Fundamental limit to qubit control with coherent field

The ultimate accuracy as regards controlling a qubit with a coherent field is studied in terms of degradation of the fidelity by employing a fully quantum mechanical treatment. While the fidelity error accompanied by $\pi /2$ pulse control is shown to be inversely proportional to the average photon number in a way similar to that revealed by J. Gea-Banacloche's [Phys. Rev. A 65, 022308 (2002)] results, our results show that the error depends strongly on the initial state of the qubit. When the initial state of the qubit is the ground state, the error is about 20 times smaller than that of the control started from the excited state, no matter how large $N$ is. This dependency is explained in the context of an exact quantum mechanical description of the pulse area theorem. Using the result, the error accumulation tendency of successive pulse controls is found to be both nonlinear and initial state dependent.

Figure 1

Solid lines are exact error rates ($1-{\mathrm{fidelity}}^2$) plotted for the average photon number of the field $N$ on a ${\log }_{10}$ scale for the initially “0” (down) qubit case and the initially “1” (up) state qubit case, which show good asymptotic agreement even for the small $N$ of $\sim 100$, with the dashed lines represent $\frac{{(\pi +2)}^2}{64N}$ and $\frac{{(\pi -2)}^2}{64N}$, respectively. The dashed lines in between represent $\frac{{\pi }^2+4}{64N}$ (Gea-Banacloche's result) and $\frac{1}{16(N+1)}$ (Ozawa's result).

Figure 2

Error rates of the $\pi$ and $\pi /2$ controls ($\theta =\pi /2$ and $\pi /4$, respectively) from the initial values ${\theta }_0=0$ and $\pi /2$ are plotted for the target values ${\theta }_0+\theta$. Open squares on the two kinked lines represent the error rates for the successive $\pi /2$ pulse controls with the average photon number $N$ in each control pulse, while open circles show the single $\pi$ pulse controls with average photon numbers $N$ (upper) and $2N$ (lower), all obtained from the quantum analysis. Filled diamonds represent the corresponding error rates obtained from the classical analysis. The error rates obtained from quantum and classical analysis show good agreement except for the kinked points.

Figure 3

Error rates of the $\pi$ and $\pi /2$ controls ($\theta =\pi /2$ and $\pi /4$, respectively) from the initial values ${\theta }_0=\pi /4$ and $3\pi /4$ are plotted for the target values ${\theta }_0+\theta$. The legend is the same as in Fig. 2. In this case, successive $\pi /2$ pulses show a linear error increase even in the quantum analysis. The quantum results are totally different from the classical counterparts and show a large discrepancy between the two initial conditions.