## 2 Silicon Nano-Ridge Qubit Device
### Electrostatic Device Simulations
The energy level splittings between the ground state and first excited state of the different valleys are shown in Figure 2g. As discussed above, the strong confinement at the \(\{111\}\) interfaces of the silicon nano-ridge decreases the energetic levels of the \(\pm k_{\mathrm{x}}^{\prime}\) and \(\pm k_{\mathrm{y}}^{\prime}\) valley compared to the tuned QD located in the top of the ridge. The ground states of \(\pm k_{\mathrm{x}}^{\prime}\) and \(\pm k_{\mathrm{y}}^{\prime}\) cross the first excited state of the \(\pm k_{\mathrm{z}}\) valley. The energy level splitting \(2.7\,\mathrm{meV}\) is larger due to the stronger confinement, but in comparison to the discussion above, limited by the two-fold degenerate ground state of the orthogonal \(\pm k_{\mathrm{x}}^{\prime}\) and \(\pm k_{\mathrm{y}}^{\prime}\) valleys. The operable temperature range for the proposed device is limited by the residual energetic splitting of the ground state in the \(\pm k_{\mathrm{z}}\) valley, which is out of the scope of this work. Previously-measured valley splittings range from \(100\,\mathrm{\SIUnitSymbolMicro eV}\) up to \(1\,\mathrm{meV}\)[22]. Thus, the device is operable at a temperature below \(100\,\mathrm{mK}\). The capacitive coupling of the middle QD and the QD of the single electron transistor is approximately \(3.4\,\mathrm{aF}\), dominantly determined by the width \(w_{\mathrm{r}}\). The simulations show the high degree of tunability of the proposed spin qubit device. Using the split gate structure, a tunnel-coupled quantum dot array can be tuned in one corner of the silicon nano-ridge, while a capacitively-coupled SET can be formed in the opposing corner, allowing spin-to-charge conversion for qubit read-out. Figure 2: Electrostatic simulation of the silicon nano-ridge qubit device. (**a**) Schematic of the different ellipsoidal energy surfaces of the degenerate valleys present in the silicon crystal lattice. For simplicity, only one quadrant of the reciprocal space is shown. For rotating angle \(\phi\) around the \(<\)001\(>\) crystal axis, the transverse effective masses mix changing the momentum ellipsoid for in-plane valleys \(k_{\mathrm{x}}^{\prime}\) and \(k_{y}^{\prime}\). (**b**) Effective masses \(m_{y}\) with \(v=\{\mathrm{x},\mathrm{y},\mathrm{z}\}\) of all crystal directions for different rotation angles. For \(\phi=45\,^{\circ}\), the effective masses \(m_{\mathrm{x}}\) and \(m_{y}\) are indistinguishable, leading to a crossing of the corresponding energetic states. (**c**) Simulated wave function of the electronic ground state for three tuned adjacent quantum dots located in the top of the schematically-shown silicon nano-ridge. All adjacent QDs are tunnel-coupled. Along three cross-sections, the electrostatic potential is shown within the overall device. The simulation is based on a 3D next-neighbor tight-binding model with a calculated electrostatic potential. \(V=0\) is defined as the conduction band minimum. (**d**) Simulated wave function of the electronic ground state for three tuned adjacent quantum dots and a single electron transistor (SET) located in opposite corners of the schematically-shown silicon nano-ridge. The electron reservoirs of the SET are sketched for clarity. Along three cross-sections, the calculated electrostatic potential is shown. \(V=0\) is defined as the conduction band minimum. (**e**) Schematic of the confining potential along the QD array for different barrier voltages \(V_{\mathrm{B,2}}\) and \(V_{\mathrm{B,3}}\). (**f**) Calculated tunnel couplings \(t_{\mathrm{c}}\) for different voltage detuning of the barrier gates. (**g**) Energy levels of the ground and excited state within each valley for different wave function localizations corresponding to (c) top-ridge state and (d) corner states. For top-ridge states, the ground and first excited orbital state are built by \(\pm k_{\mathrm{z}}\) valleys.