arxiv: 2604.24580 · v1 · submitted 2026-04-27 · 🪐 quant-ph

Recognition: unknown

A Spectral Gap Informed Parameter Schedule for QAOA

Kieran McDowall, Konstantinos Georgopoulos, Petros Wallden

Authors on Pith no claims yet

Pith reviewed 2026-05-08 04:04 UTC · model grok-4.3

classification 🪐 quant-ph

keywords QAOAspectral gapparameter scheduleGrover's problemmaximum independent setadiabatic evolutionquantum optimizationvariational algorithms

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The pith

Spectral gap informed ramps improve QAOA performance over linear schedules on Grover's problem and maximum independent set.

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

The paper introduces a method to choose QAOA parameters without variational optimization by using spectral gap data from an associated adiabatic Hamiltonian. These Spectral Gap Informed Ramps slow the evolution in regions where the gap between energy levels is smallest. On Grover's search problem, this yields better solution probabilities than linear ramps at fixed circuit depth and reaches high probabilities with shallower circuits. The approach also works for the maximum independent set problem and continues to show advantages when spectral gaps are extrapolated for larger instances and when mild noise is present. A reader should care because parameter selection is a major bottleneck in applying QAOA to real problems.

Core claim

We propose Spectral Gap Informed Ramps for QAOA (SGIR-QAOA) that use the spectral gap of the adiabatic Hamiltonian, with the QAOA mixer as initial Hamiltonian, to create smooth parameter schedules that evolve slowly where the gap is small. We demonstrate that SGIR-QAOA outperforms Linear Ramp QAOA on Grover's problem at constant depth and requires shorter depths for equivalent performance. These benefits extend to the Maximum Independent Set problem. Using extrapolated spectral gaps allows scaling to larger instances, and the performance advantage persists under mild depolarising noise.

What carries the argument

Spectral Gap Informed Ramps (SGIR-QAOA), smooth parameter schedules derived from the spectral gaps of the adiabatic Hamiltonian that adjust evolution speed to avoid regions of small gaps.

If this is right

SGIR-QAOA achieves performance improvements over LR-QAOA on Grover's at constant depth.
SGIR-QAOA requires shorter depths to reach the same optimal solution probability.
Performance benefits extend to the Maximum Independent Set problem.
The method remains effective with extrapolated spectral gap information for larger scales.
The advantage persists 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 extrapolation technique could enable SGIR-QAOA schedules for problem sizes beyond exact diagonalization.
Similar gap-based scheduling may connect QAOA to other adiabatic-inspired variational methods on combinatorial tasks.
The persistence of gains under mild noise points toward potential use on near-term devices with realistic error rates.

Load-bearing premise

Spectral gap information from the adiabatic Hamiltonian produces effective ramps when extrapolated to problem sizes where exact gaps cannot be computed, and observed improvements are not limited to the tested instances or noise models.

What would settle it

Finding that on a larger Grover instance or different MIS graph, the extrapolated SGIR-QAOA yields lower solution probability than LR-QAOA or fails to maintain advantage under the depolarizing noise model.

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 ↗

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. We go a step further and use spectral gap information from an adiabatic Hamiltonian, with the QAOA mixer Hamiltonian as our initial Hamiltonian, to make smooth ramps which we call Spectral Gap Informed Ramps (SGIR-QAOA). SGIR-QAOA schedules perform slow evolution where the spectral gap of the adiabatic Hamiltonian is small. We show that SGIR-QAOA has performance improvements over 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.

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.

Desk Editor's Note private letter to a colleague

SGIR-QAOA beats linear-ramp QAOA on small Grover and MIS instances by slowing evolution where the gap is narrow, but the scalability claim rests on unvalidated gap extrapolation.

read the letter

SGIR-QAOA improves QAOA parameter choice by building ramps that slow evolution exactly where the spectral gap of the adiabatic path is smallest. On Grover's problem it reaches higher success probability than linear-ramp QAOA at the same depth and matches the target probability at lower depth. The same pattern holds for maximum independent set, and the authors say the edge survives mild depolarizing noise when gaps are extrapolated from small instances. The new part is the direct use of gap information to shape the schedule rather than a fixed linear form. Earlier work used adiabatic inspiration but kept the ramp shape simple; here the schedule is informed by the actual gap landscape of the chosen Hamiltonian. That is a distinct move and they demonstrate it on two standard test cases. The results look solid for the sizes where exact gaps can be computed. The paper reports concrete gains and includes a noise test, which adds some practical flavor. The construction itself is reproducible in principle since it starts from the mixer as initial Hamiltonian and uses the gap to modulate the speed. The main weakness is the extrapolation step. They fit some model to small-n gaps and apply it to larger n where exact computation is impossible. No intermediate-size validation is described, so it is not clear whether the locations of the slow regions stay accurate enough. A small error near the minimal gap could push the schedule into a region where the adiabatic condition fails at the chosen depth. The abstract is careful with 'appears to persist,' but the scalability argument rests on that untested assumption. This work is aimed at researchers trying to make QAOA more deterministic and less dependent on classical optimization loops. Anyone building schedules for combinatorial problems on small to medium qubit counts could find the gap-informed construction useful. It is worth sending to peer review because the idea is fresh, the benchmarks are appropriate, and the extrapolation issue is fixable with additional checks rather than fatal to the whole approach.

Referee Report

2 major / 2 minor

Summary. The manuscript introduces Spectral Gap Informed Ramps QAOA (SGIR-QAOA), which constructs QAOA parameter schedules using spectral gap information from an adiabatic Hamiltonian (with the QAOA mixer as initial Hamiltonian) to slow evolution where the gap is small. It claims performance improvements over Linear Ramp QAOA (LR-QAOA) on Grover's problem at constant depth, shorter depths to reach the same optimal solution probability, extension of these benefits to the Maximum Independent Set (MIS) problem, and persistence of the advantages when using extrapolated spectral gaps for larger scales together with resilience under mild depolarizing noise.

Significance. If the central claims hold after validation, the work provides a concrete, non-variational method for selecting QAOA schedules that exploits spectral properties of the adiabatic path, yielding measurable gains on both a canonical problem (Grover) and a practically relevant one (MIS). The explicit use of gap information to shape the ramp, combined with the noise test and the attempt at extrapolation, strengthens the case for de-variationalizing QAOA and could guide future parameter-selection strategies.

major comments (2)

[scalability demonstration using extrapolated spectral gaps] The scalability section claims that advantages persist when spectral gaps are extrapolated to sizes where exact diagonalization is impossible, yet no cross-validation is reported that compares exact-gap and extrapolated-gap schedules on the same intermediate-size instances. Because the method deliberately slows evolution near the minimal gap, even modest extrapolation error can shift the slow-evolution region and violate the adiabatic condition at the chosen depth; this directly undermines the central scalability claim.
[Grover's problem results] In the Grover's problem results, performance improvements and depth reductions relative to LR-QAOA are asserted without accompanying quantitative metrics, error bars, or an explicit equation showing how the gap values are mapped onto the ramp parameters. The absence of these details makes it impossible to verify that the reported gains are not artifacts of the specific small instances or the particular gap-to-schedule mapping chosen.

minor comments (2)

[methods] The construction of the adiabatic Hamiltonian (mixer as initial Hamiltonian) is described only at a high level; an explicit Hamiltonian expression or pseudocode would clarify how the gap is computed and used.
[figures] Figure captions and axis labels for the probability-vs-depth plots should explicitly state the number of instances, the extrapolation ansatz, and whether error bars represent standard deviation or standard error.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their careful and insightful review of our manuscript. We have carefully considered each major comment and provide point-by-point responses below, along with our plans for revisions.

read point-by-point responses

Referee: [scalability demonstration using extrapolated spectral gaps] The scalability section claims that advantages persist when spectral gaps are extrapolated to sizes where exact diagonalization is impossible, yet no cross-validation is reported that compares exact-gap and extrapolated-gap schedules on the same intermediate-size instances. Because the method deliberately slows evolution near the minimal gap, even modest extrapolation error can shift the slow-evolution region and violate the adiabatic condition at the chosen depth; this directly undermines the central scalability claim.

Authors: We recognize the importance of validating the extrapolation procedure. To address this, we will perform and report cross-validation on intermediate system sizes (where exact diagonalization remains feasible) by comparing the performance of schedules based on exact gaps versus extrapolated gaps. This will confirm that any extrapolation errors do not compromise the adiabaticity at the depths used, thereby supporting the scalability demonstration. The revised manuscript will include these results. revision: yes
Referee: [Grover's problem results] In the Grover's problem results, performance improvements and depth reductions relative to LR-QAOA are asserted without accompanying quantitative metrics, error bars, or an explicit equation showing how the gap values are mapped onto the ramp parameters. The absence of these details makes it impossible to verify that the reported gains are not artifacts of the specific small instances or the particular gap-to-schedule mapping chosen.

Authors: We agree that additional details are necessary for full transparency and reproducibility. In the revision, we will provide the explicit mapping (equation or algorithm) used to translate spectral gap information into the QAOA parameter schedule. We will also include quantitative metrics for the performance improvements, such as success probabilities at various depths, accompanied by error bars derived from multiple independent runs or statistical analysis. These additions will allow independent verification of the reported advantages on Grover's problem. revision: yes

Circularity Check

0 steps flagged

No circularity: parameter schedules constructed from independently computed spectral gaps

full rationale

The paper's core derivation begins with the adiabatic Hamiltonian (mixer as initial, problem as final) and extracts its spectral gaps via direct diagonalization or computation. These gaps are used to construct the SGIR-QAOA ramps by slowing evolution where the gap is minimal. Performance is then evaluated separately on Grover and MIS instances. No equation reduces a claimed QAOA improvement to a quantity defined by the authors' own prior results, a fit to QAOA success probabilities, or a self-citation chain. Extrapolation of gaps for larger instances is presented as a practical extension rather than a load-bearing premise that collapses the central claims; benefits are shown where exact gaps are available, keeping the method externally falsifiable.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The approach assumes the adiabatic Hamiltonian with the QAOA mixer as initial Hamiltonian is a suitable reference for gap computation, and that gap-based modulation produces better performance than linear interpolation. No explicit free parameters are named in the abstract, but the extrapolation procedure for large instances implicitly introduces at least one modeling choice for how gaps are estimated.

axioms (1)

domain assumption The adiabatic Hamiltonian formed by interpolating the QAOA mixer and problem Hamiltonian has spectral gaps that can be used to guide a non-adiabatic but still effective QAOA schedule.
Invoked when the authors state they use spectral gap information from an adiabatic Hamiltonian with the QAOA mixer as initial Hamiltonian.

pith-pipeline@v0.9.0 · 5527 in / 1423 out tokens · 63026 ms · 2026-05-08T04:04:35.536618+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

21 extracted references · 8 canonical work pages · 2 internal anchors

[1]

Toward a linear-ramp qaoa protocol: evidence of a scaling advantage in solving some combinatorial optimization problems.npj Quantum Information, 11(1):131, 2025

JA Montanez-Barrera and Kristel Michielsen. Toward a linear-ramp qaoa protocol: evidence of a scaling advantage in solving some combinatorial optimization problems.npj Quantum Information, 11(1):131, 2025

2025
[2]

Training variational quantum algo- rithms is np-hard.Physical review letters, 127(12):120502, 2021

Lennart Bittel and Martin Kliesch. Training variational quantum algo- rithms is np-hard.Physical review letters, 127(12):120502, 2021

2021
[3]

Quantum variational algorithms are swamped with traps.Nature Communications, 13(1):7760, 2022

Eric R Anschuetz and Bobak T Kiani. Quantum variational algorithms are swamped with traps.Nature Communications, 13(1):7760, 2022

2022
[4]

Barren plateaus in quantum neural network training landscapes.Nature communications, 9(1):4812, 2018

Jarrod R McClean, Sergio Boixo, Vadim N Smelyanskiy, Ryan Babbush, and Hartmut Neven. Barren plateaus in quantum neural network training landscapes.Nature communications, 9(1):4812, 2018

2018
[5]

A Quantum Approximate Optimization Algorithm

J. Goldstone E. Farhi and S. Gutmann. A quantum approxi- mate optimization algorithm.arXiv preprint arXiv:1411.4028, 2014. arXiv:1411.4028

work page internal anchor Pith review arXiv 2014
[6]

Solving boolean satisfiability problems with the quantum approximate optimization algorithm.PRX Quantum, 5(3):030348, 2024

Sami Boulebnane and Ashley Montanaro. Solving boolean satisfiability problems with the quantum approximate optimization algorithm.PRX Quantum, 5(3):030348, 2024

2024
[7]

Quantum speedups in solving near- symmetric optimization problems by low-depth qaoa.arXiv preprint arXiv:2411.04979, 2024

Ashley Montanaro and Leo Zhou. Quantum speedups in solving near- symmetric optimization problems by low-depth qaoa.arXiv preprint arXiv:2411.04979, 2024

work page arXiv 2024
[8]

Lower bounding the maxcut of high girth 3-regular graphs using the qaoa,

Edward Farhi, Sam Gutmann, Daniel Ranard, and Benjamin Villalonga. Lower bounding the maxcut of high girth 3-regular graphs using the qaoa.arXiv preprint arXiv:2503.12789, 2025

work page arXiv 2025
[9]

Evidence of scaling advantage for the quantum approximate optimization algorithm on a classically intractable problem.Science Advances, 10(22):eadm6761, 2024

Ruslan Shaydulin, Changhao Li, Shouvanik Chakrabarti, Matthew De- Cross, Dylan Herman, Niraj Kumar, Jeffrey Larson, Danylo Lykov, Pierre Minssen, Yue Sun, et al. Evidence of scaling advantage for the quantum approximate optimization algorithm on a classically intractable problem.Science Advances, 10(22):eadm6761, 2024

2024
[10]

Akshay et al

V . Akshay et al. Parameter concentrations in quantum approximate optimization.Physical Review A, 104(1):L010401, 2021. DOI: 10.1103/PhysRevA.104.L010401

work page doi:10.1103/physreva.104.l010401 2021
[11]

A practically scalable approach to the closest vector problem for sieving via qaoa with fixed angles.Quantum Science and Technology, 2026

Ben Priestley and Petros Wallden. A practically scalable approach to the closest vector problem for sieving via qaoa with fixed angles.Quantum Science and Technology, 2026

2026
[12]

Quantum search by local adiabatic evolution.Physical Review A, 65(4):042308, 2002

J ´er´emie Roland and Nicolas J Cerf. Quantum search by local adiabatic evolution.Physical Review A, 65(4):042308, 2002

2002
[13]

Bias-field digitized counterdiabatic quantum optimization.Physical Review Research, 7(2):L022010, 2025

Alejandro Gomez Cadavid, Archismita Dalal, Anton Simen, Enrique Solano, and Narendra N Hegade. Bias-field digitized counterdiabatic quantum optimization.Physical Review Research, 7(2):L022010, 2025

2025
[14]

Adiabatic quantum unstructured search in parallel.arXiv preprint arXiv:2502.08594, 2025

Sean A Adamson and Petros Wallden. Adiabatic quantum unstructured search in parallel.arXiv preprint arXiv:2502.08594, 2025

work page arXiv 2025
[15]

Quantum optimization benchmarking library-the intractable decathlon,

Thorsten Koch, David E Bernal Neira, Ying Chen, Giorgio Cortiana, Daniel J Egger, Raoul Heese, Narendra N Hegade, Alejandro Gomez Cadavid, Rhea Huang, Toshinari Itoko, et al. Quantum optimiza- tion benchmark library–the intractable decathlon.arXiv preprint arXiv:2504.03832, 2025

work page arXiv 2025
[16]

Quantum computing with Qiskit

Ali Javadi-Abhari, Matthew Treinish, Kevin Krsulich, Christopher J Wood, Jake Lishman, Julien Gacon, Simon Martiel, Paul D Nation, Lev S Bishop, Andrew W Cross, et al. Quantum computing with qiskit. arXiv preprint arXiv:2405.08810, 2024

work page internal anchor Pith review arXiv 2024
[17]

Oliphant, Matt Haberland, Tyler Reddy, David Cournapeau, Evgeni Burovski, Pearu Peterson, Warren Weckesser, Jonathan Bright, St ´efan J

Pauli Virtanen, Ralf Gommers, Travis E. Oliphant, Matt Haberland, Tyler Reddy, David Cournapeau, Evgeni Burovski, Pearu Peterson, Warren Weckesser, Jonathan Bright, St ´efan J. van der Walt, Matthew Brett, Joshua Wilson, K. Jarrod Millman, Nikolay Mayorov, Andrew R. J. Nelson, Eric Jones, Robert Kern, Eric Larson, C J Carey, ˙Ilhan Polat, Yu Feng, Eric W....

2020
[18]

Quantum search with hybrid adiabatic–quantum-walk algorithms and realistic noise.Physical review A, 99(2):022339, 2019

James G Morley, Nicholas Chancellor, Sougato Bose, and Viv Kendon. Quantum search with hybrid adiabatic–quantum-walk algorithms and realistic noise.Physical review A, 99(2):022339, 2019

2019
[19]

Extrapolation method to optimize linear-ramp quantum approximate optimization algorithm parameters: Evaluation of runtime scaling.Physical Review A, 113(3):032413, 2026

Vanessa Dehn, Martin Zaefferer, Gerhard Hellstern, Karthik Jayadevan, Florentin Reiter, and Thomas Wellens. Extrapolation method to optimize linear-ramp quantum approximate optimization algorithm parameters: Evaluation of runtime scaling.Physical Review A, 113(3):032413, 2026

2026
[20]

A practical cross-platform, multi-algorithm study of quantum optimisation for configurational anal- ysis of materials.arXiv preprint arXiv:2504.06885, 2025

Kieran McDowall, Theodoros Kapourniotis, Christopher Oliver, Phalgun Lolur, and Konstantinos Georgopoulos. A practical cross-platform, multi-algorithm study of quantum optimisation for configurational anal- ysis of materials.arXiv preprint arXiv:2504.06885, 2025. APPENDIX A. SGIR-QAOA Performance withg min In Figure 9 the correlation between the performan...

work page arXiv 2025
[21]

In Figure 12 results are shown using SGIR exact and in Figure 13 the SGIR extrapolation technique is used

Dense Erd ˝os–R´enyi (ER) Random Graphs: We include here results onErd ˝os–R´enyi (ER) random graphswith edge probabilityp= 0.4. In Figure 12 results are shown using SGIR exact and in Figure 13 the SGIR extrapolation technique is used. For the SGIR extrapolated, we useg min fromn= 6−12to make an exponential fit to extrapolateg min,s=1 for largern. 5 10 15...