Symmetric spin-orbit interaction in triple quantum dot and minimisation of spin-orbit leakage in CNOT gate

We analyse spin-orbit interaction in triple quantum dots and show that a symmetric spin-orbit Hamiltonian does not follow the standard form used in double quantum dots, as a consequence of the presence of the third dot in the setup. Furthermore, CNOT implementation schemes based on the exchange interaction were studied. It was shown that an antisymmetric Dzyaloshinsky-Moriya term is the dominant source of spin-orbit leakage from the computational space. We present a simple scheme for the minimisation of leakage that can be implemented in cases where interacting spins enclose parallelogram or equilateral triangle loops.