## 2 Silicon Nano-Ridge Qubit Device
### Electrostatic Device Simulations  
We perform device simulations based on the presented device geometry to demonstrate the general functionality and identify the relevant feature sizes. Furthermore, we investigate two metallic top-gate structures. Firstly, we focus on a single set of top-gates, which crossed the silicon nano-ridge, as indicated in Figure 1. As a proof-of-principle, the electronic wave functions of a device tile consisting of three QDs are calculated within their electrostatic potential and tuned to be tunnel-coupled to all adjacent QDs. Secondly, we present a top-gate array, which is split and insulated along the symmetry axis shown in Figure 1. Signal routing from both sides of the nano-ridge is required in this case. Being able to apply different voltages to the top-gates on each side of the silicon nano-ridge, we tune one array of quantum dots and one single electron transistor (SET) in opposing corners of the silicon nano-ridge. Both gate structures are transversely symmetric and thus can be in principle extended to an arbitrarily-long tuned QD array. For all simulations, the width \(w_{\rm r}\) of the wet-etched trapezoidal silicon nano-ridge before LOCOS is set to 100 nm. The insulation of the silicon nano-ridge towards the top-gate structure consists of 10 nm of Al\({}_{2}\)O\({}_{3}\). We use a metallic top-gate structure with a 30-nm gate width and 15 nm of SiO\({}_{2}\) insulation. For the second device containing the split gate structure, gates on opposing sides are insulated by 15 nm of SiO\({}_{2}\). The electrostatic potentials are calculated by solving the Poisson equation by the COMSOL Multiphysics\({}^{\circledR}\) software package [37]. The electronic ground state energy levels and corresponding wave functions are calculated by solving the three-dimensional (3D) Schrodinger equation on a next-neighbor tight-binding model. We use the low temperature value \(E_{\rm G}=1.17\) eV for the fundamental band gap of silicon in our calculations [38]. In bulk silicon, the conduction band minimum is six-fold degenerate along the \(<\)100\(>\) crystal directions [2]. Confinement of an electron in the, e.g., [001] direction splits this degeneracy into a two-fold degenerate out-of-plane valley labeled \(\pm k_{\rm z}\) with the longitudinal effective mass \(m_{\rm 1}=0.98\,m_{0,e}\) relevant for the confinement energy and a four-fold degenerate in-plane valley labeled \(\pm k_{\rm x}\), \(\pm k_{\rm y}\) with relevant transverse effective mass \(m_{\rm t}=0.19\,m_{0,e}\), where \(m_{0,e}\) is the free electron mass. While the \(\pm k_{\rm z}\) valley is split off from the perpendicular valleys due to the confinement, potential strain, and the anisotropic masses, the mixing of the \(-k_{\rm z}\) and \(+k_{\rm z}\) valley states depends on the atomistic details of the confinement potential. We calculate the valley splitting between perpendicular valley states due to confinement and the effective anisotropic mass, but the remaining valley splitting within the \(\pm k_{\rm v}\) (\(v=x,y,z\)) valley spaces are beyond the scope of our manuscript. Notably, the silicon nano-ridge is oriented along the equivalent \(<\)110\(>\) crystal axis due to the anisotropic wet-etching. Hence, the in-plane valleys \(\pm k_{\rm x}\), \(\pm k_{\rm y}\) will mix, and calculating the approximate valley splitting by confinement requires a \(\phi=45\,^{\circ}\) rotation of the effective mass tensor \(M\) about the \(z\)-axis. Therefore, we apply a unitary rotation \(R(\phi,\theta)\) with \(\theta=0\), as depicted in Figure 2a. If the ground states are in the ellipsoidal energy surfaces \(k_{\rm z}\), there is no effect on the energy splitting of the lowest two eigenstates due to the identical in-plane effective masses \(m_{\rm x}=m_{\rm y}=m_{\rm t}\). However, if the ground state is, e.g., within the \(\pm k_{\rm y}\) valley space (e.g., due to strong confinement and thus large zero-energy in the z-direction), the effective masses are \(m_{\rm y}=m_{\rm l}\) and \(m_{\rm x}=m_{\rm z}=m_{\rm t}\).