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arXiv · 2504.01785

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

Chungwei Lin·Qi Ding·Petros T. Boufounos·Yanting Ma·Yebin Wang·Dries Sels·Chih-Chun Chien
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Abstract

Optimal control theory is applied to analyze the time-optimal solution with a single scalar control knob in a two-level quantum system without quantum decoherence. Emphasis is \change{placed} on the dependence on the maximum control strength $u_\text{max}$. General constraints on the optimal protocol are derived and used to rigorously parameterize the time-optimal solution. Two concrete problems are investigated. For generic state preparation problems, both multiple bang-bang and 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 singular control emerges. For the X-gate operation of a qubit, the optimal protocol \change{is exclusively} multiple bang-bang. The minimum gate time is about 80\% of that based on the resonant Rabi $π$-pulse over a wide range of control strength; in the $u_\text{max} \rightarrow 0$ limit this ratio is derived to be $π/4$. To develop practically feasible protocols, we present methods to smooth the abrupt changes in the bang-bang control while preserving perfect gate fidelity. \change{The presence of bang-bang segments in the 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.

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Chungwei Lin, Qi Ding, Petros T. Boufounos, Yanting Ma, Yebin Wang, Dries Sels, Chih-Chun Chien. 2025-04-02. Time-optimal single-scalar control on a qubit of unitary dynamics. https://doi.org/10.1103/physreva.111.042602

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