Controlled-NOT gate with weakly coupled qubits: Dependence of fidelity on the form of interaction

An approach to the construction of the controlled-NOT quantum logic gate for a four-dimensional coupled-qubit model with weak but otherwise arbitrary coupling has been given recently [M. R. Geller et al., Phys. Rev. A 81, 012320 (2010)]. How does the resulting fidelity depend on the form of qubit-qubit coupling? In this paper we calculate intrinsic fidelity curves (fidelity in the absence of decoherence versus total gate time) for a variety of qubit-qubit interactions, including the commonly occurring isotropic Heisenberg and XY models, as well as randomly generated ones. For interactions not too close to that of the Ising model, we find that the fidelity curves do not significantly depend on the form of the interaction, and we calculate the resulting interaction-averaged fidelity curve for the non-Ising-like cases and a criterion for determining its applicability.