Nonadiabatic corrections to fast dispersive multiqubit gates involving $Z$ control

We review a time-dependent version of the Schrieffer-Wolff transformation that accounts for real-time control of system parameters, soon to be rendered possible on a broad basis due to technical progress. The dispersive regime of $N$ multilevel systems coupled to a cavity via a Jaynes-Cummings interaction is extended to the most general case. As a concrete example we rigorously apply the technique to dispersive two-qubit gates in a superconducting architecture, showing that fidelities based on previous models are off by up to ${10}^{-2}$, which is certainly relevant for high-fidelity gates compatible with fault-tolerant quantum information devices. A closed analytic form for the error depending on the target evolution closes our work.

Figure 1

Normalized histograms for the differences in fidelity $\mathrm{\Delta }F$ with respect to $N_s=10000$ random unitaries. Neglecting the time-dependent part of the dispersive transformation ($\propto \dot{\lambda }$) leads to errors on the order of ${10}^{-3}$ (top row), which is certainly relevant for high-fidelity gates. Models as were used before [19, 20]—which assume all instances of $g,\chi$, and $\delta$ to be constant—lead to errors on the order of ${10}^{-2}$ (bottom row). (a) and (c) belong to sinusoidal pulses ${\mathrm{\Phi }}_s$, (b) and (d) to tangential ones $\left({\mathrm{\Phi }}_t\right)$ given by Eqs. (20).

Figure 2

Normalized histogram for the difference in fidelity $\mathrm{\Delta }F$ with respect to $N_s=10000$ random unitaries obtained from a second order Magnus expansion, see Eq. (27). The statistics are based on a tangential pulse and reproduce those of Fig. 1 very well.