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Hole-spin qubits in quasi-one-dimensional structures are a promising platform for quantum information processing because of the strong spin-orbit interaction (SOI). We present analytical results and discuss device designs that optimize the SOI in Ge semiconductors. We show that at the magnetic field values at which qubits are operated, orbital effects of magnetic fields can strongly affect the response of the spin qubit. We study one-dimensional hole systems in Ge under the influence of electric and magnetic fields applied perpendicular to the device. In our theoretical description, we include these effects exactly. The orbital effects lead to a strong renormalization of the g factor. We find a sweet spot of the nanowire (NW) g factor where charge noise is strongly suppressed and present an effective low-energy model that captures the dependence of the SOI on the electromagnetic fields. Moreover, we compare properties of NWs with square and circular cross sections with ones of gate-defined one-dimensional channels in two-dimensional Ge heterostructures. Interestingly, the effective model predicts a flat band ground state for fine-tuned electric and magnetic fields. By considering a quantum dot (QD) harmonically confined by gates, we demonstrate that the NW g-factor sweet spot is retained in the QD. Our calculations reveal that this sweet spot can be designed to coincide with the maximum of the SOI, yielding highly coherent qubits with large Rabi frequencies. We also study the effective g factor of NWs grown along different high-symmetry axes and find that our model derived for isotropic semiconductors is valid for the most relevant growth directions of nonisotropic Ge NWs. Moreover, a NW grown along one of the three main crystallographic axes shows the largest SOI. Our results show that the isotropic approximation is not justified in Ge in all cases.
Adelsberger et al. (Tue,) studied this question.