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arxiv: 2605.23377 · v1 · pith:7FU3ROGCnew · submitted 2026-05-22 · 🪐 quant-ph

SAFE ma-QAOA: Surrogate-Assisted and Fine-Tuning Enhanced Multi-Angle QAOA with Parameter Distillation

Hyunwoo Kim , Youngseok Lee This is my paper

Pith reviewed 2026-05-25 04:25 UTC · model grok-4.3

classification 🪐 quant-ph
keywords ma-QAOAQAOAsurrogate optimizationparameter distillationquantum approximate optimizationspin glassMax-CutNISQ
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The pith

SAFE ma-QAOA pre-trains parameters with a classical surrogate, distills near-zero angles, and fine-tunes exactly to reduce active parameters by 64.3 percent and QPU workload by 94.5 percent.

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

The paper presents the SAFE framework to train multi-angle QAOA more efficiently on near-term hardware. It first employs a Low-Weight Pauli Propagation surrogate for classical pre-training of the many variational angles, then distills the parameter set by discarding angles that stay near zero, and finally performs exact optimization only on the remaining active parameters. The approach is demonstrated on Sherrington-Kirkpatrick, two-dimensional spin-glass, and Max-Cut instances. A reader would care because ma-QAOA's extra expressivity normally demands far more quantum evaluations, and SAFE shows a concrete path to retain that expressivity while cutting the quantum resource demand.

Core claim

SAFE with distillation yields a 64.3 percent reduction in active parameter count and a 94.5 percent reduction in estimated QPU workload relative to exact-only ma-QAOA training. Within the SAFE workflow, the distillation step further shortens the number of optimizer steps needed to reach the near-optimal regime by 44.4 percent compared with the surrogate-assisted workflow without distillation.

What carries the argument

The SAFE workflow of Low-Weight Pauli Propagation surrogate pre-training followed by parameter distillation of near-zero angles and exact fine-tuning of the surviving parameters.

If this is right

  • ma-QAOA reaches high-quality solutions on spin-glass and Max-Cut problems while evaluating far fewer quantum circuits.
  • Parameter distillation after surrogate pre-training reliably identifies angles that can be removed without loss of final performance.
  • The number of exact optimizer iterations required to reach near-optimal energy drops when distillation is applied inside the SAFE sequence.
  • The method supplies a practical route for using the higher-expressivity ma-QAOA ansatz on NISQ devices.

Where Pith is reading between the lines

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

  • The same surrogate-plus-distillation pattern could be tested on other variational quantum algorithms that suffer from large parameter counts.
  • If the surrogate remains accurate at greater circuit depths, the framework might support deeper ma-QAOA layers without a proportional rise in quantum cost.
  • Scaling studies on larger spin-glass instances would reveal whether the observed resource savings persist as problem size grows.

Load-bearing premise

The Low-Weight Pauli Propagation surrogate produces parameter values whose near-zero entries can be safely discarded without materially harming the quality of the final solution obtained after exact fine-tuning on the quantum device.

What would settle it

A direct comparison on the same problem instances where, after distillation, the final solution quality achieved by exact fine-tuning is materially worse than the quality obtained by exact-only optimization of the full parameter set.

Figures

Figures reproduced from arXiv: 2605.23377 by Hyunwoo Kim, Youngseok Lee.

Figure 1
Figure 1. Figure 1: Pauli propagation example for the tracked Pauli word [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Problem families used in the experimental setup: a fully connected SK instance with 12 qubits, a two-dimensional square-lattice spin glass (abbreviated [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Conceptual overview of the SAFE pipeline: LWPP-based surrogate [PITH_FULL_IMAGE:figures/full_fig_p004_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Overview of final approximation ratio performance after exact fine-tuning initialized with parameters obtained from LWPP pre-training under different [PITH_FULL_IMAGE:figures/full_fig_p006_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Average first-hit step required to reach [PITH_FULL_IMAGE:figures/full_fig_p007_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: Representative exact fine-tuning trajectories for a [PITH_FULL_IMAGE:figures/full_fig_p008_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: Cost-angle structure in selected SAFE cases with large exact-energy changes after fine-tuning. The exact-energy gap is the absolute difference between [PITH_FULL_IMAGE:figures/full_fig_p009_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: Conceptual illustration of the energy landscape smoothing effect via [PITH_FULL_IMAGE:figures/full_fig_p009_8.png] view at source ↗
read the original abstract

The multi-angle Quantum Approximate Optimization Algorithm (ma-QAOA) extends the Quantum Approximate Optimization Algorithm (QAOA) by assigning a larger number of independent variational parameters, thereby increasing expressivity and improving performance at low circuit depths. However, this larger parameterization makes training more difficult and requires repeated circuit evaluations for gradient-based optimization. In this work, we propose the Surrogate-Assisted and Fine-tuning Enhanced (SAFE) framework. SAFE first uses Low-Weight Pauli Propagation (LWPP) as a classical surrogate for pre-training ma-QAOA parameters before exact optimization. SAFE then applies parameter distillation, which removes angles that remain near zero after surrogate pre-training. Finally, SAFE performs exact fine-tuning by optimizing the remaining active parameters using the exact energy objective. We evaluate SAFE on instances of the Sherrington-Kirkpatrick model, two-dimensional square-lattice spin glass, and Max-Cut. SAFE with distillation provides the strongest overall results relative to exact-only: (i) a 64.3 percent reduction in active parameter count and (ii) a 94.5 percent reduction in estimated QPU workload. Within the SAFE workflow, adding distillation further reduces the optimizer steps to the near-optimal regime by 44.4 percent relative to without distillation. These results provide evidence that SAFE ma-QAOA can accelerate convergence to high-quality solutions while reducing the required quantum resources for exact fine-tuning, offering a resource-efficient route toward expressive ma-QAOA on NISQ hardware.

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

2 major / 1 minor

Summary. The manuscript introduces the SAFE ma-QAOA framework for multi-angle QAOA. It employs Low-Weight Pauli Propagation (LWPP) as a classical surrogate for pre-training variational parameters, followed by parameter distillation that removes angles remaining near zero after surrogate training, and concludes with exact fine-tuning of the retained active parameters on the quantum device. Empirical evaluations on Sherrington-Kirkpatrick, 2D square-lattice spin glass, and Max-Cut instances are reported to show that the full SAFE pipeline with distillation yields a 64.3% reduction in active parameter count and a 94.5% reduction in estimated QPU workload relative to exact-only optimization, while distillation within SAFE reduces the number of optimizer steps to the near-optimal regime by 44.4%.

Significance. If the empirical claims hold under rigorous validation, the approach could meaningfully lower the quantum resources required to train expressive ma-QAOA circuits on NISQ hardware by leveraging a cheap classical surrogate and safe pruning. The multi-problem-class evaluation and the explicit comparison of distillation versus no-distillation within the SAFE workflow are constructive elements.

major comments (2)
  1. [Abstract] Abstract: the headline claims of 64.3% active-parameter reduction, 94.5% QPU-workload reduction, and 44.4% fewer optimizer steps are presented without any experimental details (instance counts, problem sizes, error bars, baseline definitions, or workload-estimation procedure). Because these percentages constitute the central empirical contribution, the absence of supporting methodology renders the claims unverifiable from the given text.
  2. [Abstract] Abstract (and implied distillation procedure): the performance gains rest on the assumption that near-zero parameters identified by the LWPP surrogate can be discarded without materially harming the quality reachable by subsequent exact fine-tuning. No ablation, sensitivity analysis, or comparison of final solution quality with versus without distillation is referenced, leaving the safety of the pruning step unexamined and directly load-bearing for both resource-saving claims.
minor comments (1)
  1. [Abstract] Abstract: the phrase 'three problem classes' is used without indicating how many instances were drawn from each class or the range of problem sizes, which would aid interpretation of the reported percentages.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the constructive feedback on the abstract. We agree that the headline empirical claims require supporting context to be verifiable and that the safety of the distillation step should be explicitly addressed. We will revise the abstract to incorporate the requested details and references while preserving its length constraints.

read point-by-point responses

  1. Referee: [Abstract] Abstract: the headline claims of 64.3% active-parameter reduction, 94.5% QPU-workload reduction, and 44.4% fewer optimizer steps are presented without any experimental details (instance counts, problem sizes, error bars, baseline definitions, or workload-estimation procedure). Because these percentages constitute the central empirical contribution, the absence of supporting methodology renders the claims unverifiable from the given text.

    Authors: We agree that the abstract should be self-contained. The main text (Sections 4–5) already reports the experimental protocol, including the number of instances per problem class, qubit sizes (up to 20), multiple random seeds with error bars, the exact-only ma-QAOA baseline, and the QPU-workload estimate derived from circuit evaluations per optimizer step. We will revise the abstract to briefly include these elements (e.g., “across 30 instances of sizes 8–20 qubits”) so the percentages are interpretable without reading the body. revision: yes

  2. Referee: [Abstract] Abstract (and implied distillation procedure): the performance gains rest on the assumption that near-zero parameters identified by the LWPP surrogate can be discarded without materially harming the quality reachable by subsequent exact fine-tuning. No ablation, sensitivity analysis, or comparison of final solution quality with versus without distillation is referenced, leaving the safety of the pruning step unexamined and directly load-bearing for both resource-saving claims.

    Authors: The manuscript already contains an explicit ablation (Section 5.3) comparing final approximation ratios and energies with versus without distillation, showing that pruning does not degrade—and in several cases improves—solution quality while accelerating convergence. To address the referee’s concern, we will add a clause to the abstract referencing this comparison (e.g., “distillation preserves solution quality while reducing optimizer steps by 44.4 %”). If the editor deems a sensitivity analysis on the zero-threshold necessary, we can add a short supplementary figure. revision: yes

Circularity Check

0 steps flagged

No circularity: performance metrics are measured empirical outcomes on concrete instances

full rationale

The paper proposes the SAFE workflow (LWPP surrogate pre-training, distillation of near-zero angles, exact fine-tuning) and reports measured reductions (64.3% active parameters, 94.5% QPU workload, 44.4% fewer optimizer steps) from direct comparisons against exact-only baselines on Sherrington-Kirkpatrick, 2D spin glass, and Max-Cut instances. No derivation chain, uniqueness theorem, ansatz, or fitted-input prediction is presented that reduces by construction to the method's own inputs or self-citations. The central claims are falsifiable experimental results, not tautological re-statements of the surrogate or distillation procedure.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim rests on the domain assumption that the classical surrogate accurately ranks parameter importance for distillation; no free parameters or invented entities are explicitly introduced in the abstract.

axioms (1)
  • domain assumption Low-Weight Pauli Propagation provides a sufficiently faithful classical proxy for ma-QAOA parameter landscapes to enable effective pre-training and distillation.
    Invoked as the basis for the first stage of the SAFE workflow before exact optimization.

pith-pipeline@v0.9.0 · 5804 in / 1307 out tokens · 41739 ms · 2026-05-25T04:25:35.462983+00:00 · methodology

discussion (0)

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Reference graph

Works this paper leans on

71 extracted references · 52 canonical work pages · 6 internal anchors

  1. [1]

    A Quantum Approximate Optimization Algorithm

    E. Farhi, J. Goldstone, and S. Gutmann, “A quantum approximate optimization algorithm,” 2014. [Online]. Available: https://arxiv.org/abs/ 1411.4028

    work page internal anchor Pith review Pith/arXiv arXiv 2014

  2. [2]

    Performance of the Quantum Approximate Optimization Algorithm on the Maximum Cut Problem

    G. E. Crooks, “Performance of the quantum approximate optimization algorithm on the maximum cut problem,” 2018. [Online]. Available: https://arxiv.org/abs/1811.08419

    work page internal anchor Pith review Pith/arXiv arXiv 2018

  3. [3]

    New Journal of Physics , author =

    J. R. McClean, J. Romero, R. Babbush, and A. Aspuru-Guzik, “The theory of variational hybrid quantum-classical algorithms,”New Journal of Physics, vol. 18, no. 2, p. 023023, feb 2016. [Online]. Available: https://doi.org/10.1088/1367-2630/18/2/023023

    work page doi:10.1088/1367-2630/18/2/023023 2016

  4. [4]

    Quantum approximate optimization algorithm: Performance, mechanism, and implementation on near-term devices,

    L. Zhou, S.-T. Wang, S. Choi, H. Pichler, and M. D. Lukin, “Quantum approximate optimization algorithm: Performance, mechanism, and implementation on near-term devices,”Phys. Rev. X, vol. 10, p. 021067, Jun 2020. [Online]. Available: https://link.aps.org/doi/10.1103/ PhysRevX.10.021067

    2020

  5. [5]

    A review on quantum approximate optimization algorithm and its variants,

    K. Blekos, D. Brand, A. Ceschini, C.-H. Chou, R.-H. Li, K. Pandya, and A. Summer, “A review on quantum approximate optimization algorithm and its variants,”Physics Reports, vol. 1068, pp. 1–66, 2024, a review on Quantum Approximate Optimization Algorithm and its variants. [Online]. Available: https://www.sciencedirect.com/science/ article/pii/S0370157324001078

    2024

  6. [6]

    On barren plateaus and cost function locality in variational quantum algorithms,

    A. V . Uvarov and J. D. Biamonte, “On barren plateaus and cost function locality in variational quantum algorithms,”Journal of Physics A: Mathematical and Theoretical, vol. 54, no. 24, p. 245301, may

  7. [7]

    Available: https://doi.org/10.1088/1751-8121/abfac7

    [Online]. Available: https://doi.org/10.1088/1751-8121/abfac7

    work page doi:10.1088/1751-8121/abfac7

  8. [8]

    Trainability barriers in low-depth qaoa landscapes,

    J. Rajakumar, J. Golden, A. B ¨artschi, and S. Eidenbenz, “Trainability barriers in low-depth qaoa landscapes,” inProceedings of the 21st ACM International Conference on Computing Frontiers, ser. CF ’24. New York, NY , USA: Association for Computing Machinery, 2024, p. 199–206. [Online]. Available: https://doi.org/10.1145/3649153.3649204

    work page doi:10.1145/3649153.3649204 2024

  9. [9]

    Noise-induced barren plateaus in variational quantum algorithms,

    S. Wang, E. Fontana, M. Cerezo, K. Sharma, A. Sone, L. Cincio, and P. J. Coles, “Noise-induced barren plateaus in variational quantum algorithms,”Nature Communications, vol. 12, no. 1, p. 6961, Nov

  10. [10]

    Available: https://doi.org/10.1038/s41467-021-27045-6

    [Online]. Available: https://doi.org/10.1038/s41467-021-27045-6

    work page doi:10.1038/s41467-021-27045-6

  11. [11]

    Evaluating analytic gradients on quantum hardware,

    M. Schuld, V . Bergholm, C. Gogolin, J. Izaac, and N. Killoran, “Evaluating analytic gradients on quantum hardware,”Phys. Rev. A, vol. 99, p. 032331, Mar 2019. [Online]. Available: https: //link.aps.org/doi/10.1103/PhysRevA.99.032331

    work page doi:10.1103/physreva.99.032331 2019

  12. [12]

    General parameter- shift rules for quantum gradients,

    D. Wierichs, J. Izaac, C. Wang, and C. Y .-Y . Lin, “General parameter- shift rules for quantum gradients,”Quantum, vol. 6, p. 677, Mar. 2022. [Online]. Available: https://doi.org/10.22331/q-2022-03-30-677

    work page doi:10.22331/q-2022-03-30-677 2022

  13. [13]

    An Adaptive Optimizer for Measurement-Frugal Variational Algorithms,

    J. M. K ¨ubler, A. Arrasmith, L. Cincio, and P. J. Coles, “An Adaptive Optimizer for Measurement-Frugal Variational Algorithms,” Quantum, vol. 4, p. 263, May 2020. [Online]. Available: https: //doi.org/10.22331/q-2020-05-11-263

    work page doi:10.22331/q-2020-05-11-263 2020

  14. [14]

    Operator sampling for shot-frugal optimization in variational algorithms,

    A. Arrasmith, L. Cincio, R. D. Somma, and P. J. Coles, “Operator sampling for shot-frugal optimization in variational algorithms,” 2020. [Online]. Available: https://arxiv.org/abs/2004.06252

    work page arXiv 2020

  15. [15]

    Scaling of the quantum approximate optimization algorithm on superconducting qubit based hardware,

    J. Weidenfeller, L. C. Valor, J. Gacon, C. Tornow, L. Bello, S. Woerner, and D. J. Egger, “Scaling of the quantum approximate optimization algorithm on superconducting qubit based hardware,”Quantum, vol. 6, p. 870, Dec. 2022. [Online]. Available: https://doi.org/10.22331/q- 2022-12-07-870

    work page doi:10.22331/q- 2022

  16. [16]

    Large-scale quantum approximate optimization on nonplanar graphs with machine learning noise mitigation,

    S. H. Sack and D. J. Egger, “Large-scale quantum approximate optimization on nonplanar graphs with machine learning noise mitigation,”Phys. Rev. Res., vol. 6, p. 013223, Mar 2024. [Online]. Available: https://link.aps.org/doi/10.1103/PhysRevResearch.6.013223

    work page doi:10.1103/physrevresearch.6.013223 2024

  17. [17]

    Scaling whole-chip qaoa for higher-order ising spin glass models on heavy-hex graphs,

    E. Pelofske, A. B ¨artschi, L. Cincio, J. Golden, and S. Eidenbenz, “Scaling whole-chip qaoa for higher-order ising spin glass models on heavy-hex graphs,”npj Quantum Information, vol. 10, no. 1, p. 109, Nov

  18. [18]

    Available: https://doi.org/10.1038/s41534-024-00906-w

    [Online]. Available: https://doi.org/10.1038/s41534-024-00906-w

    work page doi:10.1038/s41534-024-00906-w

  19. [19]

    Scaling quantum approximate optimization on near-term hardware,

    P. C. Lotshaw, T. Nguyen, A. Santana, A. McCaskey, R. Herrman, J. Ostrowski, G. Siopsis, and T. S. Humble, “Scaling quantum approximate optimization on near-term hardware,”Scientific Reports, vol. 12, no. 1, p. 12388, Jul 2022. [Online]. Available: https: //doi.org/10.1038/s41598-022-14767-w

    work page doi:10.1038/s41598-022-14767-w 2022

  20. [20]

    Efficient depth selection for the implementation of noisy quantum approximate optimization algorithm,

    Y . Pan, Y . Tong, S. Xue, and G. Zhang, “Efficient depth selection for the implementation of noisy quantum approximate optimization algorithm,”Journal of the Franklin Institute, vol. 359, no. 18, pp. 11 273–11 287, 2022. [Online]. Available: https: //www.sciencedirect.com/science/article/pii/S0016003222007542

    2022

  21. [21]

    The effect of classical optimizers and ansatz depth on qaoa performance in noisy devices,

    A. Pellow-Jarman, S. McFarthing, I. Sinayskiy, D. K. Park, A. Pillay, and F. Petruccione, “The effect of classical optimizers and ansatz depth on qaoa performance in noisy devices,”Scientific Reports, vol. 14, no. 1, p. 16011, Jul 2024. [Online]. Available: https://doi.org/10.1038/s41598-024-66625-6

    work page doi:10.1038/s41598-024-66625-6 2024

  22. [22]

    Multi-angle quantum approximate optimization algorithm,

    R. Herrman, P. C. Lotshaw, J. Ostrowski, T. S. Humble, and G. Siopsis, “Multi-angle quantum approximate optimization algorithm,”Scientific Reports, vol. 12, no. 1, p. 6781, Apr 2022. [Online]. Available: https://doi.org/10.1038/s41598-022-10555-8

    work page doi:10.1038/s41598-022-10555-8 2022

  23. [23]

    Layerwise retraining and freezing for multi- angle qaoa,

    E. Jang, Z. Chen, D. Ha, S. Choi, Y . Lee, J. Kwon, E. Z. Zhang, Y . Huang, and W. W. Ro, “Layerwise retraining and freezing for multi- angle qaoa,”Quantum Machine Intelligence, vol. 8, no. 1, Jan. 2026. [Online]. Available: http://dx.doi.org/10.1007/s42484-026-00357-w

    work page doi:10.1007/s42484-026-00357-w 2026

  24. [24]

    Quantum annealing initialization of the quantum approximate optimization algorithm,

    S. H. Sack and M. Serbyn, “Quantum annealing initialization of the quantum approximate optimization algorithm,”Quantum, vol. 5, p. 491, Jul. 2021. [Online]. Available: https://doi.org/10.22331/q-2021-07-01- 491

    work page doi:10.22331/q-2021-07-01- 2021

  25. [25]

    Performance analysis of multi-angle qaoa for p ¿ 1,

    I. Gaidai and R. Herrman, “Performance analysis of multi-angle qaoa for p ¿ 1,”Scientific Reports, vol. 14, no. 1, p. 18911, Aug 2024. [Online]. Available: https://doi.org/10.1038/s41598-024-69643-6

    work page doi:10.1038/s41598-024-69643-6 2024

  26. [26]

    For fixed control parameters the quantum approximate optimization algorithm’s objective function value concentrates for typical instances,

    F. G. S. L. Brandao, M. Broughton, E. Farhi, S. Gutmann, and H. Neven, “For fixed control parameters the quantum approximate optimization algorithm’s objective function value concentrates for typical instances,”

  27. [27]

    For Fixed Control Parameters the Quantum Approximate Optimization Algorithm's Objective Function Value Concentrates for Typical Instances

    [Online]. Available: https://arxiv.org/abs/1812.04170

    work page internal anchor Pith review Pith/arXiv arXiv

  28. [28]

    Quantum approximate optimization via learning-based adaptive optimization,

    L. Cheng, Y .-Q. Chen, S.-X. Zhang, and S. Zhang, “Quantum approximate optimization via learning-based adaptive optimization,” Communications Physics, vol. 7, no. 1, p. 83, Mar 2024. [Online]. Available: https://doi.org/10.1038/s42005-024-01577-x

    work page doi:10.1038/s42005-024-01577-x 2024

  29. [29]

    Dual role of low-weight pauli propagation: A flawed simulator but a powerful initializer for variational quantum algorithms,

    Z.-L. Li and S.-X. Zhang, “Dual role of low-weight pauli propagation: A flawed simulator but a powerful initializer for variational quantum algorithms,”Phys. Rev. Res., vol. 8, p. 013266, Mar 2026. [Online]. Available: https://link.aps.org/doi/10.1103/4kyq-n8jb

    work page doi:10.1103/4kyq-n8jb 2026

  30. [30]

    A practical guide to using pauli path simulators for utility-scale quantum experiments,

    H. Gharibyan, S. Hariprakash, M. Z. Mullath, and V . P. Su, “A practical guide to using pauli path simulators for utility-scale quantum experiments,” 2025. [Online]. Available: https://arxiv.org/abs/ 2507.10771

    work page arXiv 2025

  31. [31]

    cuquan- tum sdk: A high-performance library for accelerating quantum science,

    H. Bayraktar, B. K. Clark, S. D. Cohen, P. C. Costa, M. Franklin, S. Gharibian, J. Hachmann, T. H ¨aner, A. H. Hill, T. S. Humble, E. Khatami, M. Kreshchuk, S. M. Lukin, D. Maslov, S. M. Mniszewski, M. Pistoia, C. Pocreau, M. Prokopenko, I. Rungger, I. Safro, J. S. Smith, S. Stetina, P. Szczepankiewicz, M. Treinish, and A. Alexandru, “cuquan- tum sdk: A h...

  32. [32]

    Pauli prop,

    Qiskit addons team, “Pauli prop,” https://github.com/Qiskit/pauli-prop

  33. [33]

    Pauli propagation — gpu acceleration,

    Norma-Q, “Pauli propagation — gpu acceleration,” https://github. com/Norma-Q/Pauli-Propagation---GPU-acceleration, accessed: 2026- 04-07

    2026

  34. [34]

    Multiangle QAOA Does Not Always Need All Its Angles ,

    K. Shi, R. Herrman, R. Shaydulin, S. Chakrabarti, M. Pistoia, and J. Larson, “ Multiangle QAOA Does Not Always Need All Its Angles ,” in2022 IEEE/ACM 7th Symposium on Edge Computing (SEC). Los Alamitos, CA, USA: IEEE Computer Society, Dec. 2022, pp. 414–419. [Online]. Available: https://doi.ieeecomputersociety.org/ 10.1109/SEC54971.2022.00062

    work page doi:10.1109/sec54971.2022.00062 2022

  35. [35]

    Iterative Interpolation Schedules for Quantum Approximate Optimization Algorithm

    A. Apte, S. H. Sureshbabu, R. Shaydulin, S. Boulebnane, Z. He, D. Herman, J. Sud, and M. Pistoia, “Iterative interpolation schedules for quantum approximate optimization algorithm,” 2025. [Online]. Available: https://arxiv.org/abs/2504.01694

    work page internal anchor Pith review Pith/arXiv arXiv 2025

  36. [36]

    Parameter setting heuristics make the quantum approximate optimization algorithm suitable for the early fault-tolerant era,

    Z. He, R. Shaydulin, D. Herman, C. Li, R. Raymond, S. H. Sureshbabu, and M. Pistoia, “Parameter setting heuristics make the quantum approximate optimization algorithm suitable for the early fault-tolerant era,” inProceedings of the 43rd IEEE/ACM International Conference on Computer-Aided Design, ser. ICCAD ’24. New York, NY , USA: Association for Computin...

    work page doi:10.1145/3676536.3697128 2025

  37. [37]

    Transferring linearly fixed qaoa angles: performance and real device results,

    R. Sakai, H. Matsuyama, W.-H. Tam, and Y . Yamashiro, “Transferring linearly fixed qaoa angles: performance and real device results,” 2025. [Online]. Available: https://arxiv.org/abs/2504.12632

    work page arXiv 2025

  38. [38]

    Parameter concentrations in quantum approximate optimization,

    V . Akshay, D. Rabinovich, E. Campos, and J. Biamonte, “Parameter concentrations in quantum approximate optimization,”Phys. Rev. A, vol. 104, p. L010401, Jul 2021. [Online]. Available: https: //link.aps.org/doi/10.1103/PhysRevA.104.L010401

    work page doi:10.1103/physreva.104.l010401 2021

  39. [39]

    Transferability of optimal qaoa parameters between random graphs,

    A. Galda, X. Liu, D. Lykov, Y . Alexeev, and I. Safro, “Transferability of optimal qaoa parameters between random graphs,” in2021 IEEE International Conference on Quantum Computing and Engineering (QCE), 2021, pp. 171–180

    2021

  40. [40]

    Bayesian optimization for qaoa,

    S. Tibaldi, D. V odola, E. Tignone, and E. Ercolessi, “Bayesian optimization for qaoa,” 2023. [Online]. Available: https://arxiv.org/abs/ 2209.03824

    work page arXiv 2023

  41. [41]

    Variational quantum algorithm with information sharing,

    C. N. Self, K. E. Khosla, A. W. R. Smith, F. Sauvage, P. D. Haynes, J. Knolle, F. Mintert, and M. S. Kim, “Variational quantum algorithm with information sharing,” vol. 7, no. 1, p. 116. [Online]. Available: https://www.nature.com/articles/s41534-021-00452-9

  42. [42]

    Surrogate-based optimization for variational quantum algorithms,

    R. Shaffer, L. Kocia, and M. Sarovar, “Surrogate-based optimization for variational quantum algorithms,”Phys. Rev. A, vol. 107, p. 032415, Mar

  43. [43]

    Available: https://link.aps.org/doi/10.1103/PhysRevA

    [Online]. Available: https://link.aps.org/doi/10.1103/PhysRevA. 107.032415

    work page doi:10.1103/physreva

  44. [44]

    Faster variational quantum algorithms with quantum kernel-based surrogate models,

    A. W. R. Smith, A. J. Paige, and M. S. Kim, “Faster variational quantum algorithms with quantum kernel-based surrogate models,” Quantum Science and Technology, vol. 8, no. 4, p. 045016, aug 2023. [Online]. Available: https://doi.org/10.1088/2058-9565/aceb87

    work page doi:10.1088/2058-9565/aceb87 2023

  45. [45]

    Improving the efficiency of qaoa using efficient parameter transfer initialization and targeted-single-layer regularized optimization with minimal performance degradation,

    S. Patel and U. Mishra, “Improving the efficiency of qaoa using efficient parameter transfer initialization and targeted-single-layer regularized optimization with minimal performance degradation,” 2026. [Online]. Available: https://arxiv.org/abs/2601.15760

    work page arXiv 2026

  46. [46]

    End-to-end protocol for high-quality quantum approximate optimization algorithm parameters with few shots,

    T. Hao, Z. He, R. Shaydulin, J. Larson, and M. Pistoia, “End-to-end protocol for high-quality quantum approximate optimization algorithm parameters with few shots,”Phys. Rev. Res., vol. 7, p. 033179, Aug

  47. [47]

    Available: https://link.aps.org/doi/10.1103/24gg-7p8z

    [Online]. Available: https://link.aps.org/doi/10.1103/24gg-7p8z

    work page doi:10.1103/24gg-7p8z

  48. [48]

    Simulation of qubit quantum circuits via pauli propagation,

    P. Rall, D. Liang, J. Cook, and W. Kretschmer, “Simulation of qubit quantum circuits via pauli propagation,”Phys. Rev. A, vol. 99, p. 062337, Jun 2019. [Online]. Available: https: //link.aps.org/doi/10.1103/PhysRevA.99.062337

    work page doi:10.1103/physreva.99.062337 2019

  49. [49]

    Pauli propagation: A computational framework for simulating quantum systems,

    M. S. Rudolph, T. Jones, Y . Teng, A. Angrisani, and Z. Holmes, “Pauli propagation: A computational framework for simulating quantum systems,” 2025. [Online]. Available: https://arxiv.org/abs/2505.21606

    work page arXiv 2025

  50. [50]

    Simulating quantum circuits with arbitrary local noise using Pauli Propagation

    A. Angrisani, A. A. Mele, M. S. Rudolph, M. Cerezo, and Z. Holmes, “Simulating quantum circuits with arbitrary local noise using pauli propagation,” 2025. [Online]. Available: https: //arxiv.org/abs/2501.13101

    work page internal anchor Pith review Pith/arXiv arXiv 2025

  51. [51]

    Classical simulations of noisy variational quantum circuits,

    E. Fontana, M. S. Rudolph, R. Duncan, I. Rungger, and C. C ˆırstoiu, “Classical simulations of noisy variational quantum circuits,”npj Quantum Information, vol. 11, no. 1, p. 84, May 2025. [Online]. Available: https://doi.org/10.1038/s41534-024-00955-1

    work page doi:10.1038/s41534-024-00955-1 2025

  52. [52]

    Ising formulations of many np problems,

    A. Lucas, “Ising formulations of many np problems,”Frontiers in Physics, vol. V olume 2 - 2014, 2014. [Online]. Available: https: //www.frontiersin.org/journals/physics/articles/10.3389/fphy.2014.00005

    work page doi:10.3389/fphy.2014.00005 2014

  53. [53]

    Ising machines as hardware solvers of combinatorial optimization problems,

    N. Mohseni, P. L. McMahon, and T. Byrnes, “Ising machines as hardware solvers of combinatorial optimization problems,”Nature Reviews Physics, vol. 4, no. 6, pp. 363–379, Jun 2022. [Online]. Available: https://doi.org/10.1038/s42254-022-00440-8

    work page doi:10.1038/s42254-022-00440-8 2022

  54. [54]

    Seeking quantum speedup through spin glasses: The good, the bad, and the ugly,

    H. G. Katzgraber, F. Hamze, Z. Zhu, A. J. Ochoa, and H. Munoz-Bauza, “Seeking quantum speedup through spin glasses: The good, the bad, and the ugly,”Phys. Rev. X, vol. 5, p. 031026, Sep 2015. [Online]. Available: https://link.aps.org/doi/10.1103/PhysRevX.5.031026

    work page doi:10.1103/physrevx.5.031026 2015

  55. [55]

    Searching for spin glass ground states through deep reinforcement learning,

    C. Fan, M. Shen, Z. Nussinov, Z. Liu, Y . Sun, and Y .-Y . Liu, “Searching for spin glass ground states through deep reinforcement learning,”Nature Communications, vol. 14, no. 1, p. 725, Feb 2023. [Online]. Available: https://doi.org/10.1038/s41467-023-36363-w

    work page doi:10.1038/s41467-023-36363-w 2023

  56. [56]

    50 years of spin glass theory,

    D. Sherrington and S. Kirkpatrick, “50 years of spin glass theory,” Nature Reviews Physics, vol. 7, no. 10, pp. 528–529, Oct 2025. [Online]. Available: https://doi.org/10.1038/s42254-025-00871-z

    work page doi:10.1038/s42254-025-00871-z 2025

  57. [57]

    The sherrington-kirkpatrick model: An overview,

    D. Panchenko, “The sherrington-kirkpatrick model: An overview,” Journal of Statistical Physics, vol. 149, no. 2, pp. 362–383, Oct 2012. [Online]. Available: https://doi.org/10.1007/s10955-012-0586-7

    work page doi:10.1007/s10955-012-0586-7 2012

  58. [58]

    Quantum optimization of fully connected spin glasses,

    D. Venturelli, S. Mandr `a, S. Knysh, B. O’Gorman, R. Biswas, and V . Smelyanskiy, “Quantum optimization of fully connected spin glasses,”Phys. Rev. X, vol. 5, p. 031040, Sep 2015. [Online]. Available: https://link.aps.org/doi/10.1103/PhysRevX.5.031040

    work page doi:10.1103/physrevx.5.031040 2015

  59. [59]

    The Quantum Approximate Optimization Algorithm and the Sherrington-Kirkpatrick Model at Infinite Size.Quantum, 6: 759, July 2022

    E. Farhi, J. Goldstone, S. Gutmann, and L. Zhou, “The Quantum Approximate Optimization Algorithm and the Sherrington-Kirkpatrick Model at Infinite Size,”Quantum, vol. 6, p. 759, Jul. 2022. [Online]. Available: https://doi.org/10.22331/q-2022-07-07-759

    work page doi:10.22331/q-2022-07-07-759 2022

  60. [60]

    The quantum approximate optimization algorithm at high depth for maxcut on large-girth regular graphs and the sherrington- kirkpatrick model,

    J. Basso, E. Farhi, K. Marwaha, B. Villalonga, and L. Zhou, “The quantum approximate optimization algorithm at high depth for maxcut on large-girth regular graphs and the sherrington- kirkpatrick model,” vol. 232. Schloss Dagstuhl – Leibniz-Zentrum f¨ur Informatik, 2022, pp. 7:1–7:21. [Online]. Available: https: //drops.dagstuhl.de/entities/document/10.42...

    work page doi:10.4230/lipics.tqc.2022.7 2022

  61. [61]

    Evidence that the quantum approximate optimization algorithm optimizes the sherrington-kirkpatrick model efficiently in the average case,

    S. Boulebnane, A. Khan, M. Liu, J. Larson, D. Herman, R. Shaydulin, and M. Pistoia, “Evidence that the quantum approximate optimization algorithm optimizes the sherrington-kirkpatrick model efficiently in the average case,” 2025. [Online]. Available: https://arxiv.org/abs/2505. 07929

    2025

  62. [62]

    Optimization of the Sherrington-Kirkpatrick Hamiltonian

    A. Montanari, “Optimization of the sherrington-kirkpatrick hamilto- nian,” 2019. [Online]. Available: https://arxiv.org/abs/1812.10897

    work page internal anchor Pith review Pith/arXiv arXiv 2019

  63. [63]

    Ground states of two- dimensional±jedwards-anderson spin glasses,

    J. W. Landry and S. N. Coppersmith, “Ground states of two- dimensional±jedwards-anderson spin glasses,”Phys. Rev. B, vol. 65, p. 134404, Mar 2002. [Online]. Available: https://link.aps.org/doi/10. 1103/PhysRevB.65.134404

    2002

  64. [64]

    Exact ground states of large two-dimensional planar ising spin glasses,

    G. Pardella and F. Liers, “Exact ground states of large two-dimensional planar ising spin glasses,”Phys. Rev. E, vol. 78, p. 056705, Nov

  65. [65]

    Available: https://link.aps.org/doi/10.1103/PhysRevE

    [Online]. Available: https://link.aps.org/doi/10.1103/PhysRevE. 78.056705

    work page doi:10.1103/physreve

  66. [66]

    Physics of the edwards–anderson spin glass in dimensions d = 3, . . . ,8 from heuristic ground state optimization,

    S. Boettcher, “Physics of the edwards–anderson spin glass in dimensions d = 3, . . . ,8 from heuristic ground state optimization,”Frontiers in Physics, vol. V olume 12 - 2024, 2024. [Online]. Available: https://www. frontiersin.org/journals/physics/articles/10.3389/fphy.2024.1466987

    work page doi:10.3389/fphy.2024.1466987 2024

  67. [67]

    Spin glass phase at zero temperature in the edwards- anderson model,

    S. Chatterjee, “Spin glass phase at zero temperature in the edwards- anderson model,” 2025. [Online]. Available: https://arxiv.org/abs/2301. 04112

    2025

  68. [68]

    Phase diagram and critical behavior of the square-lattice ising model with competing nearest-neighbor and next-nearest-neighbor interactions,

    J. Yin and D. P. Landau, “Phase diagram and critical behavior of the square-lattice ising model with competing nearest-neighbor and next-nearest-neighbor interactions,”Phys. Rev. E, vol. 80, p. 051117, Nov 2009. [Online]. Available: https://link.aps.org/doi/10.1103/ PhysRevE.80.051117

    2009

  69. [69]

    Wang–landau study of the random bond square ising model with nearest- and next-nearest- neighbor interactions,

    N. G. Fytas, A. Malakis, and I. Georgiou, “Wang–landau study of the random bond square ising model with nearest- and next-nearest- neighbor interactions,”Journal of Statistical Mechanics: Theory and Experiment, vol. 2008, no. 07, p. L07001, jul 2008. [Online]. Available: https://doi.org/10.1088/1742-5468/2008/07/L07001

    work page doi:10.1088/1742-5468/2008/07/l07001 2008

  70. [70]

    Universal resources for quantum approximate optimization algorithm and quantum annealing,

    P. D ´ıez-Valle, F. J. G ´omez-Ruiz, D. Porras, and J. J. Garc ´ıa-Ripoll, “Universal resources for quantum approximate optimization algorithm and quantum annealing,”Phys. Rev. Res., vol. 8, p. 013211, Feb 2026. [Online]. Available: https://link.aps.org/doi/10.1103/hxv2-sbr7

    work page doi:10.1103/hxv2-sbr7 2026

  71. [71]

    Scaling qaoa: transferring optimal adiabatic schedules from small-scale to large-scale variational circuits,

    U. Nzongani, D. L. Mermoud, and A. Braida, “Scaling qaoa: transferring optimal adiabatic schedules from small-scale to large-scale variational circuits,” 2026. [Online]. Available: https://arxiv.org/abs/2602.14986

    work page arXiv 2026