Realization of holonomic single-qubit operations

Universal single-qubit operations based on purely geometric phase factors in adiabatic processes are demonstrated by utilizing a four-level system in a trapped single ${}^{40}$Ca${}^{+}$ ion connected by three oscillating fields. Robustness against parameter variations is studied. The scheme demonstrated here can be employed as a building block for large-scale holonomic quantum computations, which may be useful for large qubit systems with statistical variations in system parameters.

Figure 1

(a) Level scheme of a four-level system used to implement holonomic single-qubit operations. (b) Pulse sequence for holonomic single-qubit operations. (c) Level scheme used for ${}^{40}$Ca${}^{+}$ in the experiment. (d) Experimentally used pulse sequence. (e) Level scheme that implements another four-level system containing one rf and two optical transitions, with qubit levels $|0\rangle ,|1\rangle$ separated by an optical frequency.

Figure 2

Results of single-qubit rotations ($x$ and $z$ rotations). Variations of population are plotted against the phase shift in the middle of the gate pulse. The number of experiments per data points is (a and b) 200 for experiments using three optical transitions and (c and d) 100 for those using one rf and two optical transitions. The error bars represent errors in projection measurements, which are derived as standard deviations in binomial distributions with numbers of samples as given above. (a) Results for $x$ rotations using three optical transitions. Blue hollow circles, red filled circles, magenta crosses, and black asterisks represent the populations in $|0\rangle$; $|1\rangle$, $|2\rangle$; and $|u\rangle$; respectively. (b) Results for $z$ rotations using three optical transitions. The population in $|0\rangle$ is plotted. (c) Results for $x$ rotations using one rf and two optical transitions. Red filled circles (blue hollow circles) represent the populations in $|1\rangle$ and $|2\rangle$ when $|0\rangle$ ($|1\rangle$) is initially prepared. (d) Results for $z$ rotations using one rf and two optical transitions. The population in $|1\rangle$ and $|2\rangle$ is plotted.

Figure 3

Demonstration of the robustness of single-qubit operations. Each point represents the fringe contrast and shift of the population oscillation obtained by varying the rotation angle of a certain gate operation. Here the rotation angle is varied from 0 to $2\pi$ in 17 steps, and the number of experiments per angle step is 50. The errors in the population measurements are estimated in the same way as in Fig. 2, and the fringe contrasts and shifts are obtained through weighted fits considering those errors. The error bars in the figure represent errors in the fitted parameters. $\alpha =0.55$ is used here. (a) Fringe contrasts and (b) shifts for $x$ rotations as the peak Rabi frequencies are varied. Here the ratios of the peak values are fixed. The horizontal axis represents the peak value of ${\Omega }_2$, and the blue circles, red triangles, and green squares represent the results for pulse durations of 145, 290, and 435 $\mu$s, respectively. Numerically calculated results for pulse durations of 145, 290, and 435 $\mu$s are also plotted as blue solid curves, red dashed curves, and green dotted curves, respectively. (c) Fringe contrasts and (d) shifts for $z$ rotations as the peak Rabi frequencies are varied. The ratios of the peak values are fixed. Numerically calculated results are also plotted as a blue solid curve.