Parameterized 4-Qubit EWL Quantum Game Circuits with Dirac-Solow-Swan Hamiltonian Integration for Quadruple Helix Disruptive Innovation Recommender Systems

Agung Trisetyarso; Fithra Faisal Hastiadi; Kridanto Surendro

arxiv: 2605.18080 · v1 · pith:3IJ57CGQnew · submitted 2026-05-18 · 🪐 quant-ph · cond-mat.stat-mech· cs.AI· econ.EM· econ.TH

Parameterized 4-Qubit EWL Quantum Game Circuits with Dirac-Solow-Swan Hamiltonian Integration for Quadruple Helix Disruptive Innovation Recommender Systems

Agung Trisetyarso , Fithra Faisal Hastiadi , Kridanto Surendro This is my paper

Pith reviewed 2026-05-20 11:12 UTC · model grok-4.3

classification 🪐 quant-ph cond-mat.stat-mechcs.AIecon.EMecon.TH

keywords quantum game theoryEWL circuitDirac-Solow-Swan Hamiltonianquadruple helixdisruptive innovationrecommender systemscapital accumulationNISQ

0 comments

The pith

A 4-qubit EWL quantum game circuit maps real funding weights into recommender scores that drive a Dirac-Solow-Swan Hamiltonian for forecasting disruptive capital trajectories.

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

The paper builds a four-qubit version of the Eisert-Wilkens-Lewenstein quantum game whose local rotation angles are fixed by normalized dominance weights taken from actual ecContribution figures in the CORDIS Horizon Europe records. After the game runs, the four measurement probabilities are inserted directly as the diagonal entries of a Dirac potential term inside an otherwise standard Solow-Swan growth equation. The resulting time-evolution operator is then used to simulate how capital accumulates and may bifurcate when a quadruple-helix network shifts toward disruptive rather than sustaining innovation projects. Numerical runs on real collaboration graphs extracted from project COVend are reported to remain within the gate budget of current NISQ devices while reproducing observed capital trends at high fidelity.

Core claim

The authors introduce a parameterized 4-qubit EWL circuit, built from a multi-qubit entangler, four Ry rotations whose angles are set by CORDIS funding dominance weights, an inverse entangler, and full measurement; the resulting probabilities are interpreted as the diagonal Dirac potential inside a Solow-Swan Hamiltonian, thereby generating recommender scores and enabling simulation of capital accumulation and bifurcation dynamics under disruptive innovation in quadruple-helix ecosystems.

What carries the argument

The 4-qubit EWL game circuit with local Ry(θi) operators tuned by normalized ecContribution weights, whose post-measurement probabilities supply the diagonal Dirac potential of the integrated Solow-Swan Hamiltonian.

If this is right

The circuit requires only 22 gates and depth 11 and scales linearly with the number of helix communication rounds.
Measurement probabilities yield explicit recommender scores that separate disruptive from sustaining innovation trends.
Time evolution under the Dirac-Solow-Swan Hamiltonian produces concrete forecasts of capital trajectories and possible bifurcation points.
Open Qiskit code permits direct reproduction of the numerical experiments on the same real CORDIS collaboration networks.

Where Pith is reading between the lines

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

The same probability-to-potential mapping could be tested inside other growth models that already contain a potential term.
Replacing the four fixed Ry gates with a variational circuit layer might allow the model to learn the optimal mapping from funding data rather than using hand-tuned dominance weights.
Running the identical pipeline on larger qubit registers would indicate whether additional helix actors or project sub-categories improve forecast accuracy.

Load-bearing premise

That the four measurement probabilities produced by the quantum game can be inserted directly as the diagonal Dirac potential without losing the economic meaning of capital accumulation and bifurcation.

What would settle it

A side-by-side comparison, on held-out CORDIS quadruple-helix records, between the capital-accumulation curves generated by the integrated Hamiltonian and the actual year-by-year funding and collaboration shifts recorded after the training window.

Figures

Figures reproduced from arXiv: 2605.18080 by Agung Trisetyarso, Fithra Faisal Hastiadi, Kridanto Surendro.

**Figure 2.** Figure 2: Qiskit implementation of the parameterized 4-qubit EisertWilkensLewenstein [PITH_FULL_IMAGE:figures/full_fig_p013_2.png] view at source ↗

read the original abstract

We present a novel parameterized 4-qubit Eisert-Wilkens-Lewenstein (EWL) quantum game circuit for recommender systems in quadruple helix innovation ecosystems (academia, industry, government, and civil society). The local strategy operators $U_{i} = R_y(\theta_{i})$ for each helix actor are directly tuned by normalized dominance weights extracted from real participant funding data (\texit{ecContribution}) in the European Commission CORDIS Horizon Europe database (project COVend, ID 101045956). The circuit employs a multi-qubit EWL entangler followed by parameterized local rotations, inverse entangler, and full measurement, achieving only 22 gates and circuit depth 11 while scaling as $O(n)$ for $n$-round helix communications. Measurement probabilities after the quantum game serve as recommender scores for disruptive versus sustaining innovation trends. These scores are subsequently mapped into the diagonal Dirac potential of a Dirac-Solow-Swan Hamiltonian, enabling time-evolution simulation of capital accumulation and bifurcation dynamics under disruptive innovation. Numerical experiments on real CORDIS quadruple-helix collaboration networks demonstrate the circuit's NISQ compatibility and its ability to forecast disruptive capital trajectories with high fidelity. The proposed framework bridges quantum game theory, parameterized quantum circuits, and relativistic economic growth models, offering a computationally efficient tool for innovation policy and strategic decision-making in complex socio-economic ecosystems. Complexity analysis and reproducibility are provided through open Qiskit implementations.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

The 4-qubit EWL circuit is small and uses real CORDIS data, but the direct insertion of game probabilities into the Dirac-Solow-Swan potential has no visible derivation or validation.

read the letter

The main takeaway is that this paper builds a shallow 4-qubit EWL circuit whose local rotations come straight from normalized funding weights in the CORDIS database, then feeds the resulting measurement probabilities into the diagonal term of a Dirac-Solow-Swan Hamiltonian to simulate capital trajectories under disruptive innovation. The circuit itself stays under 22 gates and depth 11, which is practical for NISQ hardware, and the authors supply open Qiskit code plus a complexity claim of O(n) scaling for repeated helix interactions. That combination of a concrete parameterized game and real quadruple-helix collaboration data is the clearest new element. They also run the thing on actual project networks rather than toy examples, which gives the work some grounding. The soft spot sits in the probability-to-potential step. Nothing in the abstract or description shows a symmetry argument, first-principles derivation, or even an empirical calibration that explains why a quantum-game outcome should modulate the Dirac term instead of entering as a classical additive factor or a stochastic process. The reported high-fidelity forecasts on CORDIS data therefore sit close to post-hoc fitting, especially since the rotation angles are taken directly from the same input weights. No error bars, no baseline classical recommender, and no hold-out validation appear in the summary. For someone already working on quantum circuits for social or economic simulation, the implementation details and the small gate count could be worth a look. Most other readers will find the economic bridge too thin to use without further justification. I would send it to referees so domain experts can test whether the mapping can be made rigorous or whether it remains a heuristic overlay.

Referee Report

2 major / 3 minor

Summary. The paper introduces a parameterized 4-qubit EWL quantum game circuit for quadruple helix disruptive innovation recommender systems. Local strategy operators U_i = R_y(theta_i) are tuned using normalized dominance weights from CORDIS funding data (ecContribution). Measurement probabilities from the circuit are mapped to the diagonal Dirac potential in a Dirac-Solow-Swan Hamiltonian to simulate capital accumulation and bifurcation dynamics under disruptive innovation. The authors claim that numerical experiments on real CORDIS quadruple-helix collaboration networks demonstrate the circuit's NISQ compatibility (22 gates, depth 11, O(n) scaling) and its ability to forecast disruptive capital trajectories with high fidelity, supported by open Qiskit implementations.

Significance. If the mapping from quantum game probabilities to the economic Hamiltonian were rigorously derived and the forecasts validated against independent data with quantitative metrics, this could offer a novel interdisciplinary bridge between parameterized quantum circuits, quantum game theory, and relativistic economic growth models for innovation policy applications. The mention of open Qiskit code for reproducibility is a positive element.

major comments (2)

[Abstract / Hamiltonian Integration] Abstract and the section describing the probability-to-potential mapping: the central claim that EWL measurement probabilities can be directly inserted into the diagonal Dirac potential of the Solow-Swan Hamiltonian to produce accurate capital-accumulation and bifurcation trajectories lacks any first-principles derivation, symmetry argument, or empirical calibration. No justification is supplied for why game outcomes should modulate this specific term rather than, e.g., an additive classical utility function.
[Numerical Experiments] Numerical experiments section: the assertion of 'high fidelity' forecasting on CORDIS networks is unsupported by any reported metrics (e.g., R², MAE, error bars), baseline comparisons to classical recommenders, or validation against held-out data, rendering the performance claims unassessable.

minor comments (3)

[Abstract] The notation in the abstract appears as 'texit{ecContribution}' and should be corrected to 'textit{ecContribution}'.
[Introduction / Related Work] The manuscript would benefit from explicit references to prior literature on EWL games in socio-economic contexts and standard treatments of the Solow-Swan model to better situate the Dirac-Solow-Swan integration.
[Reproducibility Statement] Although open Qiskit implementations are stated to be provided, the text should include a specific repository URL or code availability statement for full reproducibility.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the thoughtful and constructive report. We address each major comment below, indicating the revisions we will make to improve the manuscript's rigor and clarity while preserving the core interdisciplinary contribution.

read point-by-point responses

Referee: [Abstract / Hamiltonian Integration] Abstract and the section describing the probability-to-potential mapping: the central claim that EWL measurement probabilities can be directly inserted into the diagonal Dirac potential of the Solow-Swan Hamiltonian to produce accurate capital-accumulation and bifurcation trajectories lacks any first-principles derivation, symmetry argument, or empirical calibration. No justification is supplied for why game outcomes should modulate this specific term rather than, e.g., an additive classical utility function.

Authors: We agree that the current version presents the mapping primarily as a proposed integration without a detailed first-principles derivation. The mapping is motivated by interpreting the post-measurement probabilities from the EWL game as strategic weights that modulate the effective potential to reflect disruptive innovation pressures within the quadruple-helix dynamics. In the revised manuscript we will expand the relevant section to include an explicit heuristic justification: the diagonal Dirac term is selected because it directly scales the time-evolution operator governing capital accumulation, allowing game-derived probabilities to influence bifurcation thresholds in a manner consistent with the relativistic growth model. We will also discuss why an additive classical utility is not used (it would not couple to the Hamiltonian evolution in the same way) and note the mapping as a modeling choice open to further theoretical refinement. revision: yes
Referee: [Numerical Experiments] Numerical experiments section: the assertion of 'high fidelity' forecasting on CORDIS networks is unsupported by any reported metrics (e.g., R², MAE, error bars), baseline comparisons to classical recommenders, or validation against held-out data, rendering the performance claims unassessable.

Authors: The manuscript reports qualitative agreement between simulated trajectories and observed CORDIS capital patterns, supported by the open Qiskit code. We acknowledge that quantitative metrics are required for proper evaluation. In the revised version we will add explicit values for R², MAE, and standard error bars computed from the simulations, include direct comparisons against classical baselines (e.g., matrix factorization and graph-based recommenders), and perform a held-out validation split on the CORDIS dataset to quantify forecasting performance. revision: yes

Circularity Check

1 steps flagged

Rotation angles tuned from CORDIS funding weights; game probabilities mapped to Hamiltonian yield 'forecasts' that reduce to input transformations by construction

specific steps

fitted input called prediction [Abstract]
"The local strategy operators $U_{i} = R_y(θ_{i})$ for each helix actor are directly tuned by normalized dominance weights extracted from real participant funding data (ecContribution) in the European Commission CORDIS Horizon Europe database (project COVend, ID 101045956). ... Measurement probabilities after the quantum game serve as recommender scores ... These scores are subsequently mapped into the diagonal Dirac potential of a Dirac-Solow-Swan Hamiltonian, enabling time-evolution simulation of capital accumulation and bifurcation dynamics under disruptive innovation. Numerical experiments,"

θ_i are set directly from the same funding weights that define the CORDIS networks used for the subsequent 'forecast' experiments. The circuit therefore computes a deterministic function of the input data; mapping its output probabilities into the Hamiltonian and claiming predictive fidelity on those networks reduces the claimed trajectories to a re-expression of the fitted inputs rather than an independent prediction.

full rationale

The derivation begins by extracting normalized dominance weights directly from the input CORDIS ecContribution data to set the circuit parameters θ_i. The resulting measurement probabilities are then inserted into the Dirac potential term. Because the numerical experiments are performed on the identical CORDIS quadruple-helix networks that supplied the tuning weights, the reported high-fidelity capital-accumulation trajectories are statistically forced by the fitted inputs rather than generated from an independent first-principles link. No external calibration or symmetry argument is supplied to justify why EWL probabilities should modulate the relativistic Hamiltonian term. This matches the fitted-input-called-prediction pattern and produces an overall circularity score of 8.

Axiom & Free-Parameter Ledger

1 free parameters · 2 axioms · 1 invented entities

The framework depends on data-derived parameters for circuit angles and an unvalidated mapping from quantum probabilities to economic potentials, plus standard assumptions from quantum information and neoclassical growth theory.

free parameters (1)

theta_i
Local strategy operators U_i = R_y(theta_i) are tuned by normalized dominance weights extracted from ecContribution funding data.

axioms (2)

standard math Standard quantum circuit model and entanglement operations for the EWL game
The multi-qubit entangler, local rotations, and inverse entangler assume the usual postulates of quantum mechanics and gate-based computation.
domain assumption Solow-Swan growth model admits a diagonal Dirac potential term derived from external game scores
The time-evolution simulation treats quantum-derived recommender scores as an additive potential without deriving this coupling from first principles.

invented entities (1)

Dirac-Solow-Swan Hamiltonian integration no independent evidence
purpose: To link quantum game measurement probabilities to capital accumulation and bifurcation dynamics
A new postulated coupling between the quantum circuit outputs and the economic model is introduced without independent falsifiable evidence outside the paper.

pith-pipeline@v0.9.0 · 5835 in / 1726 out tokens · 62322 ms · 2026-05-20T11:12:31.559550+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

Measurement probabilities after the quantum game serve as recommender scores... mapped into the diagonal Dirac potential of a Dirac-Solow-Swan Hamiltonian
IndisputableMonolith/Foundation/RealityFromDistinction.lean reality_from_one_distinction unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

enabling time-evolution simulation of capital accumulation and bifurcation dynamics

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

12 extracted references · 12 canonical work pages

[1]

Trisetyarso et al., ``Quantum-inspired computational intelligence framework (QICIF) for general automation systems,'' submitted to IEEE Trans

A. Trisetyarso et al., ``Quantum-inspired computational intelligence framework (QICIF) for general automation systems,'' submitted to IEEE Trans. Emerg. Tech. Comp. Int., 2026

work page 2026
[2]

Carayannis, E.G., Campbell, D.F.J.: `Mode 3' and `Quadruple Helix': toward a 21st century fractal innovation ecosystem. Int. J. Technol. Manag. 46(3--4), 201--234 (2009)

work page 2009
[3]

Harvard Business School Press, Boston (1997)

Christensen, C.M.: The Innovator's Dilemma. Harvard Business School Press, Boston (1997)

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[4]

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Kerenidis, I., Prakash, A.: Quantum recommendation systems. In: Proc. 8th Innovations in Theoretical Computer Science Conference (ITCS 2017), pp. 49:1--49:21 (2017)

work page 2017
[5]

Eisert, J., Wilkens, M., Lewenstein, M.: Quantum games and quantum strategies. Phys. Rev. Lett. 83(15), 3077--3080 (1999)

work page 1999
[6]

Book Preprint, ITB Press (2025)

Trisetyarso, A., Hastiadi, F.F.: Quantum Algorithm for Recommender Systems of Disruptive Innovation. Book Preprint, ITB Press (2025)

work page 2025
[7]

Nature 549(7671), 195--202 (2017)

Biamonte, J., et al.: Quantum machine learning. Nature 549(7671), 195--202 (2017)

work page 2017
[8]

Solow, R.M.: A contribution to the theory of economic growth. Q. J. Econ. 70(1), 65--94 (1956)

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[9]

Ludmila, Franeva and Andrei, Grib and George, Parfionov.: Contributions to Game Theory and Management (1), 130--151 (2007)

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Benjamin, Simon C and Hayden, Patrick M.: Multiplayer quantum games Physical Review A vol (1), no. (3), pp. 030301 (2001)

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[11]

Horizon Europe Projects and Organization Datasets (2023)

European Commission: CORDIS -- Community Research and Development Information Service. Horizon Europe Projects and Organization Datasets (2023). https://data.europa.eu/data/datasets/cordis-eu-research-projects-under-horizon-europe-2021-2027

work page 2023
[12]

https://qiskit.org

Qiskit Development Team: Qiskit: An open-source framework for quantum computing (2024). https://qiskit.org

work page 2024

[1] [1]

Trisetyarso et al., ``Quantum-inspired computational intelligence framework (QICIF) for general automation systems,'' submitted to IEEE Trans

A. Trisetyarso et al., ``Quantum-inspired computational intelligence framework (QICIF) for general automation systems,'' submitted to IEEE Trans. Emerg. Tech. Comp. Int., 2026

work page 2026

[2] [2]

Carayannis, E.G., Campbell, D.F.J.: `Mode 3' and `Quadruple Helix': toward a 21st century fractal innovation ecosystem. Int. J. Technol. Manag. 46(3--4), 201--234 (2009)

work page 2009

[3] [3]

Harvard Business School Press, Boston (1997)

Christensen, C.M.: The Innovator's Dilemma. Harvard Business School Press, Boston (1997)

work page 1997

[4] [4]

In: Proc

Kerenidis, I., Prakash, A.: Quantum recommendation systems. In: Proc. 8th Innovations in Theoretical Computer Science Conference (ITCS 2017), pp. 49:1--49:21 (2017)

work page 2017

[5] [5]

Eisert, J., Wilkens, M., Lewenstein, M.: Quantum games and quantum strategies. Phys. Rev. Lett. 83(15), 3077--3080 (1999)

work page 1999

[6] [6]

Book Preprint, ITB Press (2025)

Trisetyarso, A., Hastiadi, F.F.: Quantum Algorithm for Recommender Systems of Disruptive Innovation. Book Preprint, ITB Press (2025)

work page 2025

[7] [7]

Nature 549(7671), 195--202 (2017)

Biamonte, J., et al.: Quantum machine learning. Nature 549(7671), 195--202 (2017)

work page 2017

[8] [8]

Solow, R.M.: A contribution to the theory of economic growth. Q. J. Econ. 70(1), 65--94 (1956)

work page 1956

[9] [9]

Ludmila, Franeva and Andrei, Grib and George, Parfionov.: Contributions to Game Theory and Management (1), 130--151 (2007)

work page 2007

[10] [10]

Benjamin, Simon C and Hayden, Patrick M.: Multiplayer quantum games Physical Review A vol (1), no. (3), pp. 030301 (2001)

work page 2001

[11] [11]

Horizon Europe Projects and Organization Datasets (2023)

European Commission: CORDIS -- Community Research and Development Information Service. Horizon Europe Projects and Organization Datasets (2023). https://data.europa.eu/data/datasets/cordis-eu-research-projects-under-horizon-europe-2021-2027

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