Potential errors in a scheme of universal quantum gates in Kane’s model

We reinvestigate a plausible proposal for universal quantum gates in Kane’s model, in which the authors assumed that electron spin is always downward under a background magnetic field and the value of the controlling parameters is varied instantaneously. We demonstrate that a considerable error appears, for example, in the $X$ rotation. As a result, the controlled operations do not work. This failure is caused by improper choice of the computational bases; actually, the electron spin is not always downward over time during quantum operations.

Figure 1

Configurations of the $H_C$ energy level around the level crossing point $(J\simeq {\mu }_{B}B∕2)$. The four solid lines represent the lowest eigenvalues of $H_C^{(0,1)}$, $H_C^{(-1,-1)}$, $H_C^{(-1,1)}$, and $H_C^{(-2,1)}$, respectively, which are associated with the computational bases. The two dashed lines represent the second-lowest eigenvalues for each $H_C^{(0,1)}$ and $H_C^{(-1,-1)}$.

Figure 2

Profile of $A_i\left(t\right)$ $(a=0.6)$.

Figure 3

Profile of $J\left(t\right)$ $({\tau }_h=t_h∕t_a=2.0$ and ${\tau }^{\prime }=0.2$).