Toward Quadratically Faster Adiabatic State Preparation without Gap Information: The Constant-Speed Schedule

The efficiency of quantum adiabatic evolution is determined by the total evolution time $T$, which depends on the minimum spectral gap $\Delta$. While generic schedules lead to the scaling $T \sim \Delta^{-2}$, the rigorous lower bound is $\mathcal{O}(\Delta^{-1})$, revealing the possibility of a quadratic improvement through an appropriate schedule design. We introduce the constant-speed schedule, which follows the adiabatic eigenstate path at a uniform rate. We first show that this strategy improves the scaling of the upper bound on the required evolution time by one power of $1/\Delta$. We then propose a segmented constant-speed protocol, where the lengths of the path segments are computed from eigenstate overlaps calculated along the evolution, eliminating the need for prior spectral information. We benchmark the method on adiabatic unstructured search, the N$_2$ molecule, and the [2Fe--2S] cluster. Across these examples, the constant-speed schedule achieves the optimal $1/\Delta$ scaling in regimes with small gaps, demonstrating a quadratic speedup relative to the standard linear schedule.