The Constant Geometric Speed Schedule for Adiabatic State Preparation: Towards Quadratic Speedup without Prior Spectral Knowledge

The efficiency of adiabatic quantum evolution is governed by the evolution time $T$, which typically scales as $\mathcal{O}(Δ^{-2})$ with the minimum energy gap $Δ$. However, the rigorous lower bound is $\mathcal{O}(LΔ^{-1})$, where $L$ is the adiabatic path length. Although $L$ is formally upper-bounded by $\mathcal{O}(Δ^{-1})$, such a bound is often too loose in practice, and $L$ can be bounded independently of $Δ$. 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 $Δ^{-1}$, provided $L$ remains bounded independently of $Δ$. We propose a segmented CGS protocol where path segment lengths are computed from eigenstate overlaps on the fly, eliminating the need for prior spectral knowledge. Numerical tests on adiabatic unstructured search, N$_2$, and a [2Fe-2S] cluster demonstrate the optimal $Δ^{-1}$ scaling, confirming a quadratic speedup over the standard linear schedule.

💡 Research Summary

The paper addresses a central bottleneck in adiabatic quantum computation and adiabatic state preparation (ASP): the evolution time T required to achieve a target fidelity scales poorly with the minimum spectral gap Δ, typically as O(Δ⁻²). While rigorous lower bounds indicate that T must be at least O(L/Δ), where L is the geometric length of the instantaneous ground‑state path, this bound can be tight when L does not depend on Δ. The authors exploit this observation by introducing the Constant Geometric Speed (CGS) schedule, which moves along the adiabatic path at a uniform geometric speed. By re‑expressing the adiabatic error bound in purely geometric terms—instantaneous speed v(τ) and curvature κ(τ)—they show that a schedule with ∂_τ v = 0 eliminates the acceleration term and reduces every contribution to the error bound by one power of Δ. Consequently, if both the path length L and the total curvature K = ∫₀ᴸκ(l)dl remain bounded independently of Δ, the CGS schedule achieves the optimal scaling T = O(Δ⁻¹), a quadratic improvement over the standard linear schedule.

A practical obstacle is that L and K are not known a priori. The authors solve this by measuring overlaps between neighboring ground‑state wavefunctions during the evolution. The overlap |⟨Φ(s)|Φ(s+Δs)⟩|²≈1−(Δℓ)² gives the infinitesimal arc‑length Δℓ. By adjusting the physical time step Δt so that Δℓ/Δt remains constant, the algorithm enforces a constant geometric speed. Theorem 1 proves that a discretized schedule built from a monotonic sequence of s‑values and the corresponding overlap‑derived Δℓ’s converges to the continuous CGS schedule as the discretization becomes fine. The resulting segmented CGS protocol requires only a global lower bound on Δ (to filter out excited‑state contamination), which is far easier to estimate than the full gap function Δ(s). The overhead of overlap estimation scales as O(Δ⁻¹), preserving the overall Δ⁻¹ scaling of the total runtime T_tot = T + T_ov.

The authors validate the approach on three representative problems. In the adiabatic Grover search, where Δ = 1/√N, the CGS schedule reproduces the known optimal schedule (∂_τ s ∝ Δ²) without any prior knowledge of Δ(s). Numerical simulations for N = 2¹⁴ show a 92‑fold speedup over the linear schedule and confirm the T ∝ Δ⁻¹ scaling. For the nitrogen molecule (STO‑3G basis, singlet subspace), the minimum gap occurs near the start of the path for stretched bonds. The CGS schedule allocates disproportionate time to this region, achieving a 52‑fold reduction in evolution time compared to a linear schedule. Importantly, the measured path length L and curvature K remain essentially constant as the gap shrinks, confirming the theoretical premise. Finally, a strongly correlated