## 2 Silicon Nano-Ridge Qubit Device
### Electrostatic Device Simulations
The effect of valley mixing for different rotation angles \(\phi\) for a ground state in the \(\pm k_{\rm y}\) valley is shown in Figure 2b. For \(\phi=45\,^{\circ}\) (relevant for our nano-ridge orientation), we identify a mixing of the rotated \(k_{\rm y}^{\prime}\) valley as a combination of \(\pm k_{\rm x}\) and \(\pm k_{\rm y}\) where the masses of the rotated \(\pm k_{\rm x}^{\prime}\) and \(\pm k_{\rm y}^{\prime}\) valleys are equal. This effective mass alignment leads to an energy level crossing and an additional four-fold degeneracy of these energy levels. In the following, we evaluate the respective energy level configuration by calculating the ground and first excited state for each valley. Note that all energy levels are additionally two-fold spin-degenerate, since magnetic fields are not included in our simulations. First, we present the calculations on the non-split metallic top-gate array, which allows the formation of a tunnel-coupled QD array. After tuning all the voltages applied to the gates, we find an electrostatic potential for three adjacent quantum dots, the calculated ground-state wave functions of which are shown in Figure 2c. The 3D wave functions are confined along the \(z\)-direction at the top of the silicon nano-ridge by the Si/SiO\({}_{2}\) interfaces. The corresponding potential is plotted along and perpendicular to the silicon nano-ridge through the maximum of the middle QD potential and an adjacent potential barrier. The additional strong confinement of the Si/SiO\({}_{2}\) interface in combination with the tuned electrostatic confinement of the gates ensure the intended formation of only a single energy level of all valleys below the Fermi level for every quantum dot. Adjacent QDs are tunnel-coupled with individually-tunable tunnel-couplings over multiple orders of magnitude. The tunnel-coupling between adjacent quantum dots is \(t_{\mathrm{c}}=3.7\,\mu\mathrm{eV}\) for both tunnel barriers. The energy level splitting \(1.7\,\mathrm{meV}\) is determined by the ground state towards the first orbital excited state within the same \(k_{\mathrm{z}}\) valley space (Figure 2g). Note that the splitting within the mixed \(k_{\mathrm{z}}\) valley space depends on the atomistic details of the barrier and is not calculated here. Energy splittings have to be much larger compared to the typical thermal energy of \(\sim 0.01\,\mathrm{meV}\) at operating temperatures of \(\sim 100\,\mathrm{mK}\). Due to the strong confinement of the wave function along the \(z\)-axis by the interface, the energetic levels of the \(\pm k_{\mathrm{x}}^{\prime}\) and \(\pm k_{\mathrm{y}}^{\prime}\) in-plane valleys are lifted with respect to \(\pm k_{\mathrm{z}}\). Electron spin manipulation can be achieved by electric dipole spin resonance (EDSR) using a magnetic field gradient-inducing micro magnet [39]. Applying voltage differences of \(\pm 1\,\mathrm{mV}\) on the opposing side-gates shifts the electron wave function by \(2\,\mathrm{nm}\). This controlled shift in combination with an assumed \(1\,\mathrm{mT}/\mathrm{nm}\) magnetic field gradient leads to an expected Rabi frequency of \(f_{\mathrm{Rabi}}\approx 55\,\mathrm{MHz}\). Alternatively, the electron spin can be manipulated using electron spin resonance (ESR) using a strip line fabricated along the silicon nano-ridge [13]. Using the split top-gate array, the device is voltage-tuned to localize three adjacent quantum dots in one corner of the silicon ridge, as well as a single electron transistor (SET) in the opposite corner. The calculated electron wave functions and the corresponding electrostatic potential are depicted in Figure 2d. The adjacent three quantum dots are tunnel-coupled. The tunnel-coupling can be tuned over multiple orders of magnitude by changing the corresponding voltages applied to the barrier gates \(V_{\mathrm{B2}}\) and \(V_{\mathrm{B3}}\) (Figure 2e,f). This yields the possibility of performing dedicated two-qubit gates, as well as controlled electron shuttling.