The Constant Geometric Speed Schedule for Adiabatic State Preparation

The efficiency of adiabatic quantum evolution is governed by the evolution time $T$, which typically scales as $\mathcal{O}(\Delta^{-2})$ with the minimum energy gap $\Delta$. However, the rigorous lower bound is $\mathcal{O}(L\Delta^{-1})$, where $L$ is the adiabatic path length. Although $L$ is formally upper-bounded by $\mathcal{O}(\Delta^{-1})$, such a bound is often too loose in practice, and $L$ can be bounded independently of $\Delta$. This indicates the potential for a quadratic speedup through adiabatic schedule construction. Here, we introduce the constant geometric speed (CGS) schedule, which traverses the adiabatic path at a uniform rate. We show that this approach reduces the scaling of the evolution time by a factor of $\Delta^{-1}$, provided $L$ remains bounded independently of $\Delta$. We propose a segmented CGS protocol where path segment lengths are computed from eigenstate overlaps on the fly, reducing the prior spectral-knowledge requirement from the full gap function $\Delta(s)$ to just a global lower bound on the energy gap. Numerical tests on adiabatic unstructured search, N$_2$, and a [2Fe-2S] cluster demonstrate the optimal $\Delta^{-1}$ scaling, confirming a quadratic speedup over the standard linear schedule.