Analytic control methods for high-fidelity unitary operations in a weakly nonlinear oscillator

In qubits made from a weakly anharmonic oscillator the leading source of error at short gate times is leakage of population out of the two dimensional Hilbert space that forms the qubit. In this article we develop a general scheme based on an adiabatic expansion to find pulse shapes that correct this type of error. We find a family of solutions that allows tailoring to what is practical to implement for a specific application. Our result contains and improves the previously developed derivative removal by adiabatic gate technique [F. Motzoi et al., Phys. Rev. Lett. 103, 110501 (2009)] and allows a generalization to other nonlinear oscillators with more than one leakage transition.

Figure 1

Energy level diagram of the system we are considering. The qubit is formed by the $|0\rangle$ and $|1\rangle$ (green) levels and we aim to have complete control in this subspace when leakage to the $|2\rangle$ and then $|3\rangle ,$ etc., is possible (red arrows). The dotted black lines indicate the positions the energy levels would be at if the system was a harmonic oscillator of frequency $\omega$.

Figure 2

Fourier transforms of the control fields (a, c, and e) and populations (b, d, and f) of the ground (blue solid), first (red dotted), and second excited (green dash) in a simulation of a not gate with a Gaussian amplitude pulse for a $d=5$ SNO. In (a and b) $\sigma =2\pi /3{\Delta }_2$, (c and d) $\sigma =4\pi /3{\Delta }_2$, and (e and f) $\sigma =3\pi /{\Delta }_2$ and the gate time is taken to be $4\sigma$.

Figure 3

Gate error for the implementation of a not gate in a $d=5$ SNO as a function of $\sigma$, with $t_g=4\sigma$, and a Gaussian-shaped pulse. The blue dotted line is the zeroth-order solution. The black dashed line is the first-order $Z$-only correction. The red dash-dot-dot line is the first-order $Y$-only correction. The green dash-dot line is the first-order correction from the controls presented in Ref. [30]. The purple solid line is for the optimal first-order correction.

Figure 4

Gate error for the implementation of a not gate in a $d=5$ SNO as a function of $\sigma$, with $t_g=4\sigma$, and a Gaussian-shaped pulse. The blue dotted line is the zeroth-order solution. The black dashed line is the second-order $Z$-only correction. The red dash-dot-dot line is the second-order $Y$-only correction. The green dash-dot line is the second-order correction from the controls presented in Ref. [30]. The purple solid line is for the optimal first-order correction.

Figure 5

Gate error for the implementation of a not gate in a $d=5$ SNO as a function of $\sigma$, with $t_g=4\sigma$, and a Gaussian-shaped pulse. (a) Optimized first-order solutions. The blue is for optimized $\alpha$ with $\beta =\gamma ={\delta }_0=0$ (zeroth-order solution). The black dashed is for optimized $\alpha$ and $\gamma$ with $\beta ={\delta }_0=0$ ($Z$-only solution). The red dash-dot-dot line is optimized $\alpha$ and $\beta$ with $\gamma ={\delta }_0=0$ ($Y$-only solution). The purple solid line is for optimized $\alpha$, $\beta$, and $\gamma$ with ${\delta }_0=0$ (optimal firstorder solution). (b) The same as in (a) but with ${\delta }_0$ being optimized.

Figure 6

Energy-level diagrams for systems that can also be model by our theory. In (a) we consider the case when the qubit has leakage from both its $|0\rangle$ and $|1\rangle$ level. In (b) we consider the case when the qubit has leakage from its $|1\rangle$ level to more than one higher level.

Figure 7

Gate error for the implementation of a not gate when there is leakage above and below the qubit subspace. The system considered is explained in the text. The blue dotted line is the zeroth-order solution. The black dash line is the first order $Z$-only correction. The red dash-dot-dot line is the first order $Y$-only correction. The purple solid line is for the optimal first-order correction.

Figure 8

Gate error for the implementation of a not gate when there is many leakage transitions for the excited state. The system considered is explained in the text. The blue dotted line is the zeroth-order solution. The black dashed line is the first-order $Z$-only correction. The red dash-dot-dot line is the first-order $Y$-only correction. The purple solid line is for the optimal first-order correction.

Figure 9

Ratio of coupling strength of the $|1\rangle \to |2\rangle$ transition to the $|0\rangle \to |1\rangle$ transition as a function of anharmonicity for control through a resonator (solid blue) and direct drive (dashed red).