Time-optimal single-scalar control on a qubit of unitary dynamics

Optimal control theory is applied to analyze the time-optimal solution with a single scalar control knob in a pure two-level quantum system without quantum decoherence. Emphasis is put on the dependence on the maximum control strength $u_\text{max}$. General constraints on the optimal protocol are derived and are used to parameterize the optimal solution. Two concrete problems are investigated. For the generic state preparation problems, both multiple bang-bang or bang-singular-bang are legitimate and should be considered. Generally the optimal is bang-bang for small $u_\text{max}$ and there exists a state-dependent critical amplitude above which the singular control emerges. For the X-gate operation of a qubit, the optimal protocol can only be multiple bang-bang. The minimum gate time is about 80\% of that based on the resonant Rabi $\pi$-pulse over a wide range of control strength; in the $u_\text{max} \rightarrow 0$ limit this ratio is derived to be $\pi/4$. A few methods to smooth the abrupt changes in the bang-bang protocol while preserving the gate fidelity are presented. The bang-bang being part of time-optimal protocol indicates that the high-frequency components and a full calculation (instead of the commonly adopted Rotating Wave Approximation) are essential for the ultimate quantum speed limit.