Simple Pulses for Elimination of Leakage in Weakly Nonlinear Qubits

In realizations of quantum computing, a two-level system (qubit) is often singled out from the many levels of an anharmonic oscillator. In these cases, simple qubit control fails on short time scales because of coupling to leakage levels. We provide an easy to implement analytic formula that inhibits this leakage from any single-control analog or pixelated pulse. It is based on adding a second control that is proportional to the time derivative of the first. For realistic parameters of superconducting qubits, this strategy reduces the error by an order of magnitude relative to the state of the art, all based on smooth and feasible pulse shapes. These results show that even weak anharmonicity is sufficient and in general not a limiting factor for implementing quantum gates.

Figure 1

Error vs pulse duration for Gaussian ($\sigma =0.5t_g$) (blue solid line), Gaussian ($\sigma =0.5t_g$) with DRAG (red dot-dashed line), tangential ($\sigma =0.6t_g–1.4$) with DRAG (green dashed line), and GRAPE pulse (purple crosses with dotted guiding line). The standard Gaussian and DRAG pulses are continuous, while the GRAPE pulse has a pixel size of 1 ns. Parameters are $\lambda =\sqrt{2}$, $\Delta =2\pi (-400\mathrm{MHz})$ (typical for the Phase [8, 9, 10] and Transmon [11, 12, 13, 14] qubits). The top axis is set to dimensionless units, $1/|\Delta |$.

Figure 2

(a) 6 ns analog Gaussian ($\sigma =t_g/2$) with DRAG; (b) 4 ns digital GRAPE with no detuning and pixels at 1 ns. In both cases, the blue solid line is the ${\mathcal{E}}_x$ control, and the green dashed line is the ${\mathcal{E}}_y$ control. The red dot-dashed line in (a) represents explicit detuning, ${\delta }_1$. Other parameters are the same as in Fig. 1.

Figure 3

Pulse duration vs pixel width for 1 control (blue solid line) and 2 controls (red dashed line). Pulse duration is calculated as the minimal time to bring the error down to ${10}^{-5}$. The insert shows the two control GRAPE result for $t_g=1/4\Delta$ (blue solid line is ${\mathcal{E}}_x$, and green dashed line is ${\mathcal{E}}_y$). Other parameters are the same as in Fig. 1.

Figure 4

(a) Error vs pulse duration for $T_1=40\mu \mathrm{s}$ and (b) minimum error vs $T_1$ for Gaussian ($\sigma =t_g/2$) (blue solid line), Gaussian ($\sigma =t_g/2$) with DRAG (red dot-dashed line), and GRAPE (purple dotted line) with 1 ns pixels. Other parameters are the same as in Fig. 1 and pulses at the optimal times in (a) are plotted in Fig. 2.