Recent advances in hole-spin qubits

In III–V zincblende materials, for which there is no center of inversion symmetry, it is sometimes important to supplement equation (1) and, consequently, the Hamiltonian (2) for the envelope functions, with additional terms. These corrections are known as Dresselhaus spin–orbit couplings and include the k-linear term:

$$\begin{equation} H_1 = \frac{2}{\sqrt{3}} C_k \left[k_x \left\{J_x, J_y^2-J_z^2 \right\}+{\textrm{c.p.}}\right], \end{equation} \tag{ 3 }$$

and a more convoluted k-cubic contribution [36]:

$$\begin{align} H_3&= b_{41} \left[\left\{k_x, k_y^2-k_z^2 \right\} J_x +{\textrm{c.p.}}\right]+ b_{42} \left[\left\{k_x, k_y^2-k_z^2 \right\} J^3_x +{\textrm{c.p.}}\right] \nonumber \\ &\quad + b_{51} \left[\left\{k_x, k_y^2+k_z^2 \right\} \left\{J_x, J_y^2-J_z^2 \right\} +{\textrm{c.p.}}\right]+b_{52} \left[k_x^3 \left\{J_x, J_y^2-J_z^2 \right\} +{\textrm{c.p.}}\right]. \end{align} \tag{ 4 }$$

In particular, the k-cubic terms were shown early on to give a significant spin–orbit splitting in symmetric GaAs quantum wells [38]. A recent theoretical analysis [39] has assessed the relative importance of H₁ over H₃ in GaAs 2DHGs. Furthermore, a different type of coupling induced by inversion-asymmetry, but also depending on the external potential $V(\textbf{r})$, has been discussed [39]:

$$\begin{equation} H_{E}= \frac{e a_B \chi}{\sqrt{3}}\left[E_x \left\{J_y, J_z\right\} + {\textrm{c.p.}}\right], \end{equation} \tag{ 5 }$$

where $\textbf{E} = - \boldsymbol{\nabla}V$ is the local electric field, a_B is the Bohr radius and χ is a material parameter. Equation (5), which involves the gradient of the potential and includes interband matrix elements between heavy- and light-hole states, goes beyond the usual envelope function approximation. This pseudospin-electric coupling physically originates from the interaction between the electric field and the dipole operator, which takes the general form of equation (5) when expressed in the subspace of valence-band states. A first-principles evaluation gives a relatively large value $\chi \simeq 0.2$ in GaAs [39].

For Si and Ge, the above Dresselhaus terms are absent in the bulk. However, it is important to account for the finite strain when modeling, e.g. high-mobility SiGe/Ge/SiGe quantum wells [40, 41] as well as core–shell nanowires [42, 43]. In general, the effect of strain is described by the well-known Bir-Pikus Hamiltonian $H_{\varepsilon} = -(a_v+\frac54 b_v) (\varepsilon_{xx}+\varepsilon_{yy}+\varepsilon_{zz})+b_v (J_x^2 \varepsilon_{xx} + {\textrm{c.p.}})+\frac{2}{\sqrt{3}}d_v (\{J_x,J_y \}\varepsilon_{xy} +{\textrm{c.p.}})$, where $\left\{\varepsilon_{ij}\right\}$ are the strain tensor elements and $a_v,b_v,d_v$ are material-specific deformation potentials [44]. Assuming that only hydrostatic and uniaxial strains are present, i.e. $\varepsilon_{xy} = \varepsilon_{yz} = \varepsilon_{zx} = 0$ and $\varepsilon_{xx} = \varepsilon_{yy}$, the effect of H_ε is to induce different offsets for the heavy-hole ($J_z = \pm 3/2$) and light-hole ($J_z = \pm1/2$) states [40–42].

For quasi-2D holes, the potential in equation (2) may be taken as V(z) if the growth direction is along the main crystal axis. Then, for $\textbf{B} = 0$ and zero in-plane wave vector, $\textbf{k}_{\parallel} = (k_x, k_y) = (0, 0)$, equation (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 $\Delta_{\textrm{LH}}\simeq 2\gamma_2(\pi\hbar/L)^2/m_0$, assuming an infinite potential well of width L. For more realistic scenarios, the value of $\Delta_{\textrm{LH}}$ depends on the detailed shape of V(z) and, as discussed already, is modified by uniaxial strain. In general, the 4D degeneracy of the Luttinger Hamiltonian is broken by the confining potential, naturally generating a two-level pseudospin. For the low-energy states, we identify the effective spin up/down eigenstates with the $J_z = \pm 3/2$ eigenvalues, i.e. ${\left\vert{\Uparrow}\right\rangle} = {\left\vert{+3/2}\right\rangle}$ and ${\left\vert{\Downarrow}\right\rangle} = {\left\vert{-3/2}\right\rangle}$.

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 can be seen from equation (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 that generates the desired effective spin–orbit interactions and can be harnessed for quantum computing. Typically, the light-hole/heavy-hole splitting $\Delta_{\textrm{LH}}$ is much larger than the energy scales associated with the in-plane motions (such as the 2DHG Fermi energy or, for spin qubits, the quantization energy of lateral quantum dots). Therefore, the 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:

$$\begin{align} H^{\textrm{2D}}_\mathrm{SOI}& =\beta_D(\pi_+ \sigma_- +\pi_-\sigma_+) + \beta_{3} (\pi_+\pi_-\pi_+\sigma_- +\pi_-\pi_+\pi_-\sigma_+)\nonumber \\ & \quad +i\alpha_R (\pi_+^3\sigma_- - \pi_-^3\sigma_+)+ \gamma (B_+ \pi_+^2 \sigma_-+B_- \pi_-^2 \sigma_+), \end{align} \tag{ 6 }$$

where $v_\pm = v_x \pm i v_y, {\boldsymbol{v}} \in \{{\boldsymbol{\sigma}}, {\textbf{B}},{\boldsymbol{\pi}}\}$. The first line of equation (6) lists the linear and cubic Dresselhaus terms, which originate from the lack of inversion symmetry of the crystalline potential, i.e. they derive from H₁, H₃ and H_E [39]. The spin–orbit interactions listed in the second line also appear 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 α_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 equation (6), which is proportional to the in-plane component of the magnetic field 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 $\textbf{A} = (B_y z, -B_x z,0)$. 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 the coupling coefficients in equation (6) and not only α_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 $\Delta_{\textrm{LH}}$. Since $\Delta_{\textrm{LH}}$ 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 $\Delta_{\textrm{LH}}$ (e.g. due to a thin well or large strain) will suppress spin–orbit interactions. This general tendency is true, but must be carefully weighed against the strength of the inversion-symmetry breaking mechanism. An interesting example is the non-monotonic dependence of α_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 α_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 $\Delta_{\textrm{LH}}$, which eventually leads to a decrease in α_R . A different conclusion is found for the pseudospin-electric coupling contribution to β_D which, despite being a Dresselhaus term, is proportional to the applied gate field as well. Here, the growth of the off-diagonal matrix elements is found to dominate over the energy denominators, leading to a monotonic dependence with gate field [39].

The case of 1D 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 the lack of bulk inversion symmetry or through an asymmetric external gate field, e.g. described by a term $eV(z) = e E_z z$ in equation (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 equation (6)), for nanowires it depends linearly on k_x .

The different (cubic or linear) dependences 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 equation (1) that the terms with a linear dependence on the in-plane components 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 . ⁸ 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 that 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. In the first approximation, neglecting spin–orbit interactions and taking $\textbf{B} = 0$, the lowest 1D subbands are spin degenerate and have orbital wave fuction $\propto \psi_0(y) e^{ik_x x}$, 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 equation (6), we obtain an effective 1D Hamiltonian by computing expectation values with respect to the wave function along y (note that ${\left\langle{\psi_0}\right\vert} p_y{\left\vert{\psi_0}\right\rangle} =$ ${\left\langle{\psi_0}\right\vert} p_y^3 {\left\vert{\psi_0}\right\rangle} = 0$, since ${\left\vert{\psi_0}\right\rangle}$ is a bound state):

$$\begin{align} H^{\textrm{1D}}_{R}\simeq i\alpha_R {\left\langle{\psi_0}\right\vert} (p_+^3\sigma_- - p_-^3\sigma_+){\left\vert{\psi_0}\right\rangle} \simeq -6 \alpha_R {\left\langle{\psi_0}\right\vert} p_y^2{\left\vert{\psi_0}\right\rangle} p_x \sigma_y + \ldots, \end{align} \tag{ 7 }$$

where we only show the linear term, which dominates at small values of k_x . Including the cubic terms, this 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 in ${\left\langle{\psi_0}\right\vert} p_y^2{\left\vert{\psi_0}\right\rangle}$. 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 out 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 will be discussed in section 2.4, the results from experiments on 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 $\alpha_{\textrm{so}} p_x \sigma_y$ as in equation (7), is often characterized through the spin–orbit length $\lambda_{\textrm{so}} = \hbar/m^*\alpha_{\textrm{so}}$, where $m^*$ is the effective mass.

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 wave function 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 equation (2) and projecting it to the heavy-hole subspace gives:

$$\begin{equation} H_Z \simeq -\left(3\kappa +\frac{27}{4}q \right)\mu_\mathrm{B} B_z \sigma_z -\frac{3}{2} q \mu_\mathrm{B} (B_x \sigma_x - B_y \sigma_y), \end{equation} \tag{ 8 }$$

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^∘ rotational symmetries (this term changes sign under 90^∘ rotations). Since $\kappa \gg q$ (for example, $\kappa \simeq 1.2$ and $q \simeq 0.01$ in GaAs), the Zeeman splitting predicted by equation (8) is strongly anisotropic. However, it is important to note that equation (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 [51–53]. The finite correction arises in second-order perturbation theory, due to a cancellation between the large gap ($\Delta_{\textrm{LH}} \propto 1/L^{2}$) 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 [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 this in section 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 γ term of equation (6) as a perturbation and taking the average over the orbital wave function leads to:

$$\begin{equation} \delta H_Z \simeq 2\gamma (\langle p_x^2 \rangle - \langle p_y^2 \rangle )(B_x \sigma_x + B_y \sigma_y) . \end{equation} \tag{ 9 }$$

In elliptical dots, where $\langle p_x^2 \rangle \neq \langle p_y^2 \rangle$, equation (9) modifies the in-plane g-factors. It can be seen that the change of $g_{x,y}$ can be controlled through the z confinement, entering the γ coupling, as well as the xy potential, affecting $\langle p_{x,y}^2 \rangle \propto \omega_{x,y}$. Since equation (8) gives small values of $g_{x,y}$, corrections such as equation (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 dots, respectively. A detailed theoretical analysis of the g-factor, starting from the Luttinger Hamiltonian (equation (1)), finds a contribution $\propto \omega_x -\omega_y$, which is in agreement with equation (9). The interpretation given in [56] is that, since the two quantum dots are elliptical and have major axes that are approximately perpendicular to each other, the corrections to the in-plane g-factors can be comparable to or larger than 3 q. Moreover, these corrections will have opposite sign for the two dots. We note that while the effect captured by equation (9) is only relevant for elliptical quantum dots, other spin–orbit couplings (beyond the γ term of equation (6)) 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 their dependence on various gate potentials, can be found in [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 [41, 59–61].

One of the main advantages of hole-spin qubits is the potential for electrical manipulation which, as will be seen in section 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, $\textbf{E}(t) = E(t)\hat{x}$ and the associated potential $eE(t)x$. The basic mechanism for EDSR is a non-vanishing transition element ${\left\langle{\Uparrow}\right\vert} eE(t)x {\left\vert{\Downarrow}\right\rangle} \neq 0$, 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 (its purely magnetic counterpart), EDSR involves orbital excited states. 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 equation (6) 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, due to 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 equation (6) 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 [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 α_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 β_D interaction may be induced through an 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 on the term $\propto \beta_3$ and has been shown to allow for efficient EDSR spin manipulation in Ge quantum dots [40].

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 the off-diagonal matrix elements of the spin–orbit coupling have a very strong effect on the 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 that the contact hyperfine interaction vanishes, leaving only the weak 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 responsible for qubit relaxation. For holes, the relaxation time T₁ 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 T₁ in a GaAs quantum dot find a dependence close to $\propto\!\!1/B^5$, favoring the first scenario [67]. However, note that linear spin–orbit coupling gives a $\propto\!\!1/B^5$ dependence as well [65]. Thus, the data might be compatible with a dominant β_D term [39], see equation (6). 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. As in the case of 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 the classical charge noise (see, for example [20, 59, 68]). 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.

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 (S) degrees of freedom to the nuclear spin I_l :

$$\begin{equation} \mathcal{H}_\mathrm{hf}=\sum_l\left(\mathcal{H}^l_\mathrm{c}+\mathcal{H}^l_\mathrm{dip}+\mathcal{H}^l_\mathrm{orb}\right). \end{equation} \tag{ 10 }$$

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

$$\begin{equation} \mathcal{H}_\mathrm{c}^l=\frac{\mu_0}{4\pi}\frac{8\pi}{3}|\gamma_S|\gamma_l\delta_\mathrm{T}(\mathbf{r}-\mathbf{r}_l)\mathbf{S}\cdot\mathbf{I}_l, \end{equation} \tag{ 11 }$$

where $\gamma_S = -2\mu_\mathrm{B}$ is the electron gyromagnetic ratio, γ_l is the nuclear gyromagnetic ratio and $\delta_\mathrm{T}(\mathbf{r}) = \frac{1}{4\pi r^2}\frac{\mathrm{d} f_\mathrm{T}(r)}{\mathrm{d}r}$ is a localized function. This function is written in terms of $f_\mathrm{T}(r) = r/(r+r_\mathrm{T}/2)$, where $r_\mathrm{T} = Z\alpha^2 a_\mathrm{B}$ is the Thomson radius for a nucleus of core charge Z, with $\alpha\simeq 1/137$ being the fine structure constant and $a_\mathrm{B}$ 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:

$$\begin{equation} \mathcal{H}_\mathrm{dip}^l=\frac{\mu_0}{4\pi}|\gamma_S|\gamma_l\mathbf{S}\cdot\overleftrightarrow{D}(\mathbf{r}-\mathbf{r}_l)\cdot\mathbf{I}_l, \end{equation} \tag{ 12 }$$

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

$$\begin{equation} \mathcal{H}_\mathrm{orb}^l=\frac{\mu_0}{4\pi}|\gamma_S|\gamma_l\frac{1}{|\mathbf{r}-\mathbf{r}_l|^3}f_\mathrm{T}(|\mathbf{r}-\mathbf{r}_l|)\mathbf{L}_l\cdot\mathbf{I}_l, \end{equation} \tag{ 13 }$$

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 more 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 $r_\mathrm{T}\ne 0$ if relativistic solutions for the electronic states are to be found, since these relativistic solutions may vary rapidly on the scale $r\sim r_\mathrm{T}$ [34].

Making use of the microscopic Hamiltonian $\mathcal{H}_\mathrm{hf}$ for a qubit described by two energetically well-isolated states ${\left\vert{\Uparrow}\right\rangle},{\left\vert{\Downarrow}\right\rangle}$, we derive an effective Hamiltonian via the projector $P = \sum_{s = \Uparrow,\Downarrow}{\left\vert {s}\left\rangle\right\langle{s} \right\vert}$:

$$\begin{equation} H_\mathrm{hf} = P\mathcal{H}_\mathrm{hf}P = \sum_l\left( \mathbf{s}\cdot\overleftrightarrow{A}_l\cdot\mathbf{I}_l+\mathbf{B}_l\cdot\mathbf{I}_l\right). \end{equation} \tag{ 14 }$$

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

$$\begin{equation} {\langle {\sigma \mathbf{r}}\mid{s} \rangle} \simeq \sqrt{v_0} \sum_{j_z=-3/2}^{3/2}F^{(s)}_{j_z}(\mathbf{r})u_{j_z} (\mathbf{r},\sigma), \end{equation} \tag{ 15 }$$

where $u_{j_z} (\mathbf{r},\sigma)$ are the lattice-periodic Bloch amplitudes at crystal momentum $\textbf{k} = 0$. 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 Ω, 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 $\sum_{j_z}\int d^3 r|F^{(s)}_{j_z}(\textbf{r})|^2 = 1$. When the qubit states can be expressed as in equation (15), the hyperfine tensor elements can be calculated in a straightforward two-step process. First, a microscopic description of the electronic structure of 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.

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 (equation (2)) for the envelope functions [33, 34]:

$$\begin{equation} \mathrm{H}^l = \left[\left(\frac{1}{3}A_\parallel^{i_l}-\frac{3}{2}A_\perp^{i_l}\right)\mathbf{J}\cdot\mathbf{I}_l+\frac{2}{3}A_\perp^{i_l}\boldsymbol{\mathcal{J}}\cdot\mathbf{I}_l\right] v_0 \delta({\textbf{r}}-{\textbf{r}}_l). \end{equation} \tag{ 16 }$$

Here, the Dirac delta function bears no relation to the Fermi contact interaction since equation (16) is only determined by the dipolar and orbital terms. The presence of the delta function reflects the fact that the envelope function is approximately constant over the scale of a unit cell. Thus, the hyperfine interaction acts on the four-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, equation (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 equation (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]:

$$\begin{equation} H_\mathrm{hf}^\mathrm{HH} =\sum_lv_0\left|F(\mathbf{r}_l)\right|^2\left[A^{i_l}_{\parallel}s_zI^{z}_l+A^{i_l}_{\perp}\left(s_xI^{x}_l-s_yI^{y}_l\right)\right]. \end{equation} \tag{ 17 }$$

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

$$\begin{equation} H^\mathrm{LH}_{\mathrm{hf}} =\frac13 \sum_lv_0\left|F(\mathbf{r}_l)\right|^2\left[\left(A^{i_l}_{\parallel} - 4 A^{i_l}_{\perp}\right) s_zI^{z}_l+ \left(2A^{i_l}_{\parallel} + A_{\perp}^{i_l}\right)\left(s_xI^{x}_l+s_yI^{y}_l\right)\right], \end{equation} \tag{ 18 }$$

which corrects the expression given in appendix D of [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 ($\propto s_zI^z_l$) dominates for all nuclear species listed in table 1. For light holes instead, the in-plane coupling ($\propto s_xI^x_l, s_yI^y_l$) is always larger. Interestingly, for the Ga site of GaAs, an approximate cancellation of the Ising term occurs in equation (18), based on the density-functional theory results of [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 [34], as well as the valence-band hyperfine parameters obtained in [86] for germanium. As a 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 [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 the 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 similarly 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. To date, the transverse 'flip-flop' components of the hyperfine tensor have been largely inaccessible, with most measurements focusing on the longitudinal Overhauser field [74, 75, 88, 89]. Hole-spin echo envelope modulations [90] should be highly sensitive to these transverse terms and may provide a path to extracting these parameters in the near future.

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 figure 1(a). As explained in section 1, see equation (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 figure 1(b) and involve single-hole spin states ${\left\vert{\Uparrow}\right\rangle},{\left\vert{\Downarrow}\right\rangle}$ and trion states ${\left\vert{\Uparrow\Downarrow\uparrow}\right\rangle},{\left\vert{\Uparrow\Downarrow\downarrow}\right\rangle}$, where an additional electron–hole pair is present in the quantum dot. By driving resonantly the ${\left\vert{\Downarrow}\right\rangle}\leftrightarrow{\left\vert{\Uparrow\Downarrow\downarrow}\right\rangle}$ transition, the state ${\left\vert{\Uparrow}\right\rangle}$ can be initialized with high fidelity through optical pumping [93]. Readout can be performed in a similar manner, by monitoring the population of ${\left\vert{\Downarrow}\right\rangle}$ through fluorescent light emitted by ${\left\vert{\Uparrow\Downarrow\downarrow}\right\rangle}$. As the transition ${\left\vert{\Uparrow}\right\rangle}\leftrightarrow{\left\vert{\Uparrow\Downarrow\uparrow}\right\rangle}$ remains undriven, ${\left\vert{\Uparrow}\right\rangle}$ does not give a readout signal. Finally, the spin manipulation can be achieved through short (a few ps) off-resonant optical pulses [91, 92, 94]. As specified in figure 1(b), in the Voigt geometry all four transitions between ${\left\vert{\Uparrow}\right\rangle},{\left\vert{\Downarrow}\right\rangle}$ and ${\left\vert{\Uparrow\Downarrow\uparrow}\right\rangle},{\left\vert{\Uparrow\Downarrow\downarrow}\right\rangle}$ are dipole-allowed for light with linear polarization along x or y. The selection rules are such that a circularly polarized broadband pulse will drive the stimulated Raman transition ${\left\vert{\Uparrow}\right\rangle} \leftrightarrow {\left\vert{\Downarrow}\right\rangle}$, due to the two available Λ-systems (for example, ${\left\vert{\Uparrow}\right\rangle}\leftrightarrow{\left\vert{\Uparrow\Downarrow\downarrow}\right\rangle} \leftrightarrow {\left\vert{\Downarrow}\right\rangle}$). These pulses implement effective Rabi oscillations between ${\left\vert{\Uparrow}\right\rangle}$ and ${\left\vert{\Downarrow}\right\rangle}$, with a rotation angle that 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].

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 [75, 98, 99]. However, shorter values are observed in pulsed experiments. As an 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]. Consistent with early reports [91, 92], the high-field dephasing is attributed to voltage fluctuations affecting the g-factor, while nuclear spins likely dominate the noise at low fields [68]. Coherence can be effectively prolonged through dynamical decoupling, giving $T^{\textrm{HE}}_2$ ∼ 1–2 µs for the Hahn-echo decay time [68, 91]. The coherence time $T_2^{\textrm{CP}} \simeq 4\,\mu$s was achieved through a Carr and Purcell (CP) decoupling sequence of $N_\pi = 8$ pulses [68]. Further increasing N_π is not beneficial, due to limitations in the π-pulse fidelity. Several recent experiments have reported a π-pulse fidelity around $F_\pi\simeq 90\%$ [68, 96, 97].

For self-assembled quantum dots, scaling-up of single qubits through fabrication is challenging. While quantum-dot molecules formed by holes (in two vertically stacked dots) are actively studied [101–103] 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 a heralded protocol, based on the emission of Raman photons [104]. Applying weak pulses resonant with the ${\left\vert{\Downarrow}\right\rangle}\leftrightarrow{\left\vert{\Uparrow\Downarrow\downarrow}\right\rangle}$ transition results in a small probability of spin flip, ${\left\vert{\Downarrow}\right\rangle}\rightarrow{\left\vert{\Uparrow}\right\rangle}$, accompanied by the emission of a single Raman photon. If two distant quantum dots initialized in the ${\left\vert{\Downarrow}\right\rangle}$ state are simultaneously driven in this manner, a Raman photon can be emitted by either of the dots (emission of two Raman photons is highly unlikely). The experimental apparatus is carefully designed to erase the 'which-path' information, so that the detection of a single Raman photon heralds the desired entangled state, ideally of the form $({\left\vert{\Uparrow}\right\rangle}{\left\vert{\Downarrow}\right\rangle}+ e^{-i\theta}{\left\vert{\Downarrow}\right\rangle} {\left\vert{\Uparrow}\right\rangle})/\sqrt{2}$ [104]. In the actual experiment, a fidelity ∼55 % could be achieved [104].

Self-assembled quantum dots are excellent quantum emitters of single photons [105–107]. A number of interesting protocols are based on this property, making it possible to generate spin-photon entanglement as well as various classes of entangled photon states. The basic process is the same type of Raman scattering useful for remote entanglement. Upon application of a driving pulse on the 'diagonal' transition ${\left\vert{\Downarrow}\right\rangle}\leftrightarrow{\left\vert{\Uparrow\Downarrow\uparrow}\right\rangle}$, see figure 1(b), Raman scattering with probability p₁ leads to a final state of the form $\sqrt{1-p_1}{\left\vert{\Downarrow}\right\rangle} {\left\vert{0}\right\rangle} +$ $\sqrt{p_1}{\left\vert{\Uparrow}\right\rangle} {\left\vert{1}\right\rangle}$, where the states ${\left\vert{0}\right\rangle},{\left\vert{1}\right\rangle}$ refer to the emitted photon [108]. Crucially, the process where a photon is emitted but the hole spin returns to the initial state ${\left\vert{\Downarrow}\right\rangle}$ should be suppressed, which is achieved by making the ratio $C = \gamma_v/\gamma_d$ between the vertical/diagonal decay rates of the ${\left\vert{\Uparrow\Downarrow\uparrow}\right\rangle}$ trion as large as possible. In free space C = 1, but a Purcell enhancement of $\gamma_v$ by a factor of about four could be achieved by embedding the quantum dot in a micropillar cavity [108, 109]. 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 ${\left\vert{\Downarrow}\right\rangle} {\left\vert{0}\right\rangle}$, leading to a state of the form $\sqrt{1-p_1-p_2}{\left\vert{\Downarrow}\right\rangle} {\left\vert{00}\right\rangle} + {\left\vert{\Uparrow}\right\rangle}(\sqrt{p_2}e^{i\phi} {\left\vert{10}\right\rangle} + \sqrt{p_1}{\left\vert{01}\right\rangle} )$. Here, we indicate with ${\left\vert{n_2n_1}\right\rangle}$ the photon occupations in the temporal modes after the first (n₁) and second (n₂) pulses. The states ${\left\vert{01}\right\rangle}\equiv{\left\vert{e}\right\rangle}$ and ${\left\vert{10}\right\rangle}\equiv{\left\vert{l}\right\rangle}$ ('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 φ, controlled by the intensities and relative phases of the two pulses [108]. Continuing the process allows us to generate time-bin encoded W-states, and sequences of up to four pulses were implemented [109].

These protocols can also take advantage of active spin manipulation. A recent demonstration of the spin-photon entanglement [97] starts from $({\left\vert{\Downarrow}\right\rangle} + {\left\vert{\Uparrow}\right\rangle} )/\sqrt{2}$, obtained after applying a $\pi/2$ spin rotation to ${\left\vert{\Downarrow}\right\rangle}$. By driving the 'vertical' transition ${\left\vert{\Uparrow}\right\rangle}\leftrightarrow{\left\vert{\Uparrow\Downarrow\uparrow}\right\rangle}$, a state of the form $({\left\vert{\Downarrow}\right\rangle} {\left\vert{0}\right\rangle} + {\left\vert{\Uparrow}\right\rangle} {\left\vert{1}\right\rangle})/\sqrt{2}$ is obtained. Afterwards, a π rotation of the hole spin swaps ${\left\vert{\Uparrow}\right\rangle},{\left\vert{\Downarrow}\right\rangle}$ and a second excitation pulse is applied, with a controlled phase difference φ. Ideally, the result is a maximally entangled state $( e^{i \phi}{\left\vert{\Uparrow}\right\rangle} {\left\vert{l}\right\rangle}-{\left\vert{\Downarrow}\right\rangle} {\left\vert{e}\right\rangle})/\sqrt{2}$ between the hole-spin and the time-bin qubit. In practice, the state is obtained with a $F_{\textrm{Bell}} \simeq 68\%$ 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 Greenberger–Horne–Zeilinger (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 ${\left\vert{\Uparrow}\right\rangle}_z,{\left\vert{\Downarrow}\right\rangle}_z$, i.e. with the quantization axis along z. Furthermore, the in-plane magnetic field should be sufficiently small, allowing us to neglect its effect during a fast process of optical excitation of the trion and subsequent decay. Under these conditions, linearly polarized resonant pulses induce the entanglement process ${\left\vert{\Uparrow}\right\rangle} = ({\left\vert{\Uparrow}\right\rangle}_z+{\left\vert{\Downarrow}\right\rangle}_z)/\sqrt{2} \to ({\left\vert{\Uparrow}\right\rangle}_z {\left\vert{\sigma^-}\right\rangle}+{\left\vert{\Downarrow}\right\rangle}_z {\left\vert{\sigma^+}\right\rangle})/\sqrt{2}$, 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 1D 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 $\xi_{\textrm{LE}} \simeq 10$ photons is obtained at B = 0.1 T, substantially extending the previous $\xi_{\textrm{LE}} \simeq 3.2$ [113]. Furthermore, relying on the hole spin allows this protocol to reach GHz photon generation rates with a much higher photon indistinguishability [112].

An example of a planar double quantum dot in GaAs is shown in figure 2(a). The holes are strongly confined along the z direction (perpendicular to the 2DHG) due to the barrier induced by the band offset at the GaAs/AlGaAs heterointerface, together with the voltage applied to a global top gate (not shown in figure 2(a)). In addition, there is a smooth in-plane potential generated by voltages applied to the top gates labeled in figure 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. The charge state is usually specified with the occupation number of each dot, e.g. $(n_{\textrm{L}},n_{\textrm{R}})$ 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_CS through a nearby quantum point contact, serving as a charge sensor. In addition to charge sensing, the system can be probed by measuring the current I_DQD through the device (see figure 2(a)), under various operating conditions. This tunneling spectroscopy approach is very versatile. The 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_DQD to probe the excited states. From the spectral properties of the double quantum dot, various important parameters can be extracted, such as tunneling amplitudes and the g-factors.

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The (1,0) and (0,1) charge configurations, where only one hole is present, are already of considerable interest, due to a nontrivial interplay of Zeeman and tunneling interactions [55, 67, 118, 119]. The (1,0) configuration is realized at sufficiently large negative detuning $\epsilon = \epsilon_{\textrm{L}}-\epsilon_{\textrm{R}}$ (where $\epsilon_{\mathrm{L/R}}$ are on-site energies). In this case, the ground state at zero magnetic field is a Kramers doublet commonly denoted as ${\left\vert{L\Uparrow}\right\rangle},{\left\vert{L\Downarrow}\right\rangle}$. Instead, a positive detuning gives the (0,1) single-hole states ${\left\vert{R\Uparrow}\right\rangle},{\left\vert{R\Downarrow}\right\rangle}$. Tunneling becomes important around $\epsilon$ ≈ 0, where the hole is in a coherent superposition of L/R states. In the device of figure 2(a), it was demonstrated that the tunnel amplitudes can be modulated through gate C in a large interval between $0.1 ~\mathrm{and}~ 100\,\mu$eV [23]. Furthermore, in terms of the above basis states, tunneling is characterized by a 'spin-conserving' (t_N) and 'spin-flip' (t_F) amplitude. This should be expected in general since, as discussed at length, ${\left\vert{L\Uparrow}\right\rangle},{\left\vert{L\Downarrow}\right\rangle}$, ${\left\vert{R\Uparrow}\right\rangle},{\left\vert{R\Downarrow}\right\rangle}$ are not pure spin states in the presence of strong spin–orbit interactions. Remarkably, t_N and t_F were found to be of similar magnitude. For example, for an applied magnetic field $B_z \simeq1.4$ T, a value $t_{\textrm{F}} \simeq 0.2\,\mu$eV, slightly larger than $t_{\textrm{N}}\simeq 0.15\,\mu$eV, was found in [118]. In the strong-tunneling regime, approximately equal values $t_{\textrm{N}}\simeq t_{\textrm{F}}\simeq 100\,\mu$eV were reported around $B_z\simeq 1$ T [55], while $t_{\textrm{N}} \gt t_{\textrm{F}}$ for larger B_z . The relative strength of t_N and t_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].

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, and thus has a high energy. Furthermore, the transition from triplet states to the singlet, $T(1,1) \to S(2,0)$, is forbidden when the total spin is conserved. As a consequence, when the the double dot gets trapped in a $T(1,1)$ state, the current I_DQD from the right to the left reservoir is blocked [121].

However, for holes the large values of t_F imply a strong violation of the spin selection rule. Because spin-flip tunneling is not generally blocked, the study of the excited states through 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 $\vec{B}_{\textrm{SO}}$ 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 [120], where the direction of $\vec{B}_{\textrm{SO}}$ 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, T₁ [67].

As discussed in section 1.4, 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 that have a predominantly heavy-hole character, i.e. atomic angular momentum $J_z \simeq \pm 3/2$ (for a more precise description, see section 1.3). 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 equation (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 [117], similar to what is found in 2DHGs [124–126]. 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 [119], it was shown that the value of $g_z$ depends on the voltage applied to the middle gate (labeled C in figure 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. However, measurements were performed in the regime of large detuning, where the modulation of tunneling amplitude has a 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 ∼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 ${\left\vert{L \Uparrow}\right\rangle},{\left\vert{L \Downarrow}\right\rangle},{\left\vert{R \Uparrow}\right\rangle},{\left\vert{R \Downarrow}\right\rangle}$.

The ${\left\vert{L \Uparrow}\right\rangle} \to {\left\vert{L \Downarrow}\right\rangle}$ spin relaxation time has been measured in [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. However, to access short relaxation times, it is important to overcome bandwidth limitations in charge sensing, and in [67] it was necessary to rely on a more sophisticated latched charge readout. In short (see [67] for details), the readout protocol transfers a hole from the ${\left\vert{L \Uparrow}\right\rangle}$ state to a metastable (0,1) charge configuration (with ∼50% fidelity) via resonant spin-flip tunneling. In contrast, holes in the ${\left\vert{L \Downarrow}\right\rangle}$ 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], consistent with transport measurements in the Pauli-spin blockade [120]. The measured values of T₁ are in good agreement with theoretical values of the Dresselhaus spin–orbit coupling [39]. We note that the hole-spin relaxation times are much shorter than in electron-based GaAs quantum dots, where an extremely long $T_1 \simeq 60$ s can be demonstrated around 0.6 T [127].

In this system, a robust EDSR signal was detected with a continuous drive [55, 119]. The measurements of $g_z$ were 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 [25, 128]. 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 ${\left\vert{\Downarrow\Downarrow}\right\rangle}$ (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 the total spin is conserved, the S and $T_-$ states must cross in energy as a function of detuning. For hole systems, the relatively large value of t_F gives a robust $S-T_-$ anti-crossing 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 $S-T_-$ anti-crossing can be obtained [23].

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 ${\left\vert{\Downarrow\Downarrow}\right\rangle}$ and ${\left\vert{\Downarrow\Uparrow}\right\rangle}$ (due to the different g-factors of the two dots). ${\left\vert{\Downarrow\Downarrow}\right\rangle}$ 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 section 2.2). On the other hand, ${\left\vert{\Downarrow\Uparrow}\right\rangle}$ 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, ${\left\vert{\Downarrow}\right\rangle}\to (1,1)$ and ${\left\vert{\Uparrow}\right\rangle}\to (2,0)$, with the spin in the left dot acting as an ancilla qubit. As described for GaAs planar dots, by changing the detuning from (1,1) to (2,0) an $S-T_-$ anti-crossing 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].

In the same experiment [138], coherent single-spin manipulation was performed through EDSR, with Rabi frequencies $f_\mathrm{R} = 57\,$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 of ∼150–800 ns. The coherence time can be extended to $T^{\textrm{HE}}_2 \sim$ 1–5 µs with a Hahn echo sequence [16, 20, 138] and up to ∼100 µs by applying $N_\pi = 1500$ Carr–Purcell–Meiboom–Gill (CPMG) refocusing pulses [20]. Spin lifetime measurements have obtained T₁ = 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 [20]. A more detailed analysis of single-qubit gates has 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 $f_\mathrm{R} \simeq 10$ 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 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 Ge planar dots, a tunable $J/h$ in the range ∼35–65 MHz was realized around the symmetric point [16]. The 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 ${\left\vert{\Uparrow \Downarrow}\right\rangle}$ and ${\left\vert{\Downarrow \Uparrow}\right\rangle}$ is much larger than J. In practice, the EDSR resonant frequencies of neighboring dots can differ by a few hundred MHz (the difference is around 260 MHz in [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 ${\left\vert{0}\right\rangle},{\left\vert{1}\right\rangle}$ of the qubit, in the following discussions the spin quantization axis is chosen conventionally along z (even if the external magnetic field is physically applied in-plane).

For an interaction of the form $J \sigma_{z,1} \sigma_{z,2}$, the EDSR frequency of qubit 1 depends on the state of qubit 2 (either ${\left\vert{\Downarrow}\right\rangle} \equiv {\left\vert{0}\right\rangle}$ or ${\left\vert{\Uparrow}\right\rangle} \equiv {\left\vert{1}\right\rangle}$). In the presence of an always-on exchange interaction, applying an EDSR drive at one of the two available frequencies leads to a conditional rotation (CROT) of qubit 1, with qubit 2 acting as a control [16, 20]. A controlled-X (CX) gate is a conditional π-rotation, which could be realized with a gate time as fast as 55 ns [16]. For the four-qubit quantum processor of [20], CX gates were executed on all four pairs of neighboring qubits, with gate times under 100 ns. More advanced controlled rotations involve turning on two or three exchange couplings simultaneously (e.g. $J_{12}, J_{14} \neq 0$, where 1 is the target qubit, while 2 and 4 are the control qubits). In this case, the EDSR resonance of the target qubit splits into four or eight lines, depending on the number of control qubits [20].

In addition to controlled rotations, faster (∼10 ns) two-qubit gates are implemented through a pulsed exchange interaction. With an interaction $\propto \sigma_{z,1} \sigma_{z,2}$, a J(t) pulse will induce a relative phase between parallel (${\left\vert{\Downarrow\Downarrow}\right\rangle}$, ${\left\vert{\Uparrow \Uparrow}\right\rangle}$) and antiparallel (${\left\vert{\Downarrow\Uparrow}\right\rangle}$, ${\left\vert{\Uparrow \Downarrow}\right\rangle}$) states, i.e. a CPHASE gate. For a π phase difference (and after correcting for single-qubit phases), a controlled-Z gate (CZ) can be implemented. All possible CZ gates between qubit pairs were demonstrated, with operation times below 10 ns [20]. Similarly, the CPHASE gate leads to a controlled-S gate (CS), where the phase difference is $\pi/2$. The recent demonstration of a phase flip code on three qubits required implementing a three-qubit Toffoli-like gate. As shown in figure 3(c), this three-qubit gate was constructed by combining CZ, CS⁻¹ and single-qubit gates [21]. Finally, a SWAP gate exchanging antiparallel spin states, ${\left\vert{\Downarrow\Uparrow}\right\rangle} \leftrightarrow {\left\vert{\Uparrow \Downarrow}\right\rangle}$, was realized through an oscillating exchange pulse resonant with the ${\left\vert{\Downarrow\Uparrow}\right\rangle} -{\left\vert{\Uparrow \Downarrow}\right\rangle}$ transition [21]. Implementation of this gate relies on the presence of flip-flop terms that couple the two antiparallel states. For convenience, we present a summary of all these two-qubit gates, together with the basic mechanism behind their realization, in table 2.

Rather elaborate quantum algorithms can be implemented using the set of gates described above. In [20], four qubits are entangled into a GHZ state and subsequently brought back to a product state. Although the fidelity for generating the GHZ state was not reported, parity readouts at intermediate steps and in the final state demonstrate that coherence is maintained throughout the whole protocol. Error correction codes have been explored in [21]. The most sophisticated scheme (a phase flip code on three qubits) is reproduced in figure 3(b). In both cases, an interesting point is that coherence can be improved by including refocusing π-pulses at selected steps of the time evolution, which demonstrates the benefits of combining quantum algorithms with dynamical decoupling. Furthermore, the great flexibility in controlling the exchange interactions J_ij for the four pairs of neighboring qubits has recently allowed the simulation of resonating valence bond (RVB) states of the $2\times2$ array [143]. When operated under equal exchange couplings, $J_{12} = J_{34} = J_{14} = J_{23}$, the s-wave and d-wave symmetry RVB states are eigenstates of the system. Explicitly, these states can be written as $|s\rangle = ({\left\vert{S_{12}}\right\rangle}{\left\vert{S_{34}}\right\rangle}-{\left\vert{S_{14}}\right\rangle}{\left\vert{S_{23}}\right\rangle})/\sqrt{3}$ and ${\left\vert{d}\right\rangle} = {\left\vert{S_{12}}\right\rangle}{\left\vert{S_{34}}\right\rangle}+{\left\vert{S_{14}}\right\rangle}{\left\vert{S_{23}}\right\rangle}$, where ${\left\vert{S_{ij}}\right\rangle}$ is a singlet of spins i and j. Preparation of these correlated spin states, as well as coherent oscillations in the 2D subspace spanned by them, have been demonstrated [143].

So far we have discussed qubits formed by individual spins. However, it is also possible to define a qubit based on two suitable states of multiple coupled spins. In double quantum dots, one option is the $S-T_-$ qubit mentioned at the end of section 2.2. Another attractive possibility is to consider the unpolarized spin states $({\left\vert{\Uparrow\Downarrow}\right\rangle} \mp {\left\vert{\Downarrow\Uparrow}\right\rangle})/\sqrt{2}$, forming the so-called $S-T_0$ qubit [144]. To operate this two-level system, one can ramp the detuning rapidly from $S(2,0)$ to the (1,1) region, traversing the $S-T_-$ anticrossing diabatically, to reach $S(1,1)$. However, the singlet is not an eigenstate of the system. In hole systems, if the difference in g-factors is large, the singlet can be significantly coupled to the unpolarized triplet $T_0(1,1)$ in the following way [145]: $H_{ST} = \frac12 \Delta g\mu_\mathrm{B} B\left({\left\vert {S}\left\rangle\right\langle{T_0} \right\vert}+{\left\vert {T_0}\left\rangle\right\langle{S} \right\vert}\right)-J(\epsilon){\left\vert {S}\left\rangle\right\langle{S} \right\vert}$, where B is the external magnetic field, $\Delta g$ is the difference in g-factors between the two quantum dots, and $J(\epsilon)$ is the exchange energy. Crucially, $J(\epsilon)$ strongly depends on detuning. This dependence enables the control of the axis of rotation and the precession frequency of the effective qubit. Readout is performed by returning to the (2,0) region and relying on the Pauli-spin blockade.

Coherent $S-T_0$ oscillations in a planar Ge double quantum dot can approach a frequency of 100 MHz at $B_z = 3$ mT [145]. The dephasing time can reach $T_2^* \simeq 1\,\mu$s for relatively large detuning and a smaller $B_z = 0.5$ mT [145]. The latter regime mitigates the effect of fluctuations in J and $\Delta g_z$, induced by charge noise. On the other hand, the best quality factor Q = 52 was achieved at B = 3 mT, when the oscillation frequency is larger. Coherent two-axis control of the $S-T_0$ qubit is possible, and a CPMG decay time $T_2^{\,\textrm{CPMG}}\simeq 4.5\,\mu$s was obtained by applying a sequence with $N_\pi = 2$ (where $N_{\mathrm\pi}$ is the number of π-pulses). The coherence time can be further prolonged to $T_2^{\textrm{CPMG}}\simeq 135\,\mu$s, for $N_\pi = 512$. Furthermore, the interplay between $S-T_0$ and $S-T_-$ coherent oscillations in different regimes was analyzed by [56]. This study of coherent singlet-triplet oscillations also led to a full characterization of the device. Site dependent and anisotropic g-factors were found, with values $g_z \simeq 7$ and $g_x \simeq 0.26$ for the left dot and $g_z \simeq 5$ and $g_x\simeq -0.16$ for the right dot [56]. As previously shown in [145], the difference in g_z -factors can be modulated through the center gate, from $\Delta g_z\sim 1.5$ to above 2.2. The detailed form of the spin-flip tunneling term was determined as $t_x \sigma_x + t_y \sigma_y$, with $t_x \simeq 0.13\,\mu$eV and $t_y\simeq -0.37\,\mu$eV [56]. These values give the total strength $t_{\textrm{F}} = \sqrt{t_x^2+t_y^2} \simeq 0.39\,\mu$eV (while the spin-conserving tunneling amplitude is $t_{\textrm{N}}\simeq 11\,\mu$eV). More recently, coherent $S-T_-$ oscillations were also demonstrated in a four-qubit array [143], for all four pairs of neighboring quantum dots. At an external field of 1 mT, the oscillation frequency is between 1–2.6 MHz, depending on the pair, with typical dephasing times of several µs (the longest dephasing time, obtained for one of the four pairs, is $11\,\mu$s [143]).

These nanowires, schematically illustrated in figure 4(a), have a cylindrical cross-section with a Ge core, of typical diameter ∼10 nm, and a Si shell providing strong confinement due to the large offset between the valence bands of silicon and germanium. Details on the fabrication can be found in [152–154]. The Fermi energy lies entirely within the germanium valence band, and free holes are confined to the Ge core [155, 156]. Transport measurements, thanks to the high-quality interface between the undoped Ge/Si materials, obtained ballistic transport up to room temperature and a mobility of 600 cm²(Vs)⁻¹ [156–158]. The strong spin–orbit interaction in these nanowires can be characterized through the spin–orbit length λ_so (see at the end of section 1.3), which was found to be between 20–65 nm in various devices [158–160] and could be tuned by a factor of 5 using a back gate [158]. In a recent experiment, a tunable value of λ_so was estimated (assuming a heavy-hole effective mass) to range from 28 nm to an extremely short value of 3 nm [17].

Quantum dots can be formed through electrostatic potentials from metallic gates next to the core/shell nanowire [147, 161, 162]. However, it has proved difficult to reach the single-hole regime in these devices, and typical occupations are of the order of tens of holes. For example, the working point chosen in [163] is around the charge state ${\sim}(70,10)$ and about 15 holes were confined in the dots of [160]. This does not prevent experiments analogous to the ones in the single-hole regime. If the charge state of a double quantum dot $(n_{\textrm{L}},n_{\textrm{R}})$ is such that $n_{\mathrm{L/R}}$ give 'closed-shell' configurations, then $(n_{\textrm{L}}+1,n_{\textrm{R}}+1)$ behaves effectively as $(1,1)$, i.e. as a system with a single hole-spin trapped in each quantum dot [163].

Early studies of spin coherence were based on detuning pulses [163, 164]. As in lateral quantum dots, initialization and readout can be achieved through the Pauli-spin blockade [164]. Using this scheme, spin-lifetime measurements are possible. Values in the range $T_1 \simeq$ 0.2–0.6 ms (depending on the strength of the magnetic field) were found [164]. However, a much shorter $T_1 \simeq$ 0.2–0.8 µs was later reported in [163], possibly due to a strong dependence of T₁ on the device and/or detuning. In the same experiment, a dephasing time $T_2^* \simeq 180$ ns was extracted. The exponential (rather than Gaussian) decay of coherence suggests a negligible influence of the nuclear spins [163].

Coherent manipulation through EDSR has recently been achieved [17]. Rabi oscillations reaching $f_\mathrm{R} \simeq 250$ MHz (at the largest applied drive) and two-axis control were demonstrated. The dephasing time $T_2^* \simeq 11$ ns extracted from a Ramsey sequence is much shorter than previous reports, but can be extended to $T^{\textrm{HE}}_2 \simeq 250$ ns using Hahn echo. While these figures of merit are obtained at a specific working point, [17] also finds that $f_\mathrm{R}$ and the decay time $T_2^{\textrm{Rabi}}$ of the Rabi oscillations depend sensitively on the voltage, V_M , applied at one of the gates. For a small change $\Delta V_M \simeq 30$ mV, the Rabi frequency could be increased from $f_\mathrm{R} = 31$ to 219 MHz, which is accompanied by a reduction in $T_2^{\textrm{Rabi}}$ from 59 to 7 ns. The increase in $f_\mathrm{R}$ and the reduction in $T_2^{\textrm{Rabi}}$ are attributed to the strong effect of V_M on the spin–orbit coupling in the nanowire [42]. By optimizing the values of the applied gate voltages and for a large drive amplitude, a Rabi frequency $f_\mathrm{R} \simeq 435$ MHz can be reached [17].

In these nanowires, initial steps towards coupling hole-spin qubits to microwave photons were taken. A coherent charge coupling of 55 MHz between a double quantum dot (acting as a charge qubit) and a superconducting transmission line resonator was realized [165].

Germanium hut wires, discovered in the early 1990s [166], are self-assembled structures obtained by growing germanium on a silicon substrate. Due to the $4\%$ lattice mismatch between the germanium and silicon crystals, first pyramid-shaped and later hut-shaped germanium islands are formed on top of the Si(001) substrate, depending on the duration of the growth [167, 168]. By using a catalyst-free method, high-quality Ge hut wires, with length–width ratio exceeding 1000 could be fabricated [169]. The geometry of these nanowires is schematically shown figure 4(b). They have typical height and width of 2 and 18 nm [169, 170], respectively. To minimize the surface energy, the slope between the side surfaces of the hut and the substrate plane is fixed at 11.3^∘, corresponding to $\{105\}$ facets [167, 169, 171, 172]. Hut wires with similar structure have also been found for $\mathrm{Si}_{1-x}\mathrm{Ge}_x$ instead of Ge [173].

As in core/shell nanowires, quantum dots based on hut wires are currently operated in the multi-hole regime. Here, however, due to the relatively 'flat' geometry, the holes can have a predominantly heavy-hole character, reflected by the strong anisotropy of the g-factors. Typical values are $g_z \simeq$ 2–4.5 for the direction perpendicular to the substrate, and $g_{x,y} \lesssim 0.5$ for the in-plane direction [148, 174–176]. The precise values depend on the device, as well as the quantum dot occupation. A maximum ratio $g_z/g_y \simeq 18$ was observed inside a certain Coulomb diamond, but the anisotropy tends to decrease for Coulomb diamonds with more holes, presumably due to a larger admixture with light holes [174].

Insight into the spin–orbit coupling has been gained from measurements of the leakage current in the Pauli-spin blockade regime [18, 176]. In one experiment, the spin-preserving tunnel coupling was found to be in the range from $t_{\textrm{N}} \simeq 0.4\,\mu$eV to $14.5\,\mu$eV, while the spin-flip tunneling amplitude remains relatively constant at a larger value $t_{\textrm{F}} \sim 20\,\mu$eV. This translates to a tunable spin–orbit length in the wide range $\lambda_{\textrm{so}} \sim$ 2–48.9 nm [18]. The shortest value, $\lambda_{\textrm{so}} \sim 2$ nm, is compatible with EDSR measurements, where a spin–orbit length as short as $\lambda_{\textrm{so}} \simeq 1.4$ nm was estimated from the dependence of the Rabi frequency on the drive amplitude [18]. By studying the anisotropic dependence of the leakage current on the applied magnetic field, an in-plane spin–orbit field forming an angle of 31^∘ with the nanowire was found [175]. The nontrivial direction indicates a significant contribution from Dresselhaus spin–orbit coupling, which may be induced by interface inversion asymmetry.

Readout of the hole spin in a double quantum dot can be realized, as described before, through the Pauli-spin blockade [148]. Alternatively, relying on two neighboring wires in a 'T' configuration [170], the same type of readout as Elzerman et al [177] was demonstrated. In the latter scheme, the alignment of the chemical potential of an external reservoir with a single quantum dot is such that the upper (lower) energy state of the Zeeman-split doublet can (cannot) tunnel out into the lead. In [178], one of the nanowires in the T configuration is both capacitively and tunnel coupled to a single quantum dot (realized at the end of the other wire). Accordingly, this wire can act as both a reservoir and a charge sensor, detecting tunneling events associated with the higher-energy ${\left\vert{\Uparrow}\right\rangle}$ state. Using this readout technique, spin lifetimes increasing with an approximate $B_z^{-3/2}$ dependence were measured. They range from $T_1 \simeq 15\,\mu$s (at $B_z = 1.5$ T) to $90\,\mu$s (at $B_z = 0.5$ T) [178].

Coherent control of the hole-spin qubit is possible through EDSR. A first demonstration was able to achieve a Rabi frequency of 140 MHz and, through a Ramsey sequence, a dephasing time $T_2^* \simeq 130$ ns was measured [148]. More recently, Rabi frequencies as large as $f_\mathrm{R} = 542$ MHz have been attained for EDSR signals mediated by the strong spin–orbit coupling (spin–orbit length $\lambda_{\textrm{so}}\sim$ 1–2 nm) mentioned above [18]. In this case, values of the coherence time were found in the range $T^*_2 \simeq$ 42–84 ns, (depending on the power applied to implement the $\pi/2$ pulses of the Ramsey sequence). With a Hahn echo sequence, the coherence time can be prolonged from $T^*_2 \simeq 65$ ns to $T^{\textrm{HE}}_2 \simeq 523$ ns.

A potential advantage of hut wires over core/shell nanowires is that they grow directly on the 2D substrate, facilitating the fabrication of more complex arrangements of multiple wires. In addition to the spontaneously formed 'T' configurations exploited for charge sensing, a method to fabricate structured arrays in a controlled manner, where the hut wires are formed along patterned trenches, is of particular interest [179]. Scalability can also be facilitated by coupling such spin qubits to superconducting transmission lines. Initial experiments demonstrated charge coupling of single [180] and double quantum dots [181] to microwave resonators, with a coupling strength $g_c/2\pi = 148$ MHz (15 MHz) in the former (latter) case. Estimates based on the charge coupling still give relatively small values for the spin-photon coupling, around 2–4 MHz [180, 181].

Similar to the quantum dots discussed in previous sections, the properties of hole-spin qubits in Si reflect the presence of a strong spin–orbit interaction, which can be studied through the leakage current in the Pauli-spin-blockade regime [185, 188, 194], as well as through spectroscopic studies [22, 192] (where the Si quantum dots are coupled to resonators). A spin–orbit length of $\lambda_{\textrm{so}} \sim 110$ nm was estimated in the planar double dot of [185], and $\lambda_{\textrm{so}} \simeq 48$ nm in the FinFET of [194]. In a nanowire device of [192], a spin–orbit gap of $19\,\mu$eV was measured at the $S-T_-$ anti-crossing, which reflects the strength of the spin-flip tunneling amplitude t_F. This energy scale is similar to the spin-conserving tunneling amplitude ($t_{\textrm{N}}, t_{\textrm{F}} \sim 10\,\mu$eV in that experiment).

A detailed analysis of the spin-flip tunneling amplitude was performed recently in a nanowire-FET double quantum dot [22], where t_F is a crucial parameter to achieve strong coupling of a single-hole spin (hosted in a double quantum dot) to a superconducting resonator. The strength of the spin-flip tunneling amplitude depends on the relative angle between $\hat{n}_{\textrm{SO}}$ (the direction of the spin–orbit field) and $\hat{n} = \overleftrightarrow{g}\cdot \textbf{B}$ (where $\overleftrightarrow{g}$ is the g-tensor, assumed equal for the two quantum dots; $\hat{n}$ is the natural quantization direction for the hole-spins). Therefore, t_F has a strongly anisotropic dependence on the magnetic field. In [22], where the total tunneling amplitude is $t_c = \sqrt{t_{\textrm{N}}^2+t_{\textrm{F}}^2} \simeq 40\,\mu$eV, the spin-flip amplitude is very small when $\hat{n}$ and $\hat{n}_{\textrm{SO}}$ are almost parallel, giving $t_{\textrm{N}} \simeq t_c \gg t_{\textrm{F}}$. Instead, t_F approaches its maximum value when $\hat{n}$ and $\hat{n}_{\textrm{SO}}$ are nearly perpendicular. The latter regime gives $t_{\textrm{F}} \simeq t_{c} \gg t_{\textrm{N}}$ in this particular experiment, suggesting that the spin–orbit length is close to the distance between the two dots ($\lambda_{\textrm{so}}\sim 80$ nm).

The effective g-tensor in these Si quantum dots is anisotropic, although not as strongly as for other types of hole-spin qubits, and can be significantly affected by small variations of the gate voltages [59, 187, 190, 191]. For example, in a quantum dot based on a nanowire FET, it is possible to modulate the g-factor in the in-plane direction perpendicular to the channel from g = 1.85 to 2.6 [190]. A similar degree of tunability was demonstrated for quantum dots in the planar geometry [187]. In general, the principal axes of the g-tensor are not aligned with the crystallographic directions and do not have a simple correspondence with the geometry of the device. This is true for nanowire FET devices [59, 191] but also, more surprisingly, for planar quantum dots [187]. In the latter case, despite the expectation of a predominant heavy-hole character, the largest g-factor is obtained along a direction forming an angle φ₀ from the z axis. Considering two distinct gate voltages, ($g_{\textrm{max}} = 3.9$, $\phi_0 = 42^\circ$) and ($g_{\textrm{max}} = 1.7$, $\phi_0 = 70^\circ$) were measured, showing that not only the value of the g-factor but also the direction of the principal axes is electrically tunable. The occurrence of large tilt angles was attributed to nonuniform strain induced by the metallic electrodes, which can have a strong influence on the light-hole/heavy-hole mixing [187]. As a reference for the typical anisotropy obtained in these devices, at the same two working points, [187] finds a minimum g-factor of 1.4 (giving $g_{\textrm{max}}/g_{\textrm{min}} \simeq 2.8$) and 0.3 (giving $g_{\textrm{max}}/g_{\textrm{min}} \simeq 5.7$).

Coherent spin manipulation through EDSR was demonstrated in 2016 with a nanowire FET device [189]. Based on a double quantum dot with a ${\sim}(10,30)$ occupation, Rabi oscillations with a maximum frequency $f_\mathrm{R} = 85$ MHz were realized. The dephasing time of $T_2^* \simeq 60$ ns is obtained from the Ramsey fringes, while the Hahn echo amplitude decays with $T_2^{\textrm{HE}} \simeq 245$ ns. These coherence times were significantly improved in a recent experiment based on the same type of quantum dots, but where optimal operation points minimizing the effect of charge noise were identified [59] (see below for more details).

Another notable development is the recent demonstration of coherent spin manipulation in a FinFET device [81], performed in the large-temperature regime (1.5–4.2 K). Operating the qubits above 1 K would enable a higher cooling power, which is an obvious advantage for designing large-scale devices with tightly integrated quantum hardware and classical control electronics. Rabi oscillations with frequencies up to $f_\mathrm{R} = 147$ MHz were demonstrated in this device and, from the Ramsey fringes, a coherence time of $T_2^* \simeq 200$ ns was obtained at 1.5 K for one of the two qubits. The coherence time is reduced to $T_2^* \simeq 90$ ns at 4.2 K. Correspondingly, through randomized benchmarking, a single-qubit gate fidelity $F = 98.9\%$ was found at 1.5 K, which decreases to $F = 97.9\%$ at 4.2 K. The dephasing time depends on the magnetic field, and a maximum value $T_2^* \simeq 440$ ns can be obtained, comparable to the dephasing timescale induced by nuclear spins. Furthermore, dynamical decoupling leads to $T_2^{\textrm{CPMG}} \simeq 5.4\,\mu$s, which has been reached by applying a CPMG sequence with $N_\pi = 32$.

Since the g-factor of hole-spin qubits is sensitive to small fluctuations in the voltage V of a nearby gate, identifying sweet spots at which $\mathrm{d}g/\mathrm{d}V = 0$ is highly valuable. Experimentally, measurements of the voltage-dependent g-factor has first provided evidence for such a stationary point in a planar CMOS quantum dot [187]. More recently, the impact of sweet spots on the coherence of a hole-spin qubit was investigated for a nanowire-FET device [59]. In this experiment, the direction of the external magnetic field was varied, and two orientations were identified at which the Larmor frequency of the qubit becomes insensitive (to linear order) to fluctuations in the accumulation gate voltage. Due to the influence of a second more distant gate, the optimal orientation slightly deviates from one of the two sweet spots. The improvement in dephasing time (extracted from the Ramsey oscillations) is from $T_2^* \simeq 3\,\mu$s (for a magnetic field along the nanowire FET) to $T_2^* \simeq 6\,\mu$s (at the optimal point). By applying a Hahn echo sequence, an even more significant enhancement, from $T_2^{\textrm{HE}} \simeq 15\,\mu$s to $T_2^{\textrm{HE}} \simeq 85\,\mu$s, can be demonstrated. The longest coherence time is achieved with a CPMG sequence of $N_\pi = 256$ π-pulses, yielding $T_2^{\textrm{CPMG}} \simeq 0.4$ ms [59].

A large spin-photon coupling has recently been reported for a hole-spin qubit in a nanowire FET coupled to a superconducting resonator [22]. Similar to experiments with electron-spin qubits [6, 7], the coupling is based on the large dipole moment associated with the bonding/anti-bonding states of a single hole in a double quantum dot. In the absence of an applied magnetic field, a strong charge-photon coupling of $513\,\mathrm{MHz}$ was deduced from the dispersive shift of the cavity frequency induced by the double quantum dot around zero detuning. At finite magnetic field, the spin–photon interaction depends crucially on the strength of the spin-flip tunneling amplitude t_F. Consequently, the spin–photon interaction has a strongly anisotropic dependence on B (see section 2.5.1). A maximal spin–photon coupling of $g_s/2\pi = 330\,\mathrm{MHz}$ was achieved, which is much larger than the decay rate of the cavity ($\kappa/2\pi = 14$ MHz) and the spin decoherence rate ($\gamma_s/2\pi = 2.5$–17 MHz). The cooperativity can be large as well and reaches $4g_s^2/\kappa\gamma_s = 1600$ for a magnetic field parallel to the nanowire. Furthermore, there is a significant ∼1 MHz spin–photon coupling also in the single-dot limit [188] (reaized at large detuning).

Finally, we address boron acceptor dopants in silicon, which represent another option for hole-spin qubits. Theses dopants have been investigated in natural [195, 196] and isotopically purified silicon [197], as well as gated structures [198–200]. Characterizations of the electronic states of a single dopant or two coupled acceptors [196, 199] can be obtained from scanning tunneling spectroscopy [195, 196, 201] or transport measurements [198, 199]. Coherent spin manipulation was performed with pulsed electron spin resonance in an isotopically purified system [197], showing a long CPMG coherence time of $9.2~\mathrm{ms}$. For these donors, the existence of sweet spots protecting the qubit from charge noise [202], as well as spin-echo envelope modulations as a sensitive probe of the anisotropic hyperfine interaction [90], were recently discussed theoretically.