $t_{\rm N}$ and $t_{\rm F}$ were found to be of similar magnitude. For example, for an applied magnetic field $B_{z}\simeq$1.4 T, a value $t_{\rm F}\simeq 0.2~{}\mu$eV, slightly larger than $t_{\rm N}\simeq 0.15~{}\mu$eV, was found in Ref. [118]. In the strong-tunneling regime, approximately equal values $t_{\rm N}\simeq t_{\rm F}\simeq 100~{}\mu$eV were reported around $B_{z}\simeq 1$ T [55], while $t_{\rm N}>t_{\rm F}$ for larger $B_{z}$. The relative strength of $t_{\rm N}$ and $t_{\rm F}$ has a significant dependence on $B_{z}$, as the two tunneling amplitudes are affected differently by orbital effects of an out-of-plane magnetic field [118].

Pauli spin blockade. Insight into the microscopic form of the spin-orbit coupling has been gained by studying the Pauli spin blockade [120]. Pauli spin blockade is a fundamental transport phenomenon for double quantum dots where, in the ideal case of conserved total spin, transitions from the $T(1,1)$ spin triplets to the (2,0) charge configurations are not allowed. Starting from the (1,1) charge state, the $T(1,1)\to T(2,0)$ process between triplets is normally not available at small detuning. In fact, $T(2,0)$ involves an excited orbital of the left dot, to satisfy the Pauli principle, thus has a high energy. Furthermore, the transition from triplet states to the singlet, $T(1,1)\to S(2,0)$, is forbidden when total spin is conserved. As a consequence, when the the double dot gets trapped in a $T(1,1)$ state, the current $I_{\rm DOT}$ from the right to the left reservoir is blocked [121].

For holes, however, the large values of $t_{\rm F}$ imply a strong violation of the spin selection rule. Because spin-flip tunneling is not generally blocked, the study of the excited states though tunneling spectroscopy is facilitated by the existence of additional transport channels, which are normally suppressed [117, 122]. An important exception occurs when the external magnetic field $\vec{B}$ is along the effective spin-orbit coupling field $\vec{B}_{\rm SO}$ associated with the tunneling Hamiltonian [123]. Then, the total spin along that direction is conserved, recovering the Pauli spin blockade. A study of the anisotropic dependence of the leakage current on $\vec{B}$ was performed in Ref. [120], where the direction of $\vec{B}_{\rm SO}$ was found to be consistent with a dominant Dresselhaus mechanism. As explained further below, this finding is in agreement with recent measurements of the relaxation time, $T_{1}$ [67].

$g$-factors. As we have discussed in Sec. 1.2, an important consequence of strong spin-orbit coupling is a significant dependence of hole $g$-factors on the confinement potential. For planar quantum dots, the tight confinement along $z$ results in hole carriers which have predominant heavy-hole character, i.e., atomic angular momentum $J_{z}\simeq\pm 3/2$ (for a more precise description, see Sec. 1.1). Because the heavy-hole states differ in angular momentum by $\Delta J_{z}\simeq 3$, the coupling between these states via an in-plane magnetic field (and the associated Zeeman interaction) is highly suppressed. Accordingly, as in Eq. (8), the in-plane heavy-hole effective $g$-factors are small. Indeed, a very small value $g_{\parallel}=-0.04\pm 0.04$ was measured in Ref. [117], similar to what is found in 2DHGs [124, 125, 126]. Instead, the out-of-plane heavy-hole effective $g$-factor of an (isolated) dot was measured in the range $g_{z}\simeq 1.1-1.4$ [119].

Variations of $g_{z}$ are related to the influence of the confinement potential. This influence can be leveraged to enable electrical control of the effective $g$-factors through gate voltages. In Ref. [119], it was shown that the value of $g_{z}$ depends on the voltage applied to the middle gate [labelled C in Fig. 2 (a)]. It is instructive to look at the mechanism behind such variations, since the primary purpose of gate C is to modify the tunneling barrier between dots. Measurements, however, were performed in a regime of large detuning, where the modulation of tunneling amplitude has negligible effect on the hybridization of charge states. Therefore, the observation of a tunable value $g_{z}\simeq 1.1-1.4$ does not reflect a large variation of the tunneling amplitude, but is due to relatively minor changes in the confinement potential of a single dot. For the same reason, there is typically a $\sim 5\%-10\%$ difference in $g$-factors of the two (isolated) dots [119]. More dramatic effects can be observed when the charge states are hybridized. A strongly reduced value $g_{z}=0.85$ was obtained in the regime of large tunnel couplings and zero detuning [55], where the Zeeman-split eigenstates are nontrivial mixtures of all single-hole states $\ket{L\Uparrow},\ket{L\Downarrow},\ket{R\Uparrow},\ket{R\Downarrow}$.

Spin readout and relaxation time. The $\ket{L\Uparrow}\to\ket{L\Downarrow}$ spin relaxation time has been measured in Ref.  [67] as function of $B_{z}$ using a specialized spin-readout protocol. In general, spin readout in gated quantum dots is based on spin-to-charge conversion, i.e., it relies on processes where different spin states lead to distinct charge states. An explicit example (based on Pauli spin blockade) will be described in the next section. To access short relaxation times, however, it is important to overcome bandwidth limitations in charge sensing, and in Ref. [67] it was necessary to rely on a more sophisticated latched charge readout. In short (see Ref. [67] for details), the readout protocol transfers a hole in the $\ket{L\Uparrow}$ state to a metastable (0,1) charge configuration (with $\sim 50\%$ fidelity) via resonant spin-flip tunneling. In contrast, holes in the $\ket{L\Downarrow}$ state are made to tunnel out of the left dot, leaving the system in the $(0,0)$ charge state. By measuring the final charge state, it is possible to infer the spin of the initial state.

Relaxation times range from $T_{1}\simeq 0.3~{}\mu$s (at $B_{z}=1.5$ T) to $T_{1}\simeq 60~{}\mu$s (at $B_{z}=0.5$ T) and follow an approximate $\propto B_{z}^{-5}$ dependence. This behavior supports a dominant Dresselhaus spin-orbit coupling [65, 66], consistently with transport measurements in the Pauli spin blockade [120]. The measured values of $T_{1}$ are in good agreement with theoretical values of the Dresselhaus spin-orbit coupling [39]. We note that hole-spin relaxation times are much shorter than in electron-based GaAs quantum dots, where an extremely long $T_{1}\simeq 60$ s could be demonstrated around 0.6 T [127].

Spin manipulation. In this system, a robust EDSR signal was detected with a continuous drive [55, 119]. The measurements of $g_{z}$ were actually performed through EDSR. Unfortunately, controlled single-spin rotations have not yet been demonstrated. However, coherent oscillations between singlet and triplet states have been observed [23], demonstrating that the double quantum dot is suitable to realize a $S-T_{-}$ qubit [128, 25]. More precisely, deep in the (1,1) region, the ground state at finite $B_{z}$ is a fully polarized triplet $T_{-}(1,1)$, i.e., the spin configuration is $\ket{\Downarrow\Downarrow}$ (minimizing the Zeeman energy). On the other hand, at sufficiently large negative detuning the ground state becomes $S(2,0)$ since, as discussed for the Pauli spin blockade, the triplet states $T(2,0)$ have higher energy. It follows that, when total spin is conserved, the $S$ and $T_{-}$ states must cross in energy as function of detuning. For hole systems, the relatively large value of $t_{\rm F}$ gives a robust $S-T_{-}$ anticrossing point, which can be exploited to induce Landau-Zener transitions. Starting from the singlet ground state and applying Gaussian-shaped detuning pulses, a Landau-Zener-Stückelberg-Majorana interference pattern induced by the $S-T_{-}$ anticrossing could be obtained [23].

As for GaAs quantum dots, development of hole-spin qubits in planar Ge was made possible by the availability of high-quality 2DHGs. Doped quantum wells can reach high mobilities exceeding $10^{6}$ cm²(Vs)^-1 [129, 130], and the first demonstration of a gate-controlled quantum dot [131] was based on an undoped device with 500,000 cm²(Vs)^-1 mobility. Further advances in fabrication are reported in Refs. [132, 133, 134]. Remarkably, in the short time span from 2018 to 2021 experiments have advanced from a tunable single dot to a fully controllable four-qubit processor [20, 19]. A reason behind this rapid progress is the small in-plane effective mass $\sim 0.05m_{0}$ [135, 40] which facilitates fabrication (planar Ge quantum dots can be relatively large).

The four-qubit system is shown in Fig. 3, together with a recently implemented algorithm [21]. To operate the quadruple dot, it is helpful to define suitable linear combinations of the physically applied voltages which result in changes of a single on-site energy or tunnel coupling, without affecting other physical parameters of the qubits array (the so-called ‘virtual gates’ [136]). Efficient strategies to control a larger number of qubits, such as a $4\times 4$ array [137], are being developed. In the following, we will describe in some detail how coherent manipulation of single-hole spins is realized in the quadrupole quantum dot, as well as the operation of singlet-triplet qubits in Ge double quantum dots.

Spin readout. Single-shot spin readout in planar Ge quantum dots can be achieved through the same mechanism responsible for the Pauli spin blockade [138]. Consider a double quantum dot where, deep in the (1,1) region, the low-energy states are of the form $\ket{\Downarrow\Downarrow}$ and $\ket{\Downarrow\Uparrow}$ (due to the different $g$-factors of the two dots). $\ket{\Downarrow\Downarrow}$ is the $T_{-}(1,1)$ triplet which, by detuning the system to the (2,0) region, remains blocked in the same charge configuration (for a brief description of the Pauli spin blockade, see Sec. 2.2). On the other hand, $\ket{\Downarrow\Uparrow}$ has a finite overlap with the $S(1,1)$ singlet, and can evolve to the $S(2,0)$ ground state for an appropriate choice of the detuning ramp rate. This gives a spin-to-charge conversion process for the hole-spin in the right dot, $\ket{\Downarrow}\to(1,1)$ and $\ket{\Uparrow}\to(2,0)$, with the spin in the left dot acting as an ancilla qubit. As described for the GaAs planar dots, in changing the detuning from (1,1) to (2,0) a $S-T_{-}$ anticrossing point is encountered. While this is a source of readout errors, the reduced visibility ($\nu\simeq 56\%$) was mainly attributed to triplet spin relaxation during charge sensing [138], suggesting that a latched readout scheme could be advantageous [139, 140]. Indeed, later experiments adopted a latched Pauli spin blockade readout [20].

Single-spin manipulation and coherence. In the same experiment [138], coherent single-spin manipulation was performed through EDSR, with Rabi frequencies $f_{R}=57~{}$MHz and $93$ MHz for the two dots. Through a Ramsey sequence, a coherence time $T_{2}^{*}=380~{}$ns was extracted, likely limited by charge noise affecting the $g$-factor [138]. More generally, $T_{2}^{*}$ in various quantum dots was found in a range $\sim 150-800$ ns. The coherence time can be extended to $T^{\rm HE}_{2}\sim 1-5~{}\mu$s with a Hahn echo sequence [138, 16, 20] and up to $\sim 100~{}\mu$s by applying $N_{\pi}=1500$ Carr-Purcell-Meiboom-Gill (CPMG) refocusing pulses [20]. Spin lifetime measurements have obtained $T_{1}=1-16$ ms [20]. The quality of the single-qubit gates is remarkable, as it was reported in the range $F=99.4-99.9\%$ for the four quantum dots of Ref. [20]. A more detailed analysis of single-qubit gates recently appeared [19], showing that the strength of the EDSR drive can be optimized to almost reach the $F=99.99\%$ threshold (for one of the four qubits). While it is possible to obtain large values of the Rabi frequency, exceeding 100 MHz, faster spin manipulation comes at the price of unwanted errors and the optimal $F$ is realized at relatively small $f_{R}\simeq 10$ MHz. In the same array, the optimized fidelity of two other qubits exceeds $99.9\%$ and is close to $99.9\%$ for the last qubit. Interestingly, the fidelity shows little degradation when driving multiple qubits, indicating highly local cross-talk. A large value of $F\simeq 99.9\%$ is maintained upon simultaneous drive of any pair of neighboring qubits, and the fidelity is reduced to $\simeq 99\%$ when all four qubits are driven simultaneously [19].

Two-qubit gates. Two-qubit gates can be realized via the exchange interaction. Because the exchange interaction is induced by a finite tunneling amplitude, only neighboring dots are coupled strongly enough to implement two-qubit gates effectively. Furthermore, the coupling strength, $J$, can be controlled through the tunnel barrier between the dots. Of special interest is the (1,1) configuration with equal on-site energies for the two quantum dots. As first demonstrated for electron-spin qubits [141, 142], at this symmetric point the two-qubit gates are more resilient to charge noise, since the value of $J$ becomes insensitive (at leading order) to small variations of the detuning $\epsilon$. In demonstrating a controllable exchange interaction in Ge planar dots, a tunable $J/h$ in the range $\sim 35-65$ MHz was realized around the symmetric point [16]. Tunability of the exchange coupling in the range $J/h\simeq 8-54$ MHz was reported by a more recent experiment [143].

The discussion of various two-qubit gates can be simplified by neglecting the flip-flop terms, e.g., setting $J\boldsymbol{\sigma}_{1}\cdot\boldsymbol{\sigma}_{2}\approx J\sigma_{z,1}\sigma_{z,2}$, where $z$ is the direction of the magnetic field. This approximation becomes accurate when the difference in Zeeman energy between $\ket{\Uparrow\Downarrow}$ and $\ket{\Downarrow\Uparrow}$ is much larger than $J$. In practice, EDSR resonant frequencies of neighbouring dots can differ by a few hundred MHz (the difference is around 260 MHz in Ref. [16]), which is indeed larger than typical values of $J/h$. Note that, since the eigenstates of the Zeeman Hamiltonian are identified with the logical states $\ket{0},\ket{1}$ of the qubit, here the spin quantization axis is chosen conventionally along $z$ (even if the external magnetic field is physically applied in-plane).