Theoretical analysis of hole spin qubits in gate-defined germanium quantum dots

Holes in germanium heterostructures offer a promising platform for scalable quantum computing devices, as the strong spin-orbit coupling in the valence band enables purely electrical control of qubit states and facilitates fast gate operations. However, the complexity of the valence band structure, which underlies the spin-orbit interaction, presents significant theoretical challenges. As a result, most theoretical studies rely on numerical solutions to k·p theory and the Luttinger-Kohn Hamiltonian, or atomistic tight-binding models. While these numerical approaches have yielded valuable insights into hole spin qubit devices, they often lack the intuitive understanding that an analytical model can provide. In this work, we present an analytical framework that captures the essential physical properties of germanium hole systems, with particular emphasis on spin-orbit physics and three-dimensional quantum dot confinement. This framework not only offers a more intuitive understanding of the underlying physics but also serves as a practical tool for optimizing qubit performance through control of gate-induced confinement and magnetic field orientation.