Spectral Gap Informed Ramp QAOA

Kieran McDowall; Konstantinos Georgopoulos; Petros Wallden

Spectral gap information from an adiabatic Hamiltonian can be used to construct QAOA parameter schedules that outperform linear ramps.

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T0 review · grok-4.3

2026-07-01 08:53 UTC pith:LMFZ7Z5Z

load-bearing objection SGIR-QAOA adds gap information to linear ramp schedules but the abstract supplies no numbers, no construction rule, and no verification of the claimed gains. the 2 major comments →

arxiv 2604.24580 v2 pith:LMFZ7Z5Z submitted 2026-04-27 quant-ph

Spectral Gap Informed Ramp QAOA

Kieran McDowall , Konstantinos Georgopoulos , Petros Wallden This is my paper

classification quant-ph

keywords QAOAspectral gapadiabatic algorithmGrover's problemMaximum Independent Setparameter schedulesquantum optimizationde-variationalisation

verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

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

The paper proposes SGIR-QAOA as a way to choose variational parameters for QAOA without expensive optimization. It extracts spectral gap details from an adiabatic path that starts with the QAOA mixer Hamiltonian and uses them to slow down evolution where the gap is narrow. This produces better solution probabilities than linear ramp schedules on Grover's problem at fixed depth and reaches the same probabilities with fewer layers. The advantage carries over to the maximum independent set problem and holds when spectral gaps are extrapolated for larger sizes or when mild noise is added.

Core claim

SGIR-QAOA incorporates spectral gap information from an adiabatic Hamiltonian, with the QAOA mixer as the initial Hamiltonian, to construct smooth parameter schedules that perform slow evolution where the spectral gap is small, resulting in performance improvements over LR-QAOA on Grover's problem at constant depth and requiring shorter depths for the same solution probability.

What carries the argument

Spectral gap informed ramp schedule that adjusts evolution speed based on the gap size of the adiabatic Hamiltonian.

Load-bearing premise

Spectral gap information extracted from the adiabatic Hamiltonian can be used to build parameter schedules that outperform linear ramps without requiring full variational optimization or new fitting parameters.

What would settle it

Demonstrating that on Grover's problem, at a fixed depth where LR-QAOA reaches a certain probability, SGIR-QAOA does not exceed that probability or requires equal or greater depth.

Watch this falsifier — get emailed when new claim-graph text bears on it.

If this is right

SGIR-QAOA achieves higher solution probabilities than LR-QAOA on Grover's problem at constant circuit depth.
SGIR-QAOA reaches the same optimal solution probability as LR-QAOA using shorter circuit depths.
Performance benefits of SGIR-QAOA extend to the Maximum Independent Set problem.
SGIR-QAOA remains effective when spectral gap information is extrapolated for problem sizes too large for exact computation.
Advantages of SGIR-QAOA persist under mild depolarising noise.

Where Pith is reading between the lines

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

The method could reduce the need for classical optimization loops in QAOA for other combinatorial problems.
Extrapolation of spectral gaps suggests the approach may scale without computing the full spectrum at every size.
Persistence under noise indicates potential robustness on near-term hardware.
If the spectral gap can be approximated efficiently, SGIR-QAOA might apply to larger instances than full adiabatic simulation allows.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit.

Desk Editor's Note

SGIR-QAOA adds gap information to linear ramp schedules but the abstract supplies no numbers, no construction rule, and no verification of the claimed gains.

read the letter

The paper's main move is to build QAOA parameter schedules from the spectral gap of an adiabatic path that starts with the mixer Hamiltonian. It slows the evolution where the gap shrinks and claims this beats plain linear-ramp QAOA on Grover at fixed depth while reaching the same probability with fewer layers; the same pattern is said to hold for MIS, extrapolated gaps, and mild depolarising noise.

The construction itself is a straightforward extension of LR-QAOA. It gives an explicit, non-variational rule that uses one spectral quantity rather than running an optimizer, and the choice of test problems is reasonable.

The soft spot is that the abstract contains none of the supporting evidence. There are no probability values, no depth comparisons, no description of how the gap is turned into the beta and gamma sequences, and no account of how gaps are computed or extrapolated for larger instances. Without those details the performance claims cannot be checked.

This is aimed at people already working on fixed or informed QAOA schedules. A reader following that literature would see the motivation and the intended comparison, but would need the full text to judge whether the advantage is real or whether the gap extraction adds hidden costs or fitting steps.

The work shows clear engagement with the parameter-choice problem, so it deserves peer review once the methods and data are available.

Referee Report

2 major / 0 minor

Summary. The manuscript introduces Spectral Gap Informed Ramp QAOA (SGIR-QAOA), a variant of QAOA that constructs smooth parameter schedules by incorporating spectral gap information from an adiabatic Hamiltonian (using the QAOA mixer as the initial Hamiltonian). It claims that SGIR-QAOA outperforms Linear Ramp QAOA (LR-QAOA) on Grover's problem at constant depth, achieves the same optimal solution probability at shorter depths, extends these benefits to the Maximum Independent Set (MIS) problem, scales via extrapolated spectral gaps, and maintains advantages under mild depolarising noise.

Significance. If the numerical claims hold with the stated construction details, the work would supply a concrete method for de-variationalizing QAOA parameter selection using only adiabatic spectral-gap data, potentially reducing optimization overhead on combinatorial problems while preserving performance under noise; the absence of any equations, fitting procedures, or quantitative results in the provided text, however, leaves both the magnitude of improvement and the parameter-free character of the schedules unverified.

major comments (2)

[Abstract] Abstract: the claims of performance improvements over LR-QAOA on Grover's problem at constant depth, shorter depths for equivalent probability, and extension to MIS are asserted without any numerical values, error bars, depth comparisons, or description of how the spectral gap is computed or extrapolated, rendering the central empirical claims impossible to assess from the manuscript as supplied.
[Abstract] Abstract: the schedule construction is described only qualitatively ('performs slow evolution where the spectral gap of the adiabatic Hamiltonian is small') with no explicit mapping from gap values to (β,γ) schedules, no equations, and no statement of whether new fitting parameters are introduced; this directly bears on the weakest assumption that gap information alone suffices without full variational optimization or hidden parameters.

Simulated Author's Rebuttal

2 responses · 1 unresolved

We thank the referee for the detailed comments. We address each major comment below. Only the abstract was excerpted for this review, which limits our ability to reference specific equations or numerical data from the body of the manuscript.

read point-by-point responses

Referee: [Abstract] Abstract: the claims of performance improvements over LR-QAOA on Grover's problem at constant depth, shorter depths for equivalent probability, and extension to MIS are asserted without any numerical values, error bars, depth comparisons, or description of how the spectral gap is computed or extrapolated, rendering the central empirical claims impossible to assess from the manuscript as supplied.

Authors: We agree that the abstract, being a concise summary, does not contain specific numerical values, error bars or depth comparisons. The full manuscript presents these quantitative results from numerical simulations on Grover's problem and MIS, including success probabilities and comparisons to LR-QAOA, as well as the method for computing and extrapolating the spectral gap. We will revise the abstract to include one or two key quantitative statements highlighting the observed improvements. revision: partial
Referee: [Abstract] Abstract: the schedule construction is described only qualitatively ('performs slow evolution where the spectral gap of the adiabatic Hamiltonian is small') with no explicit mapping from gap values to (β,γ) schedules, no equations, and no statement of whether new fitting parameters are introduced; this directly bears on the weakest assumption that gap information alone suffices without full variational optimization or hidden parameters.

Authors: The abstract uses qualitative language for brevity. The manuscript provides the explicit construction that maps gap values to the QAOA schedules without introducing new fitting parameters, relying solely on the spectral gap of the adiabatic Hamiltonian (with the QAOA mixer as initial Hamiltonian). We will revise the abstract to state that the schedules are constructed directly from the gap information without additional variational parameters. revision: yes

standing simulated objections not resolved

The full manuscript text (beyond the abstract) is not provided, preventing inclusion of specific numerical results, equations, or section references in this response.

Circularity Check

0 steps flagged

No circularity detectable; abstract contains no equations or derivation chain

full rationale

The abstract describes SGIR-QAOA as incorporating spectral gap information to construct schedules and claims performance improvements, but provides no equations, parameter mappings, fitting procedures, or self-citations. No load-bearing step can be quoted that reduces a claimed result to its inputs by construction, as required for any circularity finding. The paper's claims rest on unspecified numerical evidence, but absence of detail precludes identifying self-definitional, fitted-prediction, or self-citation circularity.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract-only review supplies no explicit free parameters, axioms, or invented entities; the central claim rests on the unstated assumption that an adiabatic Hamiltonian's spectral gap can be evaluated or extrapolated at the relevant system sizes.

pith-pipeline@v0.9.1-grok · 5733 in / 1237 out tokens · 27366 ms · 2026-07-01T08:53:04.401114+00:00 · methodology

0 comments

read the original abstract

A challenge with the Quantum Approximate Optimisation Algorithm (QAOA), and variational algorithms in general, is finding good variational parameters, a task which in itself can be NP-hard. Recent work has sought to de-variationalise QAOA by picking well-informed guesses for the variational parameters. The Linear Ramp QAOA (LR-QAOA) achieves this by using parameter schedules inspired by the quantum adiabatic algorithm. In this work, we propose Spectral Gap Informed Ramp QAOA (SGIR-QAOA), a new QAOA variant that incorporates spectral gap information from an adiabatic Hamiltonian, with the QAOA mixer Hamiltonian as the initial Hamiltonian, to construct smooth parameter schedules. SGIR-QAOA performs slow evolution where the spectral gap of the adiabatic Hamiltonian is small. We show that SGIR-QAOA has performance improvements over the LR-QAOA on Grover's problem at constant depth and that SGIR-QAOA requires shorter depths to achieve the same optimal solution probability. We then show that these performance benefits extend to a problem with potential practical applications - the Maximum Independent Set (MIS) problem. Finally, we demonstrate the scalability of the SGIR-QAOA method using extrapolated spectral gap information for scales that the spectral gap cannot be exactly evaluated, and show that the advantage appears to persist under mild depolarising noise.

Figures

Figures reproduced from arXiv: 2604.24580 by Kieran McDowall, Konstantinos Georgopoulos, Petros Wallden.

**Figure 1.** Figure 1: (Top) Eigenvalue spectrum from HAd for Grover’s problem with the marked solution state |x⟩ = |000000⟩ (n = 6). (Bottom) The optimal Roland Cerf (RC) adiabatic schedule. III. SPECTRAL GAP INFORMED RAMP - QAOA Here we introduce the Spectral Gap Informed Ramp – QAOA (SGIR–QAOA) method. Given that QAOA is a Trot- view at source ↗

**Figure 2.** Figure 2: (Top) Plotting Equation 7 for Grover’s problem k = 0 n = 4, with the marked solution state |x⟩ = |0000⟩ discritised by p = 10. (Bottom) An example of the corrsponding SGIR schedule. The eigenvalue spectrum in view at source ↗

**Figure 4.** Figure 4: Solving Grover’s problem where the QAOA depth view at source ↗

**Figure 3.** Figure 3: Solving Grover’s problem with our QAOA methods at constant QAOA depth p = 10. At each n the experiment is repeated 10 times with a different random marked solution. The random QAOA parameters are changed for each of these instances. The y-axis is logarithmic. We further explore the performance of SGIR–QAOA in view at source ↗

**Figure 5.** Figure 5: Solving MIS degree 3 graphs with LR–QAOA and exact SGIR–QAOA. At each n the experiment is repeated 10 times with a different randomly generated instance. A log scale is used on the y-axis. The error in the fit for the exponential scaling coefficient is included in the legend view at source ↗

**Figure 6.** Figure 6: Solving the MIS problem with degree 3 graphs where the QAOA depth p required to reach an optimal solution threshold P th s at different problem sizes is plotted. 10 15 20 n (problem size) 10−6 10−5 10−4 10−3 10−2 10−1 100 Ps p = 10 Random guess: 2−n LR-QAOA: 2−(0.56±0.02)n Extrapolated SGIR-QAOA: 2−(0.46±0.02)n view at source ↗

**Figure 8.** Figure 8: , 10 random n = 10 MIS instances are solved at varying QAOA depth p under depolarising noise of strength pnoise = 0.001 and with noiseless statevector simulation as a comparison, with the penalty term now set at λ = 100 and the parameter grid set to 20 × 20. Under noise, the peak in the optimal solution probability for SGIR–QAOA is higher and occurs at a lower QAOA depth p than that of noiseless LR–QAOA. In view at source ↗

**Figure 10.** Figure 10: Solving MIS degree 3 graphs with extrapolated SGIR–QAOA at larger n than in view at source ↗

**Figure 9.** Figure 9: The percentage improvement of SGIR-QAOA against LRQAOA, p = 10, with the minimum spectral gap of the problem (Pearson correlation coefficient ρ = −0.68). B. Maximum Independent Set - Extras In view at source ↗

**Figure 11.** Figure 11: An extension of view at source ↗

**Figure 12.** Figure 12: Solving the MIS problem with ER graphs, probability of edges = 0.4. At each n the experiment is repeated 10 times with a different randomly generated instance. A log scale is used on the y-axis. The error in the fit for the exponential scalaing coefficient is included in the legend. 5 10 15 20 n (problem size) 10−6 10−5 10−4 10−3 10−2 10−1 Ps p = 10 Random guess: 2−n LR-QAOA: 2−(0.35±0.02)n Extrapolated S… view at source ↗

**Figure 13.** Figure 13: Solving the MIS problem with ER graphs, probability of edges = 0.4. Using extrapolated gaps to calculate the SGIR schedule. Using n = 5 − 12 to extrapolate gmin which always occurs at s = 1. At s = 0 we take the gap E2 − E0 = g2 = 4, as we know this analytically. For s between 0 and 1 we take the average gap values from n = 5−12. At each n the experiment is repeated 10 times with a different randomly gene… view at source ↗

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Forward citations

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