Effect of the Dzyaloshinski-Moriya term in the quantum ${\left(\text{SWAP}\right)}^{\alpha }$ 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 ${\left(\text{SWAP}\right)}^{\alpha }$ two-qubit quantum gate. In this work, we study the effect that the Dzyaloshinski-Moriya term (DM) has on the ${\left(\text{SWAP}\right)}^{\alpha }$ 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 ${\left(\text{SWAP}\right)}^{\alpha }$ 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 $\alpha$ state.

Figure 1

(Color online) Probabilities for the states $|\uparrow ,\downarrow ⟩$ and $|\downarrow ,\uparrow ⟩$ as a function of time, for $\alpha =\frac{1}{4}$, $\frac{1}{2}$, and 1. The values of ${\lambda }_r$ for the reference curves are given in Table and are included in the figure for the same values for $\alpha$ with ${\beta }_0=0$. By considering a nonzero DM amplitude, ${\beta }_0=0.4$, and the same reference value of ${\lambda }_r$ as a gate error occurs. We use the scheme presented in Eq. (3.13) to correct the pulse width to the values of ${\lambda }_c$ also presented in Table , producing curves with the correct probabilities.

Figure 2

(Color online) The time evolution for the on-site magnetization, ${\sigma }_z\left(t\right)$, for values of $\alpha =\frac{1}{4}$, $\alpha =\frac{1}{2}$, and $\alpha =1$. The same guidelines as in Fig. 1 are followed: (a) a reference curve, ${\lambda }_r$ and ${\beta }_0=0$, is presented for each case, (b) the error curve appears by “turning on” the DM amplitude with the same value for ${\lambda }_r$ and ${\beta }_0=0.4$, and (c) the corrected curve with ${\lambda }_c$ and $\beta =0.4$ is determined.

Figure 3

(Color online) The time evolution for the concurrence $C\left(t\right)$, for values of $\alpha =\frac{1}{4}$, $\alpha =\frac{1}{2}$, and $\alpha =1$. The same guidelines as in Fig. 1 are followed: (a) a reference curve, ${\lambda }_r$ and ${\beta }_0=0$, is presented for each case, (b) the error curve appears by “turning on” the DM amplitude with the same value for ${\lambda }_r$ and ${\beta }_0=0.4$, and (c) the corrected curve with ${\lambda }_c$ and $\beta =0.4$ is determined.

Figure 4

(Color online) The time evolution of the fidelity for $\alpha =\frac{1}{4}$, $\alpha =\frac{1}{2}$, and $\alpha =1$. The same guidelines as in Fig. 1 are followed: (a) a reference curve, ${\lambda }_r$ and ${\beta }_0=0$, is presented for each case, (b) the error curve appears by “turning on” the DM amplitude with the same value for ${\lambda }_r$ and ${\beta }_0=0.4$, and (c) the corrected curve with ${\lambda }_c$ and $\beta =0.4$ is determined. The final values of the fidelity does not reach the optimum of 1 except for $\alpha =1$.

Figure 5

(Color online) The control parameter, inverse pulse width ${\lambda }_c$, as a function of the DM amplitude ${\beta }_0$, under the pulse condition (3.13), for three values of $\alpha =1/4$, 1/2, and 1. The pulse parameter ${\lambda }_c$ can be fitted with a quadratic behavior of the amplitude ${\beta }_0$.