Effective spin Hamiltonian of a gated triple quantum dot in the presence of spin-orbit interaction

We derive and study the effective spin Hamiltonian of a gated triple quantum dot that includes the effects of spin-orbit interaction and an external magnetic field. In the analysis of the resulting spin interaction in linear and in general triangular geometry of the dots, we show that the pairwise spin interaction does depend on the position of the third dot. The spin-orbit induced anisotropy, in addition to changing its strength, also changes its symmetry with the motion of the third quantum dot outside the linear arrangement. Our results present a simplified model that may be used in the design of quantum computers based on three-spin qubits.