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arxiv: 2606.18914 · v2 · pith:OCCRVEZ7new · submitted 2026-06-17 · 🪐 quant-ph

Benchmark of Pauli Correlation Encoding for different optimisation problems

Fernando Alonso , Colom\'an Sampr\'on , Jacobo Veiga , Mariamo Mussa Juane , Andr\'es G\'omez This is my paper

Pith reviewed 2026-06-26 20:30 UTC · model grok-4.3

classification 🪐 quant-ph
keywords quantum optimizationPauli Correlation Encodingcombinatorial optimizationNISQencoding schemesbenchmark evaluationpost-processing
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The pith

Pauli Correlation Encoding matches or exceeds reference solutions on multiple combinatorial optimization problems using a polynomial number of qubits.

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

The paper evaluates a quantum-classical framework that encodes binary decision variables through correlations among Pauli operators rather than assigning one qubit per variable. This allows representation of m variables with a number of qubits that grows only polynomially. The authors apply the method to three standard optimization problems drawn from an established benchmark set and vary the compression order, hyperparameters, and post-processing steps. They also examine how finite-shot sampling and hardware noise alter the optimization trajectory. Across these tests the framework produces solutions that are competitive with the reference and, in several instances, match or surpass them.

Core claim

The Pauli Correlation Encoding framework, when applied to combinatorial optimization problems, achieves performance that is competitive with established benchmarks and in several cases yields equivalent or superior solutions, even when accounting for shot noise and hardware imperfections.

What carries the argument

Pauli Correlation Encoding, a scheme that represents multiple binary variables using a polynomial number of qubits through correlations in Pauli operators.

If this is right

  • Increasing the compression order of the encoding changes solution quality in a manner that depends on the structure of the particular optimization problem.
  • Post-processing of the quantum measurement outcomes improves the final solution quality beyond what the raw optimizer returns.
  • Finite-shot estimation and realistic hardware noise degrade the accuracy of the cost function but do not eliminate the ability to reach competitive solutions.
  • Hyperparameter choices must be tuned to the problem at hand for the encoding to reach its best reported performance.

Where Pith is reading between the lines

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

  • The polynomial qubit scaling may permit treatment of larger problem sizes on near-term hardware than direct one-qubit-per-variable mappings allow.
  • The same encoding approach could be tested on additional classes of combinatorial problems to determine its breadth of applicability.
  • If the method continues to perform well under noise, it may serve as a useful bridge between current NISQ devices and future fault-tolerant quantum optimizers.

Load-bearing premise

The encoding can be implemented and optimized on the test problems such that performance comparisons to the reference remain fair without hidden classical advantages or post-selection biases.

What would settle it

Applying the same framework to the identical benchmark instances and obtaining solutions that are consistently and substantially worse than the reference under matched conditions would falsify the competitiveness result.

Figures

Figures reproduced from arXiv: 2606.18914 by Andr\'es G\'omez, Colom\'an Sampr\'on, Fernando Alonso, Jacobo Veiga, Mariamo Mussa Juane.

Figure 1
Figure 1. Figure 1: Schematic of the quantum circuit used to construct the PQC. [PITH_FULL_IMAGE:figures/full_fig_p005_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Scaling of the number of qubits as a function of the number of variables for MCP. [PITH_FULL_IMAGE:figures/full_fig_p008_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Scaling of the number of layers and parameters under the assumed configuration. [PITH_FULL_IMAGE:figures/full_fig_p009_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Results obtained using regularisation for the 10 MCP instances. [PITH_FULL_IMAGE:figures/full_fig_p010_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Scaling of the number of qubits as a function of the number of variables for the BPP. [PITH_FULL_IMAGE:figures/full_fig_p013_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: Heat map of solutions for instance N = 8 of the BPP, as a function of α and β, averaged over five random initialisations. Finally, 50 random initialisations of all different instances have been run for compression orders k ∈ {2, 3, 4} using the optimal α and β configurations 3 . The results are shown in Figures 7 and 8, which report the percentage of feasible solutions and the frequency with which the mini… view at source ↗
Figure 7
Figure 7. Figure 7: Comparison of the percentages of feasible and best feasible solutions obtained with [PITH_FULL_IMAGE:figures/full_fig_p016_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: Comparison of the best solutions obtained with and without regularisation across the [PITH_FULL_IMAGE:figures/full_fig_p016_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: Scaling of the number of qubits as a function of the number of variables for the TSP. [PITH_FULL_IMAGE:figures/full_fig_p019_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: Heat map of solutions for instance N = 8 of the TSP, as a function of α and β, averaged over five random initialisations. Finally, 50 independent random initialisations were performed for each problem instance and for compression orders k ∈ 2, 3, 4, using the optimal hyperparameter configuration for α and β 4 . 4For further details regarding the hyperparameter configuration, see Appendix B. 20 [PITH_FULL… view at source ↗
Figure 11
Figure 11. Figure 11: Comparison of the percentages of feasible and best feasible solutions obtained with [PITH_FULL_IMAGE:figures/full_fig_p022_11.png] view at source ↗
Figure 12
Figure 12. Figure 12: Comparison of the best solutions obtained with and without regularisation across [PITH_FULL_IMAGE:figures/full_fig_p022_12.png] view at source ↗
Figure 13
Figure 13. Figure 13: Comparison of the percentage improvemente of the solutions obtained with and [PITH_FULL_IMAGE:figures/full_fig_p023_13.png] view at source ↗
Figure 14
Figure 14. Figure 14: MAE as a function of the number of shots for 10 MCP instances with k = 2. 24 [PITH_FULL_IMAGE:figures/full_fig_p024_14.png] view at source ↗
Figure 15
Figure 15. Figure 15: MAE as a function of the number of shots for 10 MCP instances with k = 3. (a) Ideal - k = 4 (b) Noisy - k = 4 [PITH_FULL_IMAGE:figures/full_fig_p025_15.png] view at source ↗
Figure 16
Figure 16. Figure 16: MAE as a function of the number of shots for 10 MCP instances with k = 4. (a) Ideal - k = 2 (b) Noisy - k = 2 [PITH_FULL_IMAGE:figures/full_fig_p025_16.png] view at source ↗
Figure 17
Figure 17. Figure 17: MaxErr as a function of the number of shots for 10 MCP instances with k = 2. 25 [PITH_FULL_IMAGE:figures/full_fig_p025_17.png] view at source ↗
Figure 18
Figure 18. Figure 18: MaxErr as a function of the number of shots for 10 MCP instances with k = 3. (a) Ideal - k = 4 (b) Noisy - k = 4 [PITH_FULL_IMAGE:figures/full_fig_p026_18.png] view at source ↗
Figure 19
Figure 19. Figure 19: MaxErr as a function of the number of shots for 10 MCP instances with k = 4. In the ideal emulation scenario, from 105 shots onwards, the error reaches convergence on the order of 10−3 and even 10−4 in some cases. However, increasing the compression order k introduces a certain level of instability across different seeds for shot counts of 106 and 107 , although the error remains consistently within the r… view at source ↗
Figure 20
Figure 20. Figure 20: Scaling of the transpiled circuit depth as a function of the number of variables for [PITH_FULL_IMAGE:figures/full_fig_p027_20.png] view at source ↗
Figure 21
Figure 21. Figure 21: Solution ratio for k = 2 on ideal backend, using shot-based executions for MCP. 27 [PITH_FULL_IMAGE:figures/full_fig_p027_21.png] view at source ↗
Figure 22
Figure 22. Figure 22: Solution ratio for k = 2 on noisy backend, using shot-based executions for MCP. However, reproducing the full optimisation with shot-based sampling is extremely costly: when using Differential Evolution, the number of evaluations is given by the product of the popsize and the number of circuit parameters, resulting in tens of millions of executions in the most complex cases. For this reason, an estimate o… view at source ↗
Figure 23
Figure 23. Figure 23: Estimated execution time as a function of the number of parameters for different [PITH_FULL_IMAGE:figures/full_fig_p028_23.png] view at source ↗
Figure 24
Figure 24. Figure 24: Heat map of metrics over the relaxed variables for instance [PITH_FULL_IMAGE:figures/full_fig_p040_24.png] view at source ↗
Figure 25
Figure 25. Figure 25: Heat map of metrics over the relaxed variables for instance [PITH_FULL_IMAGE:figures/full_fig_p041_25.png] view at source ↗
Figure 26
Figure 26. Figure 26: Heat map of rate of feasible solutions over the relaxed variables for instance [PITH_FULL_IMAGE:figures/full_fig_p042_26.png] view at source ↗
read the original abstract

The continuous progress of quantum technologies has spurred the exploration of their potential applications across diverse fields, particularly in combinatorial optimisation. In this work, we study a quantum-classical optimisation framework based on Pauli Correlation Encoding, an encoding scheme that can represent m binary variables using a polynomial number of qubits. To evaluate the performance of the method, we use three classical optimisation problems against the instances of the QOPTLib benchmark. The study includes an analysis of the impact of the compression order of the encoding scheme, the problem structure, and hyperparameter selection on solution quality, as well as the role of post-processing in improving performance. Additionally, we study the effect of shot-based execution and hardware noise, showing how these factors influence both the accuracy of expected value estimation and the overall dynamics of the optimisation process. The results indicate that the proposed PCE-based framework achieves competitive performance against the benchmark and, in several cases, obtains equivalent or even superior solutions, highlighting its potential as an efficient encoding strategy for quantum optimisation in the NISQ and near fault-tolerant era.

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 / 2 minor

Summary. The manuscript proposes a quantum-classical hybrid optimization framework based on Pauli Correlation Encoding (PCE), which encodes m binary variables using a polynomial number of qubits. It benchmarks the approach on three combinatorial optimization problems drawn from the QOPTLib library, analyzing the effects of compression order, problem structure, hyperparameter choice, classical post-processing, finite-shot sampling, and hardware noise on solution quality. The central claim is that the PCE framework achieves competitive performance against the QOPTLib reference and, in several cases, yields equivalent or superior solutions.

Significance. If the performance attribution holds after appropriate controls, the work would contribute a qubit-efficient encoding strategy relevant to NISQ-era combinatorial optimization. The inclusion of shot-noise and hardware-noise analyses is a positive feature that directly addresses practical deployment constraints.

major comments (2)
  1. [Results / benchmark comparisons] Results section (benchmark comparisons): The claim that PCE obtains equivalent or superior solutions requires an explicit control in which the identical classical post-processing pipeline (whose benefit is acknowledged in the abstract) is also applied to the QOPTLib reference solutions. Without this, any reported edge cannot be securely attributed to the encoding rather than to post-processing.
  2. [Methods / experimental protocol] Methods / experimental protocol: The manuscript does not state the number of QOPTLib instances per problem class, the number of independent runs per instance, or the statistical tests used to support the 'competitive or superior' claim. These quantities are load-bearing for the central empirical assertion.
minor comments (2)
  1. [Encoding scheme] Notation for the compression order parameter is introduced without an explicit equation reference in the main text; a numbered definition would improve clarity.
  2. [Figures] Figure captions for the noise-sensitivity plots should include the precise noise model parameters and the number of shots used.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the constructive feedback. We address the two major comments below and will revise the manuscript accordingly to strengthen the empirical claims and experimental protocol.

read point-by-point responses

  1. Referee: [Results / benchmark comparisons] Results section (benchmark comparisons): The claim that PCE obtains equivalent or superior solutions requires an explicit control in which the identical classical post-processing pipeline (whose benefit is acknowledged in the abstract) is also applied to the QOPTLib reference solutions. Without this, any reported edge cannot be securely attributed to the encoding rather than to post-processing.

    Authors: We agree that this control is necessary for a fair attribution of performance differences. The current manuscript applies post-processing only to PCE outputs. We will add an explicit comparison in which the identical post-processing pipeline is applied to the QOPTLib reference solutions and report the updated results. revision: yes

  2. Referee: [Methods / experimental protocol] Methods / experimental protocol: The manuscript does not state the number of QOPTLib instances per problem class, the number of independent runs per instance, or the statistical tests used to support the 'competitive or superior' claim. These quantities are load-bearing for the central empirical assertion.

    Authors: We acknowledge this omission. The revised manuscript will explicitly state: (i) the number of QOPTLib instances per problem class, (ii) the number of independent runs performed per instance, and (iii) the statistical tests (e.g., paired t-test or Wilcoxon signed-rank test with appropriate multiple-comparison correction) used to support claims of competitive or superior performance. revision: yes

Circularity Check

0 steps flagged

No circularity: empirical benchmark against external library

full rationale

The paper is an empirical benchmark study comparing a PCE-based quantum-classical optimization framework to the independent QOPTLib reference library on three classical problems. No derivation chain, first-principles predictions, or fitted parameters are claimed; performance is assessed via direct solution quality metrics on external instances. Post-processing is discussed as an implementation detail but does not create self-referential reductions. The central claims rest on observable comparisons to a public external benchmark rather than any quantity defined by the authors' own inputs or prior self-citations.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Only abstract available; no explicit free parameters, axioms, or invented entities are stated. The encoding itself is treated as given from prior work.

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discussion (0)

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