Single-qubit quantum gate at an arbitrary speed
Quantum information processing comprises physical processes, which obey the quantum speed limit (QSL): high speed requires strong driving. Single-qubit gates using Rabi oscillation, which is based on the rotating wave approximation (RWA), satisfy this bound in the form that the gate time $T$ is inversely proportional to the Rabi frequency $Ω$, characterizing the driving strength. However, if the gate time is comparable or shorter than the qubit period $T_{0} \equiv 2π/ ω_{0}$, the RWA actually breaks down since the Rabi frequency has to be large compared to the qubit frequency $ω_{0}$ due to the QSL, which is given as $T \gtrsim π/Ω$. We show that it is possible to construct a universal set of single-qubit gates at this strong-coupling and ultrafast regime, by adjusting the central frequency $ω$ and the Rabi frequency $Ω$ of the driving pulse. We observe a transition in the scaling behavior of the central frequency from the long-gate time regime ($T \gg T_{0}$) to the short-gate time ($T \ll T_{0}$) regime. In the former, the central frequency is nearly resonant to the qubit, i.e., $ω\simeq ω_{0}$, whereas in the latter, the central frequency is inversely proportional to the gate time, i.e., $ω\sim π/T$. We identify the transition gate time at which the scaling exponent $n$ of the optimal central frequency $ω\sim T^{n}$ changes from $n=0$ to $n=-1$.