Figure 1

Parallel geometry: four quantum dots (yellow discs) aligned along the $x$ axis in the presence of an external magnetic field $\mathbf{B}$ that is applied parallel to the SOI vector $\mathit{\beta }$ (red arrow), which must be perpendicular to the line of the dots and which we take to be the $z$ direction. At each dot, there is a local magnetic field ${\mathbf{B}}_i$ (blue arrows), also assumed to be parallel to $\mathbf{B}$, but with alternating orientations as indicated. The direction of $\mathbf{B}$ defines the spin quantization axis. The dots are defined electrostatically by metallic gates (light green structures). Each dot contains a spin-1/2, and the exchange ($J_{ij}$) and the SOI-induced (${\mathit{\beta }}_{ij}$) interactions between the spins can be controlled by changing the electrostatic potential between the dots. The dots 1 and 2 (from left to right) define the first ST qubit, and the dots 3 and 4 the second ST qubit.Reuse & Permissions

Figure 2

Perpendicular geometry: similar to the setup shown in Fig. 1, but with the difference that here the SOI vector $\mathit{\beta }$ (red arrow) is perpendicular to the magnetic fields $\mathbf{B}$ and ${\mathbf{B}}_i$, which define the spin quantization axis $z$.Reuse & Permissions

Figure 3

The Bloch sphere is defined with the north pole corresponding to $\mid \uparrow \downarrow \rangle$ and the south pole corresponding to $\mid \downarrow \uparrow \rangle$. The effect of the unitary evolution operator $U_0^C$ [see Eq. (17)] on a state in the space $\{\mid \uparrow \downarrow \rangle ,\mid \downarrow \uparrow \rangle \}$ is equivalent to the rotation on the Bloch sphere around the vector $J_{23}{\mathbf{e}}_x-{\beta }_{23}{\mathbf{e}}_y+\Delta b_{23}{\mathbf{e}}_z$. The conditional phase gate $C_{23}$ corresponds to the full rotation on the Bloch sphere (shown by blue and green circles).Reuse & Permissions

Figure 4

Two schemes for the swap gate ${\pi }_{12}$: $\mid \uparrow \downarrow \rangle \to \mid \downarrow \uparrow \rangle$, composed of three consecutive rotations on the Bloch sphere where the offset induced by $\Delta b_{12}$ and ${\beta }_{12}$ is fully compensated. (a) First, starting from the north pole, we turn on $J_{12}$ and ${\beta }_{12}$ to induce rotation around $J_{12}{\mathbf{e}}_x-{\beta }_{12}{\mathbf{e}}_y+\Delta b_{12}{\mathbf{e}}_z$ (upper blue arc) until we reach the equator where we turn off $J_{12}$ and ${\beta }_{12}$. Second, we let the state precess around the $z$ axis in the equatorial plane until the mirror point of the starting point on the equator is reached (brown arc). Third, we induce once more rotation around $J_{12}{\mathbf{e}}_x-{\beta }_{12}{\mathbf{e}}_y+\Delta b_{12}{\mathbf{e}}_z$ (lower blue arc) until we reach the south pole. Lower panel: associated rectangular gate pulses and switching times $T_{\pi }^{(a1,a2,a3)}$ for the three steps. (b) Alternative scheme where during the second step the state precesses along the equator until it reaches the point diametrically opposite to its starting point (brown arc).Reuse & Permissions

Figure 5

An alternative scheme for the conditional phase gate $C_{23}$ (see also Fig. 3). Instead of the rotation with the pulse defined by Eqs. (18) and (35), we consider a sequence of three pulses. During the first and the third pulses (blue arcs), the state precesses quickly acquiring the $\pi /4$ phase [see Eq. (35)]. During the second pulse $J_{23}=0$, the state precesses around the $z$ axis over a shorter path than the one in Fig. 3. As a result, the switching is faster.Reuse & Permissions

Figure 6

Schematic setup for 2D architecture. Two dots define an ST qubit (black dotted ellipses). An external magnetic field $\mathbf{B}$ and local magnetic fields (blue arrows) are parallel. In the case of the coupling between two qubits from the same row (red solid ellipse), the SOI vector ${\mathit{\beta }}_{\parallel }$ is parallel to the magnetic field $\mathbf{B}$, corresponding to the parallel geometry (see Fig. 1). In the case of coupling between two qubits from two neighboring rows (red dashed ellipse), the SOI vector ${\mathit{\beta }}_{\perp }$ is perpendicular to the magnetic field $\mathbf{B}$, corresponding to the perpendicular geometry (see Fig. 2).Reuse & Permissions