Pith. sign in

REVIEW 2 major objections 4 minor 5 cited by

Warm-started XY-mixers keep the biased state as ground state, and iterative updates raise optimal-solution odds by orders of magnitude in constrained QAOA.

Reviewed by Pith at T0; open to challenge. T0 means a machine referee read the full paper against a public rubric. the ladder, T0–T4 →

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗

T0 review · grok-4.5

2026-07-13 13:59 UTC pith:MTCQVB4F

load-bearing objection Clean math fix for warm-started XY-mixers plus a working iterative loop and a real 144-qubit demo; the optimize-once schedule is a real but secondary caveat, not a collapse of the claim. the 2 major comments →

arxiv 2604.02083 v2 pith:MTCQVB4F submitted 2026-04-02 quant-ph

Constrained Quantum Optimization via Iterative Warm-Start XY-Mixers

David Bucher , Maximilian Janetschek , Michael Poppel , Jonas Stein , Claudia Linnhoff-Popien , Sebastian Feld This is my paper
classification quant-ph
keywords QAOAXY-mixerwarm-startone-hot constraintsMax-k-CutTSPNISQiterative bias
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.

Hard constraints in combinatorial problems force ordinary QAOA either to enlarge the search space with penalties or to use XY-mixers that stay inside the feasible Hamming-weight-1 subspace. Prior warm-starting of those mixers used a biased initial state but left the mixer itself unchanged, breaking the ground-state alignment required by the adiabatic theorem. This paper constructs an explicit warm-started XY-mixer Hamiltonian whose unique ground state inside the one-hot sector is precisely the biased W-state, proves that property for fully-connected and for connected topologies, and supplies a two-Pauli-rotation circuit. The same mixer is then driven by an iterative classical loop that re-weights the bias from the Boltzmann-weighted samples of the previous round. On Max-k-Cut and TSP instances the loop multiplies the probability of sampling an optimum by one to two orders of magnitude; on 144-qubit hardware-tailored instances the same pipeline, repaired by a greedy descent, recovers true optima on a real quantum processor.

Core claim

The operator formed by embedding single-qubit warm-start mixers into every edge of a connected one-hot topology has the probability-weighted W-state as its unique ground state of energy −1 inside the Hamming-weight-1 subspace; when this mixer is iterated with sample-based probability updates, the resulting IWS-QAOA finds optimal solutions far more frequently than ordinary XY-QAOA.

What carries the argument

The warm-started XY-mixer HP = (1/(k−1)) ∑ Hij(qij) with qij = Pi/(Pi+Pj); Proposition 1 and its corollaries establish that |WP⟩ is its unique ground state of energy −1, while a two-Pauli-rotation circuit realises the corresponding time evolution.

Load-bearing premise

A single linear parameter schedule optimised only for the uniform initial distribution stays near-optimal after every later bias update.

What would settle it

Re-optimise the linear schedule after each probability update on the same Max-k-Cut or TSP instances; if the reported Popt gains disappear or reverse, the single-optimisation claim is false.

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

If this is right

  • Any one-hot-constrained QAOA can keep adiabatic guarantees while still biasing the search.
  • Hardware-efficient XY topologies remain valid warm-start mixers once the degree-correction terms of Corollary 1.2 are included.
  • Sample-based iterative warm-starting can replace classical SDP or continuous relaxations for problems whose relaxed optima lie at integer vertices.
  • Greedy post-processing of noisy one-hot measurements is sufficient to recover global optima on present-day 100-plus-qubit devices when the underlying circuit is shallow.

Where Pith is reading between the lines

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

  • The same construction should extend immediately to other fixed-Hamming-weight sectors once the appropriate warm-start blocks are written.
  • If the Boltzmann update is replaced by a diversity-preserving rule, the method could trade exploitation for broader exploration and reduce trapping in local minima.
  • Because the mixer alignment is topology-independent once the degree corrections are present, the technique can be ported to any sparse hardware graph that admits a connected matching decomposition.

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

2 major / 4 minor

Summary. The manuscript formulates a warm-started XY-mixer Hamiltonian HP (Eq. 12) for one-hot constraints and proves that the biased |WP⟩ state is its unique ground state of energy −1 inside the Hamming-weight-1 subspace (Proposition 1 and Corollaries 1.1–1.3). It supplies an exact two-Pauli-rotation circuit for the warm-start XY block (Proposition 2) and embeds the construction into Iterative Warm-Starting QAOA (IWS-QAOA, Algorithm 2), which updates a Boltzmann-weighted probability distribution from previous samples. Numerical simulations on Max-k-Cut and TSP instances report orders-of-magnitude gains in Popt relative to standard XY-QAOA; hardware-tailored 144-qubit instances on ibm_boston, repaired by greedy steepest-descent post-processing, recover optimal solutions on three of five instances.

Significance. If the claims hold, the work supplies a theoretically aligned mixer for constrained warm-start QAOA that previous biased-|W⟩ constructions lacked, together with a shallow NISQ circuit and a classical iterative loop that does not require problem-specific SDP solvers. The ground-state proofs are self-contained linear algebra (eigenvalue calculation, number-operator commutator, Perron–Frobenius), the circuit identity is verified algebraically in the single-excitation subspace, and the empirical claims are measured against external baselines (plain XY-QAOA, classical random IWS, SCIP optima). Successful recovery of optima on 144-qubit hardware-tailored instances places the method among the larger-scale demonstrations of XY-mixers on superconducting devices.

major comments (2)
  1. Sec. III C and Algorithm 2 freeze a single linear-schedule optimization of {β0, Δβ, γ0, Δγ} performed only on the uniform distribution P(0). Sec. IV C and Fig. 1 examine landscapes only for the ideal (fully concentrated) bias on one N=7 TSP instance; they do not re-optimize or map the landscape under the intermediate distributions that arise during IWS. If the high-quality region of the (Δβ, Δγ) plane moves appreciably once the bias concentrates, the reported Popt gains and the hardware optima (which further reuse averaged parameters from smaller instances) become contingent on an untested transfer. A short re-optimization check on intermediate P(t) for at least one MkC and one TSP instance would close this gap.
  2. Hardware results (Sec. V, Table III) rely on a classical greedy steepest-descent repair of the penalized QUBO (Eq. 32). Fig. 9 shows that raw feasibility is already low (f ≈ 11–22 % per constraint), so the quantum circuit contributes only a noisy seed. The manuscript does not quantify how much of the final optimality is attributable to the quantum samples versus the classical post-processor alone (beyond the rnd-PP baseline). An ablation that feeds the same post-processor with samples drawn from the final IWS distribution without the QAOA circuit would clarify the quantum contribution.
minor comments (4)
  1. The dual use of the symbol β for both QAOA mixer angles and the Boltzmann inverse temperature (Eq. 23) is flagged in the text but remains easy to misread; a distinct symbol for the temperature would improve clarity.
  2. Fig. 1 caption and surrounding text refer to both “WS XY” and “default XY”; the precise regularization values used for each panel should be restated in the caption for self-contained reading.
  3. Appendix A describes three state-preparation schemes; it would help the reader to state explicitly which scheme is used for the numerical simulations versus the hardware experiments.
  4. Typographical inconsistencies appear in a few places (e.g., “ibm boston” vs. “ibm_boston”, occasional missing spaces around math). A light copy-edit pass would remove them.

Circularity Check

0 steps flagged

No significant circularity: mixer ground-state claim is a direct constructive proof; empirical speed-ups are measured against independent baselines.

full rationale

The central theoretical claim (Proposition 1 and Corollaries 1.1–1.3) constructs HP (Eq. 12) so that |WP angle is an eigenstate of energy −1 inside the Hamming-weight-1 subspace, then verifies the eigenvalue equation by direct expansion, invariance under the number operator, and uniqueness via the classical Perron–Frobenius theorem. This is intentional design plus verification, not a reduction of a claimed prediction to its own inputs. The circuit decomposition (Proposition 2) is an exact algebraic identity proved in Appendix C. IWS-QAOA (Algorithm 2) updates probabilities from samples via a Boltzmann weight and reuses a single linear-schedule parameter set; the performance claims (orders-of-magnitude Popt gains, hardware optima) are empirical comparisons against external baselines (plain XY-QAOA, classical random IWS, SCIP optima) on Max-k-Cut/TSP and 144-qubit hardware-tailored instances. Hyper-parameters are chosen by a preliminary study, not fitted to the final reported ratios. Minor self-citations exist for simulation techniques and related prior work by overlapping authors, but none is load-bearing for the mixer proof or the measured gains. The optimize-once schedule assumption is a methodological limitation, not circularity. Score 1 reflects only the presence of non-load-bearing self-citations; the derivation chain is self-contained.

Axiom & Free-Parameter Ledger

5 free parameters · 4 axioms · 2 invented entities

The mathematical core rests on standard linear algebra (Perron–Frobenius, commutators with the number operator) plus the domain assumption that one-hot constraints are non-overlapping so that product XY-mixers remain valid. Free parameters are the usual QAOA / IWS hyper-parameters chosen once by a preliminary study; they are not fitted to the final performance numbers. No new physical entities are postulated.

free parameters (5)
  • regularization ε = 0.2 (sim) / 0.1 (HW)
    Clamps each probability into [ε/(k−1), 1−ε]; set to 0.2 (simulations) or 0.1 (hardware) after a preliminary study; controls exploitation vs. collapse.
  • inverse temperature β (Boltzmann update) = 15
    Controls how aggressively low-energy samples dominate the next probability vector; fixed at 15 after preliminary study.
  • shots per iteration M = 100–500
    Sample size used to re-estimate probabilities; tested at 100/200/500; smaller M accelerates iteration but raises local-minima risk.
  • linear-schedule parameters {β0, Δβ, γ0, Δγ} = instance-dependent (Table II for HW)
    Optimized once with BFGS on the uniform start and then frozen for all subsequent IWS iterations; hardware uses averages from smaller simulated instances.
  • penalty λ (TSP / post-processing QUBO) = 2 / 10
    Quadratic penalty weight for the second set of one-hot constraints (TSP) or for the classical repair objective; set to 2 (TSP) or 10 (PP).
axioms (4)
  • standard math Perron–Frobenius theorem for matrices with non-positive off-diagonal entries implies a unique positive ground-state eigenvector.
    Invoked in the final step of Proposition 1 and Corollaries 1.1–1.2 to establish uniqueness of |WP angle.
  • domain assumption One-hot constraints act on disjoint sets of binary variables, so product mixers and product |W angle states remain valid.
    Stated in Sec. II C and used throughout; XY-mixers cannot enforce overlapping constraints.
  • ad hoc to paper A single linear-schedule optimization performed on the uniform distribution remains near-optimal after probability updates.
    Explicitly assumed in Sec. III C to justify the optimize-once strategy; supported only by the landscape comparison in Fig. 1 for an ideal bias, not for intermediate IWS iterates.
  • domain assumption Hardware noise can be adequately mitigated for the reported claims by a classical greedy steepest-descent repair that never flips already-feasible one-hot blocks.
    Sec. V B; without this post-processing the raw feasibility rate is <1 % and the hardware optima would not be observed.
invented entities (2)
  • warm-started XY-mixer Hamiltonian HP (and its scaled version H̃) no independent evidence
    purpose: Provide a mixer whose unique ground state inside the Hamming-weight-1 sector is exactly the biased |WP angle, restoring adiabatic alignment.
    Defined in Eq. (12) / (20); the paper proves its spectral properties but the operator itself is a construction introduced here.
  • Iterative Warm-Starting (IWS) probability-update loop no independent evidence
    purpose: Generate successive bias distributions from Boltzmann-weighted quantum samples without requiring a classical SDP or continuous relaxation.
    Algorithm 2; combines ideas from adaptive-bias QAOA with the new mixer; the closed loop is new.

pith-pipeline@v1.1.0-grok45 · 32222 in / 3775 out tokens · 27991 ms · 2026-07-13T13:59:12.435433+00:00 · methodology

0 comments
read the original abstract

The Quantum Approximate Optimization Algorithm (QAOA) is a leading hybrid heuristic for combinatorial optimization, but efficiently handling hard constraints remains a significant challenge. XY-mixers successfully confine quantum state evolution to a feasible subspace, such as the Hamming-weight-1 sector for one-hot constraints. On the contrary, warm-starting biases the search toward promising regions based on preliminary solutions. Combining these two techniques requires maintaining the essential alignment between the initial state and the mixer Hamiltonian to preserve convergence guarantees. Previous work demonstrated warm-starting with XY-mixers via a biased initial state, but relying only on standard mixer Hamiltonians. Consequently, the initial state is no longer a ground state of the mixer. In this work, we overcome these limitations by formulating a warm-started XY-mixer Hamiltonian for one-hot constraints and proving its ground-state properties. Furthermore, we provide a shallow circuit implementation suitable for NISQ implementations. We embed the warm-starting into a classical heuristic that iteratively updates the bias based on previous samples, called Iterative Warm-Starting (IWS). Extensive numerical simulations on Max-$k$-Cut and Traveling Salesperson Problem instances demonstrate that IWS-QAOA significantly accelerates the solution-finding process, increasing the probability of sampling optimal solutions by orders of magnitude compared to standard XY-QAOA. Finally, we validate our approach on the ibm_boston QPU using hardware-tailored 144-qubit problem instances. By coupling IWS-QAOA with a greedy steepest-descent post-processing strategy to repair infeasible measurements caused by hardware noise, we successfully identify optimal solutions on actual quantum devices.

Figures

Figures reproduced from arXiv: 2604.02083 by Claudia Linnhoff-Popien, David Bucher, Jonas Stein, Maximilian Janetschek, Michael Poppel, Sebastian Feld.

Figure 1
Figure 1. Figure 1: Contour plots showing the energy landscape in terms of approximation ratio of WS-QAOA for [PITH_FULL_IMAGE:figures/full_fig_p009_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Approximation ratio (a) and approximation trace (b) of IWS-QAOA at [PITH_FULL_IMAGE:figures/full_fig_p011_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Median improvement ratio of the optimal solu [PITH_FULL_IMAGE:figures/full_fig_p011_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Approximation trace of IWS-QAOA at p = 1 as a function of the total number of shots for M ∈ {100, 200, 500} and M = 3000 on TSP instances ranging from 6 to 9 cities. Each panel displays the median over five instances, with ten runs per instance. Error bands indicate the interquartile range. The solid black line represents the median baseline performance of IWS using classical random sampling with M = 200, … view at source ↗
Figure 5
Figure 5. Figure 5: Median improvement ratio of Popt achieved by IWS￾QAOA relative to the baseline without warm-starting (WS) for the four TSP instance sizes across various M values. Er￾ror bars indicate the interquartile range. The gray bars rep￾resent IWS with classical random sampling at M = 200. For IWS-QAOA, Popt is evaluated directly from the state vector following the final iteration of IWS-QAOA. the threshold for extr… view at source ↗
Figure 6
Figure 6. Figure 6: Approximation trace of IWS-QAOA for circuit depths [PITH_FULL_IMAGE:figures/full_fig_p013_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: Median improvement ratio of Popt achieved by IWS￾QAOA relative to the non-warm-started baseline for the 9- city TSP instances across circuit depths p ∈ {1, . . . , 5}. Error bars indicate the interquartile range. For IWS-QAOA, Popt is evaluated directly from the state vector following the final iteration. instances, the weights wij are uniformly sampled from {−1, −0.9, . . . , 1}. A key advantage of our ch… view at source ↗
Figure 8
Figure 8. Figure 8: Hardware-tailored 144-qubit problem instance for the [PITH_FULL_IMAGE:figures/full_fig_p014_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: Histograms depicting the number of violated [PITH_FULL_IMAGE:figures/full_fig_p014_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: Histograms of the measurement outcomes in terms of approximation ratio from [PITH_FULL_IMAGE:figures/full_fig_p015_10.png] view at source ↗
Figure 11
Figure 11. Figure 11: Best objective value found across the 10 independent repetitions as a function of total accumulated shots, plotted [PITH_FULL_IMAGE:figures/full_fig_p016_11.png] view at source ↗

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Forward citations

Cited by 5 Pith papers

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score.

  1. Quantum Approximate Optimization via Noise-Directed Adaptive Warm-Starting

    quant-ph 2026-07 conditional novelty 6.0

    Bitflip-gauge warm-start QAOA that aligns the ansatz with amplitude-damping noise improves 100-qubit Ising approximation ratios over non-gauge iterative warm-start at no extra circuit cost.

  2. RL-Guided Quantum-ALNS for Constrained VRP

    quant-ph 2026-07 conditional novelty 6.0

    A DQN-guided controller selectively invokes quantum sampling within ALNS repair for constrained VRP, finding quantum repair admissible in ~16% of states but beneficial in 29/36 matched-budget settings.

  3. Feasibility-driven QAOA with penalty scheduling

    quant-ph 2026-06 unverdicted novelty 6.0

    Introduces Λ-lr-QAOA and piecewise-ramp QAOA that promote penalty schedules to variational parameters and use a feasibility-driven loss on budget-constrained MWIS satellite planning instances.

  4. Constrained Counterdiabatic Quantum Approximate Optimization Algorithm for Portfolio Optimization

    quant-ph 2026-05 unverdicted novelty 6.0

    CCD-QAOA incorporates counterdiabatic terms into the QAOA ansatz and shows higher approximation ratios than standard XY-mixer, Grover-mixer, and penalty QAOA for portfolio problems with budget and risk constraints.

  5. Iterative warm-start optimization with quantum imaginary time evolution

    quant-ph 2026-04 unverdicted novelty 6.0

    An iterative nonvariational quantum algorithm using warm-start states and classically computed imaginary time evolution circuits achieves median solutions within 95% of optimal for MaxCut on small 3-regular graphs usi...

Reference graph

Works this paper leans on

66 extracted references · 9 linked inside Pith · cited by 5 Pith papers

  1. [1]

    We therefore require the topologyGto be connected, ensur- ing that∃β:⟨e i|e −iβHG P |ej⟩ ̸= 0 for alli, j

    Trotterization A valid mixer must facilitate transition probabilities between every pair of states within its domain [13]. We therefore require the topologyGto be connected, ensur- ing that∃β:⟨e i|e −iβHG P |ej⟩ ̸= 0 for alli, j. Under this condition,H G P (as defined in Eq. (16)) is a valid mixer with|W P ⟩as its unique ground state according to Corol- l...

  2. [2]

    Implementation of the warm-start XY-block The circuit implementation for the time evolution of the single-qubit warm-start mixere −iHWS M (q)β is given by the decompositionR Y (α)RZ(−2β)RY (−α), where α= 2 arccos √q[16]. This protocol can be ex- tended to the XY-mixer case, which requires embed- ding these rotations into the single-excitation subspace via...

  3. [3]

    Conse- quently, we observe that the optimalβvalues for QAOA increase as p q(1−q) decreases

    Scaling the XY-block Because the XY-part ofH(q) diminishes asqap- proaches the extreme pointsq→0 orq→1, the effective mixing magnitude| ⟨01|e −iβH(q) |10⟩ |decreases. Conse- quently, we observe that the optimalβvalues for QAOA increase as p q(1−q) decreases. To counteract this and ensure consistentβparameters, we implement a scaled and shifted version ofH...

  4. [4]

    Max-Cut remains a central opti- mization problem for benchmarking QAOA and variants like warm-starting [16, 31]

    Max-k-Cut QAOA was initially developed as an approximate algo- rithm for Max-Cut [9]. Max-Cut remains a central opti- mization problem for benchmarking QAOA and variants like warm-starting [16, 31]. Max-k-Cut (MkC) is the nat- ural extension that separates the nodes intokpartitions rather than two. For a graphG(V, E) with|V|=Nand edge weightsw uv, it is f...

  5. [5]

    It is naturally formulated as a quadratic integer program: min x X u,v∈E wuv NX t=1 xu,txv,(t+1)%N s.t

    Traveling Salesperson Problem One of the most famous combinatorial optimization problems is the Traveling Salesperson Problem (TSP), which seeks to find the shortest cycle connecting all nodes in a given fully connected graphK N with edge weights wuv >0. It is naturally formulated as a quadratic integer program: min x X u,v∈E wuv NX t=1 xu,txv,(t+1)%N s.t...

  6. [6]

    2a shows the approximation ratio ofp= 1 IWS- QAOA with respect to the total shots gathered through- out the algorithm execution

    Max-k-Cut Fig. 2a shows the approximation ratio ofp= 1 IWS- QAOA with respect to the total shots gathered through- out the algorithm execution. It is apparent that IWS- QAOA—independent ofM—improves upon the base QAOA approximation ratio (which corresponds to the performance at the first data point). Furthermore, we observe that all values ofMconverge to ...

  7. [7]

    F¨ orderprogramm Quanten- technologien – von den Grundlagen zum Markt

    Traveling Salesperson Problem Fig. 4 displays the approximation trace of thep= 1 IWS-QAOA across different TSP instance sizes. In the smallest case (N= 6), all methods find the optimal solu- tion in fewer than 2000 shots. However, forN≥7, some methods begin to fail to identify the optimal solution. The non-warm-started QAOA baseline does not reach an appr...

    2000

  8. [8]

    R. P. Feynman, Simulating physics with computers, International Journal of Theoretical Physics21, 467 (1982)

    1982

  9. [9]

    Preskill, Quantum Computing in the NISQ era and beyond, Quantum2, 79 (2018)

    J. Preskill, Quantum Computing in the NISQ era and beyond, Quantum2, 79 (2018)

    2018

  10. [10]

    Acharya et al., Quantum error correction below the surface code threshold, Nature638, 920 (2025)

    R. Acharya et al., Quantum error correction below the surface code threshold, Nature638, 920 (2025)

    2025

  11. [11]

    AbuGhanem, IBM quantum computers: evolution, performance, and future directions, The Journal of Su- percomputing81, 687 (2025)

    M. AbuGhanem, IBM quantum computers: evolution, performance, and future directions, The Journal of Su- percomputing81, 687 (2025)

    2025

  12. [12]

    Abbas et al., Challenges and opportunities in quantum optimization, Nature Reviews Physics6, 718 (2024)

    A. Abbas et al., Challenges and opportunities in quantum optimization, Nature Reviews Physics6, 718 (2024)

    2024

  13. [13]

    Feld et al., A Hybrid Solution Method for the Ca- pacitated Vehicle Routing Problem Using a Quantum Annealer, Frontiers in ICT6, 10.3389/fict.2019.00013 (2019)

    S. Feld et al., A Hybrid Solution Method for the Ca- pacitated Vehicle Routing Problem Using a Quantum Annealer, Frontiers in ICT6, 10.3389/fict.2019.00013 (2019)

    doi:10.3389/fict.2019.00013 2019

  14. [14]

    Krellner et al., Solving a real-world modular logistic scheduling problem with a quantum-classical metaheuris- tics (2025), arXiv:2507.21701 [quant-ph]

    F. Krellner et al., Solving a real-world modular logistic scheduling problem with a quantum-classical metaheuris- tics (2025), arXiv:2507.21701 [quant-ph]

    Pith/arXiv arXiv 2025

  15. [15]

    Blenninger et al., Q-GRID: Quantum Optimization for the Future Energy Grid, KI - K¨ unstliche Intelligenz38, 339 (2024)

    J. Blenninger et al., Q-GRID: Quantum Optimization for the Future Energy Grid, KI - K¨ unstliche Intelligenz38, 339 (2024)

    2024

  16. [16]

    Farhi, J

    E. Farhi, J. Goldstone, and S. Gutmann, A Quan- tum Approximate Optimization Algorithm (2014), arXiv:1411.4028 [quant-ph]

    Pith/arXiv arXiv 2014

  17. [17]

    Born and V

    M. Born and V. Fock, Beweis des Adiabatensatzes, Zeitschrift f¨ ur Physik51, 165 (1928)

    1928

  18. [18]

    Lucas, Ising formulations of many NP problems, Fron- tiers in Physics2, 10.3389/fphy.2014.00005 (2014)

    A. Lucas, Ising formulations of many NP problems, Fron- tiers in Physics2, 10.3389/fphy.2014.00005 (2014)

    doi:10.3389/fphy.2014.00005 2014

  19. [19]

    Glover, G

    F. Glover, G. Kochenberger, R. Hennig, and Y. Du, Quantum bridge analytics I: a tutorial on formulating and using QUBO models, Annals of Operations Research 314, 141 (2022)

    2022

  20. [20]

    Hadfield et al., From the Quantum Approximate Opti- mization Algorithm to a Quantum Alternating Operator Ansatz, Algorithms12, 10.3390/a12020034 (2019)

    S. Hadfield et al., From the Quantum Approximate Opti- mization Algorithm to a Quantum Alternating Operator Ansatz, Algorithms12, 10.3390/a12020034 (2019)

    doi:10.3390/a12020034 2019

  21. [21]

    F. G. Fuchs, K. O. Lye, H. M. Nilsen, A. J. Stasik, and G. Sartor, Constraint Preserving Mixers for the Quan- tum Approximate Optimization Algorithm, Algorithms 15, 10.3390/a15060202 (2022)

    doi:10.3390/a15060202 2022

  22. [22]

    Z. Wang, N. C. Rubin, J. M. Dominy, and E. G. Rieffel, $XY$mixers: Analytical and numerical results for the quantum alternating operator ansatz, Physical Review A101, 012320 (2020)

    2020

  23. [23]

    D. J. Egger, J. Mareˇ cek, and S. Woerner, Warm-starting quantum optimization, Quantum5, 479 (2021)

    2021

  24. [24]

    Yu et al., Quantum approximate optimization algo- rithm with adaptive bias fields, Physical Review Research 4, 023249 (2022)

    Y. Yu et al., Quantum approximate optimization algo- rithm with adaptive bias fields, Physical Review Research 4, 023249 (2022)

    2022

  25. [25]

    R. S. d. Carmo, M. C. S. Santana, F. F. Fanchini, V. H. C. d. Albuquerque, and J. P. Papa, Warm- Starting QAOA with XY Mixers: A Novel Approach for Quantum-Enhanced Vehicle Routing Optimization (2025), arXiv:2504.19934 [quant-ph]

    Pith/arXiv arXiv 2025

  26. [26]

    M. A. Lopez-Ruiz et al., A Non-Variational Quantum Approach to the Job Shop Scheduling Problem (2025), arXiv:2510.26859 [quant-ph]

    arXiv 2025

  27. [27]

    Pelofske, A

    E. Pelofske, A. B¨ artschi, and S. Eidenbenz, Quantum An- nealing vs. QAOA: 127 Qubit Higher-Order Ising Prob- lems on NISQ Computers, inHigh Performance Com- puting, edited by A. Bhatele, J. Hammond, M. Baboulin, and C. Kruse (Springer Nature Switzerland, Cham, 2023) pp. 240–258

    2023

  28. [28]

    N. Mohseni et al., Evidence of quantum scaling advantage in approximate optimization for energy coalition forma- tion with 100+ agents, Quantum Science and Technology 11, 015009 (2025)

    2025

  29. [29]

    Mohseni et al., Constrained Quantum Optimization at Utility Scale: Application to the Knapsack Problem (2026), arXiv:2603.00260 [quant-ph]

    N. Mohseni et al., Constrained Quantum Optimization at Utility Scale: Application to the Knapsack Problem (2026), arXiv:2603.00260 [quant-ph]

    arXiv 2026

  30. [30]

    S. V. Romero et al., Bias-field digitized counterdiabatic quantum algorithm for higher-order binary optimization, Communications Physics8, 348 (2025)

    2025

  31. [31]

    Wecker, M

    D. Wecker, M. B. Hastings, and M. Troyer, Training a quantum optimizer, Physical Review A94, 022309 (2016)

    2016

  32. [32]

    K. Blekos et al., A review on Quantum Approximate Op- timization Algorithm and its variants, Physics Reports A review on Quantum Approximate Optimization Algo- rithm and its variants,1068, 1 (2024)

    2024

  33. [33]

    J. A. Monta˜ nez-Barrera and K. Michielsen, Toward a linear-ramp QAOA protocol: evidence of a scaling ad- vantage in solving some combinatorial optimization prob- lems, npj Quantum Information11, 131 (2025)

    2025

  34. [34]

    V. Dehn et al., Extrapolation method to optimize linear- ramp quantum approximate optimization algorithm pa- rameters: Evaluation of runtime scaling, Physical Review A113, 032413 (2026)

    2026

  35. [35]

    R. Tate, J. Moondra, B. Gard, G. Mohler, and S. Gupta, Warm-Started QAOA with Custom Mix- ers Provably Converges and Computationally Beats Goemans-Williamson’s Max-Cut at Low Circuit Depths, Quantum7, 1121 (2023)

    2023

  36. [36]

    R. Tate, M. Farhadi, C. Herold, G. Mohler, and S. Gupta, Bridging Classical and Quantum with SDP initialized warm-starts for QAOA, ACM Transactions on Quantum 18 Computing4, 9:1 (2023)

    2023

  37. [37]

    He et al., Regularized Warm-Started Quantum Ap- proximate Optimization and Conditions for Surpass- ing Classical Solvers on the Max-Cut Problem (2026), arXiv:2603.10191 [quant-ph]

    Z. He et al., Regularized Warm-Started Quantum Ap- proximate Optimization and Conditions for Surpass- ing Classical Solvers on the Max-Cut Problem (2026), arXiv:2603.10191 [quant-ph]

    arXiv 2026

  38. [38]

    Yu, X.-B

    Y. Yu, X.-B. Wang, N. Shannon, and R. Joynt, Warm- start adaptive-bias quantum approximate optimization algorithm, Physical Review A112, 012422 (2025)

    2025

  39. [39]

    H. Yuan, S. Yang, and C. H. W. Barnes, Iterative quan- tum optimisation with a warm-started quantum state (2025), arXiv:2502.09704 [quant-ph]

    Pith/arXiv arXiv 2025

  40. [40]

    A. G. Cadavid, A. Dalal, A. Simen, E. Solano, and N. N. Hegade, Bias-field digitized counterdiabatic quantum op- timization, Physical Review Research7, L022010 (2025)

    2025

  41. [41]

    Bucher, J

    D. Bucher, J. Stein, S. Feld, and C. Linnhoff-Popien, Penalty-free approach to accelerating constrained quan- tum optimization, Physical Review A112, 062605 (2025)

    2025

  42. [42]

    Gleixner et al., MIPLIB 2017: data-driven compila- tion of the 6th mixed-integer programming library, Math- ematical Programming Computation13, 443 (2021)

    A. Gleixner et al., MIPLIB 2017: data-driven compila- tion of the 6th mixed-integer programming library, Math- ematical Programming Computation13, 443 (2021)

    2017

  43. [43]

    Onah and K

    C. Onah and K. Michielsen, Fundamental Limitations of QAOA on Constrained Problems and a Route to Ex- ponential Enhancement (2025), arXiv:2511.17259 [quant- ph]

    Pith/arXiv arXiv 2025

  44. [44]

    Kordonowy and H

    S. Kordonowy and H. Leipold, The Lie algebra of XY-mixer topologies and warm starting QAOA for constrained optimization, npj Quantum Information 10.1038/s41534-026-01192-4 (2026)

    doi:10.1038/s41534-026-01192-4 2026

  45. [45]

    He et al., Alignment between initial state and mixer improves QAOA performance for constrained optimiza- tion, npj Quantum Information9, 121 (2023)

    Z. He et al., Alignment between initial state and mixer improves QAOA performance for constrained optimiza- tion, npj Quantum Information9, 121 (2023)

    2023

  46. [46]

    Tasaki,Physics and Mathematics of Quantum Many- Body Systems, Graduate Texts in Physics (Springer In- ternational Publishing, Cham, 2020)

    H. Tasaki,Physics and Mathematics of Quantum Many- Body Systems, Graduate Texts in Physics (Springer In- ternational Publishing, Cham, 2020)

    2020

  47. [47]

    Diestel, Colouring, inGraph Theory, edited by R

    R. Diestel, Colouring, inGraph Theory, edited by R. Di- estel (Springer, Berlin, Heidelberg, 2025) pp. 123–154

    2025

  48. [48]

    Horst and H

    R. Horst and H. Tuy, Cutting Methods, inGlobal Opti- mization: Deterministic Approaches, edited by R. Horst and H. Tuy (Springer, Berlin, Heidelberg, 1996) pp. 181– 224

    1996

  49. [49]

    M. Cain, E. Farhi, S. Gutmann, D. Ranard, and E. Tang, The QAOA gets stuck starting from a good classical string (2023), arXiv:2207.05089 [quant-ph]

    Pith/arXiv arXiv 2023

  50. [50]

    Tsvelikhovskiy, B

    B. Tsvelikhovskiy, B. Bach, J. Falla, and I. Safro, Re- ductions of QAOA Induced by Classical Symmetries: Theoretical Insights and Practical Implications (2026), arXiv:2602.16141 [quant-ph]

    Pith/arXiv arXiv 2026

  51. [51]

    Schawe and A

    H. Schawe and A. K. Hartmann, Phase transitions of Traveling Salesperson problems solved with linear pro- gramming and cutting planes, Europhysics Letters113, 30004 (2016)

    2016

  52. [52]

    Bucher et al., Towards Robust Benchmarking of Quantum Optimization Algorithms (IEEE Computer So- ciety, 2024) pp

    D. Bucher et al., Towards Robust Benchmarking of Quantum Optimization Algorithms (IEEE Computer So- ciety, 2024) pp. 159–170

    2024

  53. [53]

    Fletcher,Practical methods of optimization, online- ausg ed

    R. Fletcher,Practical methods of optimization, online- ausg ed. (Wiley, Chichester, West Sussex, U.K, 2008)

    2008

  54. [54]

    Virtanen et al., SciPy 1.0: Fundamental algorithms for scientific computing in python, Nature Methods17, 261 (2020)

    P. Virtanen et al., SciPy 1.0: Fundamental algorithms for scientific computing in python, Nature Methods17, 261 (2020)

    2020

  55. [55]

    D. Bucher et al., Efficient QAOA Architecture for Solv- ing Multi-Constrained Optimization Problems, in2025 IEEE International Conference on Quantum Computing and Engineering (QCE), Vol. 01 (2025) pp. 356–367

    2025

  56. [56]

    Stein et al., CUAOA: A Novel CUDA-Accelerated Sim- ulation Framework for the QAOA, in2024 IEEE Inter- national Conference on Quantum Computing and Engi- neering (QCE), Vol

    J. Stein et al., CUAOA: A Novel CUDA-Accelerated Sim- ulation Framework for the QAOA, in2024 IEEE Inter- national Conference on Quantum Computing and Engi- neering (QCE), Vol. 02 (2024) pp. 280–285

    2024

  57. [57]

    Lykov, R

    D. Lykov, R. Shaydulin, Y. Sun, Y. Alexeev, and M. Pis- toia, Fast Simulation of High-Depth QAOA Circuits, in Proceedings of the SC ’23 Workshops of the International Conference on High Performance Computing, Network, Storage, and Analysis, SC-W ’23 (Association for Com- puting Machinery, New York, NY, USA, 2023) pp. 1443– 1451

    2023

  58. [58]

    Golden, A

    J. Golden, A. Baertschi, D. O’Malley, E. Pelofske, and S. Eidenbenz, JuliQAOA: Fast, Flexible QAOA Simula- tion, inProceedings of the SC ’23 Workshops of the In- ternational Conference on High Performance Computing, Network, Storage, and Analysis, SC-W ’23 (Association for Computing Machinery, New York, NY, USA, 2023) pp. 1454–1459

    2023

  59. [59]

    Weidenfeller et al., Scaling of the quantum approxi- mate optimization algorithm on superconducting qubit based hardware, Quantum6, 870 (2022)

    J. Weidenfeller et al., Scaling of the quantum approxi- mate optimization algorithm on superconducting qubit based hardware, Quantum6, 870 (2022)

    2022

  60. [60]

    Matsuo, S

    A. Matsuo, S. Yamashita, and D. J. Egger, A SAT Ap- proach to the Initial Mapping Problem in SWAP Gate Insertion for Commuting Gates, IEICE Transactions on Fundamentals of Electronics, Communications and Com- puter SciencesE106.A, 1424 (2023)

    2023

  61. [61]

    J. N¨ ußlein et al., Reducing QUBO density by factoring out semi-symmetries, inProceedings of the 17th inter- national conference on agents and artificial intelligence - volume 1: QAIO(SciTePress / INSTICC, 2025) pp. 783–792

    2025

  62. [62]

    Chandarana et al., Runtime Quantum Advantage with Digital Quantum Optimization (2025), arXiv:2505.08663 [quant-ph]

    P. Chandarana et al., Runtime Quantum Advantage with Digital Quantum Optimization (2025), arXiv:2505.08663 [quant-ph]

    Pith/arXiv arXiv 2025

  63. [63]

    Kotil et al., Quantum approximate multi-objective optimization, Nature Computational Science5, 1168 (2025)

    A. Kotil et al., Quantum approximate multi-objective optimization, Nature Computational Science5, 1168 (2025)

    2025

  64. [64]

    Hojny et al.,The SCIP optimization suite 10.0, Tech- nical Report (Optimization Online, 2025)

    C. Hojny et al.,The SCIP optimization suite 10.0, Tech- nical Report (Optimization Online, 2025)

    2025

  65. [65]

    D. Cruz et al., Efficient quantum algorithms for$GHZ$ and$W$states, and implementation on the IBM quan- tum computer, Advanced Quantum Technologies2, 1900015 (2019), arXiv:1807.05572 [quant-ph]

    Pith/arXiv arXiv 2019

  66. [66]

    Perron and F

    L. Perron and F. Didier, CP-SAT (2025). 19 Appendix A: State Preparation of the Biased W-State In this section, we discuss the circuits required to construct the biased|W P ⟩state. a. Linear SynthesisThe standard|W⟩state forkqubits is constructed starting from the state|e 1⟩=|10· · ·0⟩. Following Ref. [58], we apply a sequence of gatesB ij(q) =cnot jiC(R ...

    2025