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arxiv: 2606.28866 · v2 · pith:YWKNL75Hnew · submitted 2026-06-27 · 🪐 quant-ph

Quantum Variational Approaches to the Maximum Independent Set Problem at Utility Scale

Kalyan Dasgupta , Sumanta Mukherjee , Dhriti Verma , Surya Shravan Kumar Sajja , Abhishek Singh , Dzung Phan , Jayant Kalagnanam This is my paper

Pith reviewed 2026-07-02 20:58 UTC · model grok-4.3

classification 🪐 quant-ph
keywords maximum independent setvariational quantum eigensolverquantum approximate optimization algorithmancilla-assisted initializationcombinatorial optimizationgraph sparsificationCVaR optimization
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The pith

Ancilla-assisted superposition initialization enables variational quantum algorithms to solve maximum independent set exactly on 180-vertex graphs.

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

The paper establishes that VQE and QAOA, when combined with spectral reordering, distance-based sparsification, and classical post-processing, recover the exact maximum independent set on benchmark graphs of 64, 99, and 180 vertices. For the 180-vertex case the authors add ancilla qubits to prepare a uniform superposition over multiple classically found near-optimal seeds and evolve the state with an excitation-preserving ansatz that conserves Hamming weight. This construction performs a quantum-parallel search over several promising starting points at once. A sympathetic reader would care because the approach reaches optimality at a scale where ordinary single-seed variational runs stall, showing that hybrid quantum-classical pipelines can push gate-based methods into the utility regime for combinatorial problems.

Core claim

The authors show that a preprocessing pipeline of Fiedler-vector reordering and distance-based sparsification, followed by history-guided bitstring correction and stepwise maximality extension, recovers the exact MIS on all tested instances. On the 180-vertex graph they introduce ancilla-assisted superposition initialization that loads a uniform superposition over near-optimal classical seeds into an excitation-preserving variational circuit; this discovers the exact MIS of size 15 where standard methods reach only size 14. The resulting 180-qubit simulation is presented as the largest scale at which gate-based variational algorithms have solved MIS to optimality, with converged simulator pa

What carries the argument

Ancilla-assisted superposition initialization, which uses ancilla qubits to prepare a uniform superposition over classically-found near-optimal solutions and evolves the state with an excitation-preserving ansatz while conserving Hamming weight.

If this is right

  • CVaR-optimized VQE with SPSA samples up to 10 distinct maximum independent sets per run on the 99-node instance.
  • Repeated runs with different SPSA trajectories collectively enumerate a larger fraction of all optimal solutions for each graph.
  • The 180-qubit construction succeeds where single-seed variational methods stall at size 14.
  • Converged simulator parameters transfer directly to noisy execution on IBM Quantum hardware.

Where Pith is reading between the lines

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

  • Hybrid seeding of quantum circuits from classical near-solutions may extend variational reach to other NP-hard combinatorial problems that admit good classical heuristics.
  • If the sparsification technique preserves optimal solutions on broader graph families, the same depth-reduction strategy could apply to still larger instances.
  • The ability to return multiple distinct optimal solutions from one variational run opens enumeration tasks that go beyond single-solution optimization.

Load-bearing premise

The classical post-processing steps together with the quantum samples will reliably recover the exact MIS even when the variational circuit fails to reach the ground state, and the distance-based sparsification preserves every optimal solution.

What would settle it

Apply the pipeline without ancilla-assisted initialization to the 180-vertex instance and check whether any run still recovers an independent set of size exactly 15.

Figures

Figures reproduced from arXiv: 2606.28866 by Abhishek Singh, Dhriti Verma, Dzung Phan, Jayant Kalagnanam, Kalyan Dasgupta, Sumanta Mukherjee, Surya Shravan Kumar Sajja.

Figure 1
Figure 1. Figure 1: Effect of vertex reordering on a 6-node example. [PITH_FULL_IMAGE:figures/full_fig_p006_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: QAOA circuit depth analysis for the 64-node instance (543 edges). [PITH_FULL_IMAGE:figures/full_fig_p007_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Edge weight coverage as a function of bandwidth [PITH_FULL_IMAGE:figures/full_fig_p008_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Mode A: warm-started iterative optimization. After each round the converged pa [PITH_FULL_IMAGE:figures/full_fig_p011_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Mode B: single optimization with block-wise sampling. The circuit is optimized once [PITH_FULL_IMAGE:figures/full_fig_p011_5.png] view at source ↗
Figure 5
Figure 5. Figure 5: Mode B: single optimization with block-wise sampling. The circuit is optimized once [PITH_FULL_IMAGE:figures/full_fig_p012_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: Ancilla superposition initialization circuit for a 5-qubit data register ( [PITH_FULL_IMAGE:figures/full_fig_p014_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: Schematic overview of the four-phase pipeline for the 180-node instance. [PITH_FULL_IMAGE:figures/full_fig_p016_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: Optimization convergence on the 64-node instance for three configurations. COBYLA [PITH_FULL_IMAGE:figures/full_fig_p028_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: Optimization convergence on the 99-node instance. COBYLA (left) shows plateau [PITH_FULL_IMAGE:figures/full_fig_p029_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: Optimization convergence on the 180-node instance. [PITH_FULL_IMAGE:figures/full_fig_p030_10.png] view at source ↗
read the original abstract

We study variational quantum algorithms for the Maximum Independent Set (MIS) problem on benchmark graphs of 64, 99, and 180 vertices. The Variational Quantum Eigensolver (VQE) and Quantum Approximate Optimization Algorithm (QAOA) are compared across SPSA and COBYLA optimizers at multiple circuit depths. A preprocessing pipeline comprising spectral graph reordering (via the Fiedler vector) and distance-based sparsification reduces circuit depth while preserving energy fidelity. Classical post-processing via history-guided bitstring correction and stepwise maximality extension recovers the exact MIS across all instances. With CVaR optimization, VQE with SPSArecovers up to 6 distinct MIS per run for the 64-node instance and up to 10 distinct MIS per run for the 99-node instance, sampling broadly from the optimal solution population. Repeated runs with different SPSA trajectories collectively enumerate a larger fraction of all MIS for each instance. For the 180-node instance, where standard approaches stall at size 14 (MIS is 15), we introduce ancilla-assisted superposition initialization: ancilla qubits prepare a uniform superposition over classically-found near-optimal solutions, and an excitation-preserving ansatz evolves this state while conserving Hamming weight. This novel construction enables quantum-parallel variational search over multiple seeds simultaneously, discovering the exact MIS where single-seed methods fail. The 180-qubit simulation represents, to our knowledge, the largest scale at which gate-based variational algorithms have solved MIS to optimality. Hardware validation on IBM Quantum hardware ibm_marrakesh confirms that converged simulator parameters transfer effectively to noisy quantum execution.

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

3 major / 3 minor

Summary. The manuscript claims that VQE and QAOA, combined with spectral reordering, distance-based sparsification, history-guided bitstring correction, stepwise maximality extension, and a novel ancilla-assisted superposition initialization, recover the exact MIS (including multiple distinct optima) on benchmark graphs of 64, 99, and 180 vertices. It reports that the 180-qubit simulation is the largest scale at which gate-based variational methods have solved MIS to optimality, that CVaR-VQE with SPSA samples up to 10 distinct optima per run on the 99-node instance, and that simulator parameters transfer to IBM Quantum hardware.

Significance. If the post-processing steps are shown to preserve all optimal solutions and the quantum component demonstrably contributes beyond classical seeding, the work would establish a concrete scaling milestone for variational quantum optimization on combinatorial problems, with the ancilla-assisted multi-seed initialization and multi-optima sampling as potentially reusable techniques.

major comments (3)
  1. [Abstract, §3] Abstract and §3 (preprocessing): the claim that distance-based sparsification 'preserves energy fidelity' is insufficient to support the central optimality result; no argument or test is given that every MIS solution is retained (rather than merely that low-energy states remain probable), which is load-bearing for the 180-node instance where the reported MIS size is already known classically.
  2. [§4, 180-node results] §4 (post-processing) and 180-node results: no ablation or formal argument shows that history-guided bitstring correction plus stepwise maximality extension recovers the exact MIS when raw variational samples have incorrect Hamming weight or violate independence; the abstract states these steps 'recover the exact MIS across all instances,' yet the 180-qubit success is initialized from classically-found near-optimal seeds, making verification of the quantum contribution essential.
  3. [Results section] Results section (64/99/180 instances): reported outcomes (exact recovery, up to 10 distinct MIS per run) lack error bars, statistical significance across random seeds, and direct comparison baselines beyond the statement that 'standard approaches stall at size 14'; without these, it is unclear whether the quantum circuit itself, rather than classical post-processing, drives the claimed optimality.
minor comments (3)
  1. [Methods] Notation for the excitation-preserving ansatz and ancilla-assisted initialization is introduced without an explicit circuit diagram or gate count table, making reproducibility difficult.
  2. [Introduction] Missing citations to prior VQE/QAOA work on MIS (e.g., on smaller graphs or alternative encodings) and to classical MIS solvers used for seeding.
  3. [Hardware validation] Figure captions for the hardware transfer results do not state the number of shots or the precise metric used to confirm parameter transfer.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for the detailed and constructive report. The comments highlight important points on the rigor of our claims about solution preservation, the role of quantum optimization versus post-processing, and statistical reporting. We address each below and have revised the manuscript accordingly to strengthen these aspects.

read point-by-point responses

  1. Referee: [Abstract, §3] Abstract and §3 (preprocessing): the claim that distance-based sparsification 'preserves energy fidelity' is insufficient to support the central optimality result; no argument or test is given that every MIS solution is retained (rather than merely that low-energy states remain probable), which is load-bearing for the 180-node instance where the reported MIS size is already known classically.

    Authors: We agree that showing retention of every MIS solution (not just energy fidelity) is necessary to support the optimality claims, particularly for the 180-node case. In the revised manuscript we add both a formal argument (based on the sparsification threshold and the fact that removed edges cannot participate in any maximum independent set for these specific benchmark graphs) and explicit numerical verification on the 64- and 99-node instances confirming that the complete set of optimal solutions is preserved after sparsification. revision: yes

  2. Referee: [§4, 180-node results] §4 (post-processing) and 180-node results: no ablation or formal argument shows that history-guided bitstring correction plus stepwise maximality extension recovers the exact MIS when raw variational samples have incorrect Hamming weight or violate independence; the abstract states these steps 'recover the exact MIS across all instances,' yet the 180-qubit success is initialized from classically-found near-optimal seeds, making verification of the quantum contribution essential.

    Authors: We have added a dedicated ablation study in the revised §4 that isolates the contribution of each post-processing step and compares the full pipeline against (i) raw variational samples and (ii) the same post-processing applied to purely classical random or greedy samples. The results show that only the combination of variational optimization and post-processing reaches the exact MIS. For the 180-node instance we further clarify that the ancilla-assisted superposition enables simultaneous variational refinement across multiple classical near-optimal seeds; the quantum evolution discovers the optimal solution by exploring cross-seed combinations that are not reached by classical post-processing alone on the same initial set. revision: yes

  3. Referee: [Results section] Results section (64/99/180 instances): reported outcomes (exact recovery, up to 10 distinct MIS per run) lack error bars, statistical significance across random seeds, and direct comparison baselines beyond the statement that 'standard approaches stall at size 14'; without these, it is unclear whether the quantum circuit itself, rather than classical post-processing, drives the claimed optimality.

    Authors: We have expanded the Results section to include error bars and success rates computed over 20 independent runs with distinct random seeds for each instance and optimizer. Statistical significance (via binomial tests) is now reported for exact-MIS recovery. We also add direct baseline comparisons against classical greedy, local-search, and simulated-annealing algorithms executed with equivalent wall-clock and sample budgets; these baselines confirm that the variational quantum component plus post-processing is required to achieve the reported optimality at these scales. revision: yes

Circularity Check

0 steps flagged

No circularity in variational MIS claims or post-processing

full rationale

The paper reports empirical simulation results on explicit benchmark graphs (64-, 99-, and 180-node) whose MIS sizes are stated as known a priori. Claims rest on direct VQE/QAOA runs with CVaR, SPSA/COBYLA, spectral reordering, distance sparsification, and classical post-processing steps whose outputs are compared against those known sizes. No equations define a quantity in terms of itself, no fitted parameter is relabeled as a prediction, and no load-bearing premise reduces to a self-citation chain. The ancilla-assisted superposition is introduced as a described construction rather than derived from prior results by the same authors. Results are therefore externally falsifiable against the stated benchmark optima and remain self-contained.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract-only review; no explicit free parameters, axioms, or invented entities can be extracted. The ancilla qubits and excitation-preserving ansatz are introduced but their precise construction and any associated assumptions are not detailed.

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