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arxiv: 2510.01923 · v3 · submitted 2025-10-02 · 🪐 quant-ph

The Constant Geometric Speed Schedule for Adiabatic State Preparation

Mancheon Han , Hyowon Park , Sangkook Choi This is my paper

Pith reviewed 2026-05-18 10:50 UTC · model grok-4.3

classification 🪐 quant-ph
keywords adiabatic state preparationconstant geometric speed scheduleenergy gap scalingadiabatic path lengthquadratic speedupquantum annealingmolecular simulationunstructured search
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The pith

A constant geometric speed schedule reduces adiabatic evolution time scaling from O(Δ^{-2}) to O(L Δ^{-1}) when path length stays independent of the gap.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

Adiabatic state preparation changes a Hamiltonian slowly from an easy initial form to a hard target, keeping the system in the ground state if the change is slow enough relative to the smallest energy gap Δ. The usual linear schedule forces evolution time T to grow as the square of 1/Δ. The paper replaces that schedule with one that advances at constant speed measured by the geometric distance along the path in parameter space. This change improves the scaling to T proportional to L/Δ, where L is the total path length, whenever L itself does not grow as Δ shrinks. The authors also supply a segmented version that estimates segment lengths from eigenstate overlaps, so only a global lower bound on Δ is required instead of the full gap function. Tests on search problems and small molecules confirm the improved scaling.

Core claim

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, reducing the prior spectral-knowledge requirement from the full gap function Δ(s) to just a global lower bound on the energy gap.

What carries the argument

Constant geometric speed (CGS) schedule that advances the Hamiltonian at a uniform rate measured by geometric distance along the adiabatic path in parameter space.

If this is right

  • Evolution time T scales as O(L Δ^{-1}) rather than the conventional O(Δ^{-2}).
  • Quadratic speedup over the linear schedule is realized whenever L stays bounded as Δ approaches zero.
  • The segmented protocol requires only a global lower bound on the gap instead of the full function Δ(s).
  • Optimal Δ^{-1} scaling is observed in numerical tests on unstructured search, N₂, and a [2Fe-2S] cluster.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The same constant-speed construction could be paired with existing adiabatic optimizations such as counter-diabatic driving to reduce coherence demands further.
  • Similar uniform-rate ideas may apply directly to diabatic or hybrid quantum-classical state-preparation routines that already track path geometry.
  • Hardware implementations on devices with fixed coherence times would see the largest practical gains precisely when the gap is smallest yet the path remains short.

Load-bearing premise

The adiabatic path length L can be bounded independently of the minimum energy gap Δ.

What would settle it

A concrete adiabatic problem in which the path length L grows at least linearly with 1/Δ, so that the CGS schedule loses its claimed scaling advantage over the linear schedule.

Figures

Figures reproduced from arXiv: 2510.01923 by Hyowon Park, Mancheon Han, Sangkook Choi.

Figure 1
Figure 1. Figure 1: Schematic of the constant speed schedule construction. [PITH_FULL_IMAGE:figures/full_fig_p005_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Application of the constant speed schedule to the adiabatic Grover search. Panels (a–c) show results for [PITH_FULL_IMAGE:figures/full_fig_p006_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Application of the constant speed schedule to the nitrogen molecule. (a) Low-lying energy spectra from Density Functional Theory [PITH_FULL_IMAGE:figures/full_fig_p008_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Application of the constant speed schedule to the [2Fe-2S] cluster. (a) Molecular structure. (b) Estimated adiabatic evolution time [PITH_FULL_IMAGE:figures/full_fig_p008_4.png] view at source ↗
read the original abstract

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.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

1 major / 2 minor

Summary. The manuscript introduces the constant geometric speed (CGS) schedule for adiabatic state preparation. Standard linear schedules yield evolution time T scaling as O(Δ^{-2}) with the minimum gap Δ. The paper notes the rigorous lower bound T = O(L Δ^{-1}) with adiabatic path length L, and claims that CGS achieves the optimal Δ^{-1} scaling (quadratic improvement) by traversing the path at uniform geometric speed, provided L remains bounded independently of Δ. A segmented CGS variant is proposed that determines segment lengths from on-the-fly eigenstate overlaps, requiring only a global lower bound on the gap rather than the full Δ(s) function. Numerical tests on adiabatic unstructured search, the N₂ molecule, and a [2Fe-2S] cluster are reported to confirm the optimal scaling.

Significance. If the condition that L is independent of Δ holds in the relevant regimes, the CGS schedule offers a practical route to quadratic speedups in adiabatic quantum algorithms while reducing the spectral information needed a priori. The segmented protocol is a notable practical contribution. The numerical demonstrations on both unstructured search and molecular systems provide concrete support for improved scaling in those instances.

major comments (1)
  1. [Abstract and Numerical Tests] Abstract and numerical results: The headline claim of a Δ^{-1} scaling improvement (quadratic speedup) is conditional on L remaining bounded independently of Δ. While the abstract correctly notes the formal bound L = O(Δ^{-1}) and asserts that L 'can be bounded independently' in practice, no explicit computation or plot of L versus Δ is provided for the tested systems (unstructured search, N₂, [2Fe-2S]). If L scales as Δ^{-1} in these examples, the total T reverts to O(Δ^{-2}) and the claimed improvement does not materialize. This verification is load-bearing for the central scaling result.
minor comments (2)
  1. [Methods] Clarify in the methods section how the geometric speed is normalized and how the on-the-fly overlap computation is implemented without introducing additional error sources.
  2. [Figures] Ensure all figures reporting scaling include error bars or confidence intervals from multiple runs, and label axes consistently with the symbols defined in the text.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their careful review of our manuscript on the constant geometric speed schedule for adiabatic state preparation. We address the major comment point by point below.

read point-by-point responses

  1. Referee: [Abstract and Numerical Tests] Abstract and numerical results: The headline claim of a Δ^{-1} scaling improvement (quadratic speedup) is conditional on L remaining bounded independently of Δ. While the abstract correctly notes the formal bound L = O(Δ^{-1}) and asserts that L 'can be bounded independently' in practice, no explicit computation or plot of L versus Δ is provided for the tested systems (unstructured search, N₂, [2Fe-2S]). If L scales as Δ^{-1} in these examples, the total T reverts to O(Δ^{-2}) and the claimed improvement does not materialize. This verification is load-bearing for the central scaling result.

    Authors: We thank the referee for this important observation. The numerical results demonstrate that the CGS schedule achieves T scaling as O(Δ^{-1}) for the tested systems, in contrast to the linear schedule. Given the relation T = O(L Δ^{-1}), the observed scaling implies L remains bounded independently of Δ; otherwise T would revert to O(Δ^{-2}). To make this explicit, we will add plots of the computed path length L versus Δ for the unstructured search, N₂, and [2Fe-2S] systems in the revised manuscript. These will confirm L is independent of Δ in the relevant regimes. revision: yes

Circularity Check

0 steps flagged

No significant circularity; central claim conditional on external bound for L

full rationale

The paper derives the Δ^{-1} scaling improvement for the CGS schedule directly from the known rigorous lower bound T = O(L Δ^{-1}) by constructing a uniform geometric speed traversal that saturates this bound when L is independent of Δ. This independence is explicitly treated as a provided condition rather than derived from the schedule or fitted to data; the abstract and claim acknowledge the formal O(Δ^{-1}) upper bound on L while noting it is often loose in practice. No self-definitional loops, fitted parameters renamed as predictions, or load-bearing self-citations appear in the provided derivation chain. The segmented protocol reduces spectral requirements to a global gap lower bound via on-the-fly overlaps, which is an independent algorithmic choice. Numerical demonstrations on search, N2, and [2Fe-2S] are presented as confirmation rather than inputs to the scaling result. The derivation remains self-contained against the external adiabatic theorem benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The main unproven or assumed element is the independence of the path length from the gap, which enables the quadratic speedup claim. No free parameters or new entities are introduced in the abstract.

axioms (1)
  • domain assumption The adiabatic path length L can be bounded independently of the minimum energy gap Δ
    This condition is required for the claimed reduction in evolution time scaling to Δ^{-1}.

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Reference graph

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