Effect of the Dzyaloshinski-Moriya term in the quantum (SWAP)^α gate produced with exchange coupling

The exchange coupling between two spins has been proposed as a control parameter over the states of the two-qubit space, so that it can be used to produce quantum gate operations, and in particular, the (SWAP)^α two-qubit quantum gate. In this work, we study the effect that the Dzyaloshinski-Moriya term (DM) has on the (SWAP)^α gate operation. By considering a time-dependent exchange interaction we have calculated dynamical properties for the probability of the two spin states, magnetization, concurrence, and fidelity of the required quantum gate to determine the effect of the anisotropic term on the (SWAP)^α gate operation. We present an analytical relation, valid in the subspace S_z=0, between the concurrence (measurement of entanglement) and the on-site magnetization as an observable. We also show, with calculation of the fidelity of the system, that the DM term, will add an intrinsic error to the gate that depends on its amplitude and the concurrence of the α state.