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We study the exchange interaction between two hole-spin qubits in a doublequantum dot setup in a silicon nanowire in the presence of magnetic andelectric fields. Based on symmetry arguments we show that there exists aneffective spin that is conserved even in highly anisotropic semiconductors,provided that the system has a twofold symmetry with respect to the directionof the applied magnetic field. This finding facilitates the definition of qubitbasis states and simplifies the form of exchange interaction for two-qubitgates in coupled quantum dots. If the magnetic field is applied along a genericdirection, cubic anisotropy terms act as an effective spin-orbit interactionintroducing novel exchange couplings even for an inversion symmetric setup.Considering the example of a silicon nanowire double dot, we present therelative strength of these anisotropic exchange interaction terms and calculatethe fidelity of the $\sqrt{\text{SWAP}}$ gate. Furthermore, we show that theanisotropy-induced spin-orbit effects can be comparable to that of the directRashba spin-orbit interaction for experimentally feasible electric fieldstrengths.

Exchange interaction of hole-spin qubits in double quantum dots in highly anisotropic semiconductors